Daevite Mathematics: Complex Numbers

Link to article: Daevite Mathematics: Complex Numbers.

:root {
  --header-title: "SCP FOUNDATION";
  --lgurl: url("https://scp-wiki.wdfiles.com/local--files/math-department-hub/Math%20Department%20Logo%20-%20Penumbra.png");
  /* --header-subtitle: "DEPARTMENT OF MATHEMATICS"; */
  /* --accentColor: #1eb5e8; */
  --header-subtitle: "DEPARTMENT OF APPLIED MATHEMATICS";
  --accentColor: #65cdf0;
  /* --header-subtitle: "DEPARTMENT OF ESOTERIC MATHEMATICS"; */
  /* --accentColor: #0099cc; */
  /* --header-subtitle: "DEPARTMENT OF TACTICAL MATHEMATICS"; */
  /* --accentColor: #636fee; */
  --fnColor: var(--accentColor);
}

/* Bottom Rating Module */
.bottom-rating-module {
  display: flex;
  flex-direction: column;
  align-items: center;
  margin-block: 1rem;
}

.bottom-rating-module .page-rate-widget-box {
  margin: 0;
}

/* Article-Specific Styling */
div.vertblock {
    border: solid 2px var(--darkColor);
    box-shadow: 0px -3px 0px -0.1px var(--accentColor);
}

/*
table:has(#toc) {
  margin: 0 auto !important;
}
*/

.scp-image-block img {
  filter: brightness(50%);
}
.scp-image-block img:hover {
  filter: brightness(100%);
}

.fnnum {
  font-weight: normal;
}
.fnnum::after,
.fncon::before {
    content: "ⓘ";
}
.fncon {
  background: var(--darkerColor);
  color: var(--lightColor);
  border: .1rem solid var(--accentColor);
}

a.external {
  text-decoration-line: underline;
  text-decoration-style: dashed;
}

/* Bibliography */
.bibitems {
    background-color: var(--darkerColor);
    padding-left: 1.4rem;
    padding-right: 1.4rem;
    padding-bottom: 1.5rem;
    box-shadow: -0.24rem 0px 0px 0px var(--accentColor);
}
.bibitems .title {
    color: var(--lightColor);
}
.bibitem {
  text-indent: 1.4rem hanging; 
  padding-bottom: 0.3125rem;
}
.bibitem::first-letter {
  font-weight: bold;
}

```
text-align: center;
```
```
margin:0px
```
```
margin-top: 15px;
```
```
margin-top: 15px;
```
```
width: 90%; margin: auto;
```
```
max-width: 100%;
```
```
margin-top: 15px;
```
```
margin-top: 15px;
```
```
margin-top: 15px;
```
```
margin-top: 15px;
```
```
margin-top: 15px;
```
```
font-style:italic
```
```
text-indent:1.4em hanging
```

```
darkbox
```
```
external
```
```
external
```
```
darkbox
```
```
external
```
```
darkbox
```
```
fnnum
```
```
fncon
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
scp-image-block
```
```
video
```
```
scp-image-caption
```
```
external
```
```
external
```
```
blockquote
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
external
```
```
blockquote
```
```
external
```
```
external
```
```
external
```
```
darkbox
```
```
blockquote
```
```
blockquote
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
blockquote
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
external
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
external
```
```
blockquote
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
fnnum
```
```
fncon
```
```
darkbox
```
```
fnnum
```
```
fncon
```
```
external
```
```
blockquote
```
```
blockquote
```
```
bottom-rating-module
```
```
footer-wikiwalk-nav
```

=====
[[include component:preview
text=The Daevites' mathematical aptitude was a major, nonanomalous contributor to their development as a highly advanced civilization.]]
=====

[[include :scp-wiki:theme:penumbra]]
[[include :scp-wiki:component:betterfootnotes]]

[[module css]]
:root {
  --header-title: "SCP FOUNDATION";
  --lgurl: url("https://scp-wiki.wdfiles.com/local--files/math-department-hub/Math%20Department%20Logo%20-%20Penumbra.png");
  /* --header-subtitle: "DEPARTMENT OF MATHEMATICS"; */
  /* --accentColor: #1eb5e8; */
  --header-subtitle: "DEPARTMENT OF APPLIED MATHEMATICS";
  --accentColor: #65cdf0;
  /* --header-subtitle: "DEPARTMENT OF ESOTERIC MATHEMATICS"; */
  /* --accentColor: #0099cc; */
  /* --header-subtitle: "DEPARTMENT OF TACTICAL MATHEMATICS"; */
  /* --accentColor: #636fee; */
  --fnColor: var(--accentColor);
}

/* Bottom Rating Module */
.bottom-rating-module {
  display: flex;
  flex-direction: column;
  align-items: center;
  margin-block: 1rem;
}

.bottom-rating-module .page-rate-widget-box {
  margin: 0;
}

/* Article-Specific Styling */
div.vertblock {
    border: solid 2px var(--darkColor);
    box-shadow: 0px -3px 0px -0.1px var(--accentColor);
}

/*
table:has(#toc) {
  margin: 0 auto !important;
}
*/

.scp-image-block img {
  filter: brightness(50%);
}
.scp-image-block img:hover {
  filter: brightness(100%);
}

.fnnum {
  font-weight: normal;
}
.fnnum::after,
.fncon::before {
    content: "ⓘ";
}
.fncon {
  background: var(--darkerColor);
  color: var(--lightColor);
  border: .1rem solid var(--accentColor);
}

a.external {
  text-decoration-line: underline;
  text-decoration-style: dashed;
}

/* Bibliography */
.bibitems {
    background-color: var(--darkerColor);
    padding-left: 1.4rem;
    padding-right: 1.4rem;
    padding-bottom: 1.5rem;
    box-shadow: -0.24rem 0px 0px 0px var(--accentColor);
}
.bibitems .title {
    color: var(--lightColor);
}
.bibitem {
  text-indent: 1.4rem hanging; 
  padding-bottom: 0.3125rem;
}
.bibitem::first-letter {
  font-weight: bold;
}

[[/module]]


[[>]]
[[module Rate]]
[[/>]]

[[include :scp-wiki:component:author-label-source start=--
|name=badooga]]
**Crimson Skies/Project Athena: Daevite Mathematics**

The Daevites' mathematical aptitude was a major, //nonanomalous// contributor to their development as a highly advanced civilization.

**Author Note:** This is an extensive bit of mathematics material; read it slowly if needed, and don't be afraid to scroll back up to previous sections or to sit on it and think for a bit if you don't understand something right away.

[[[badooga-author-page|More By Badooga]]]

[[include :scp-wiki:component:author-label-source end=--]]

[[div class="darkbox" style="text-align: center;"]]

+++* Accessibility Notice: Media

All images and videos have been dimmed by 50% to prevent eyestrain. To bring them to full brightness, simply hover over or tap them. Furthermore, math text may take some time to populate with data upon page load/reload.

[[/div]]

**DOI:** {{scp.hist/500219}}

**Clearance:** [[[Expanded Security Clearance Codes|G3-2/DOTT/HIST/MATH]]]

**Date:** 2023 February 17

**Authors:** Paige Buyantu ([[[math-department-hub|Department of Applied Mathematics]]]), Teresa Zhang ([[[secure-facility-dossier-site-12|Department of Anthropology]]])

**Keywords:** Arboriculture · Agriculture · Architecture · [[[daevite-hub|Daevite Empire]]] · Etymology · History of Mathematics · Hydrology · [[[divine-dossier|Parareligion]]] · Pedagogy · Thaumaturgy · Trigonometry

**Abstract:** Recent archaeological digs in Turpan, China helmed by Yezhov //et al.// have yielded a significant host of artifacts pertaining to the Daevite Empire of antiquity. Amongst these artifacts is a partially preserved mathematics textbook (circa 3,000 BCE), titled //Geometric Veneration//, intended as instructional material for children of ages 12 and older. Building off of the linguistic insights generated by Zhang //et al.//, a mathematical analysis of this textbook's contents is warranted, for its treatise on [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Complex_number"]]complex numbers[[/a]] is by far the oldest of its kind (and the //only// of its kind from antiquity). Here, we take the route of translating this text to English words //and// modern mathematical language, allowing for the direct study and comparison of Daevite mathematical knowledge, pedagogy, and application to both modern treatments of the subject matter and that of the ancient civilizations that existed during the same period. Our results indicate that the Daevites' application-centric approach to mathematics, and its inextricable ties to their culture and religion, offered them significant advancements in trigonometry that their contemporaries lacked. We then argue that these advancements—and their application to fields like agriculture and architecture—were major, //nonanomalous// contributors to the Daevites' development as a civilization that allowed them to economically and militaristically dominate their peers.

[[toc title="Table of Contents"]]

+ Introduction
The Daevite Empire was an ancient civilization that began in south-central Siberia and eventually stretched throughout Central Asia, the Middle East, and Eastern Europe. Well known by the Foundation for its [[[SCP-140|retrocausally malleable history]]] and widespread use of thaumaturgy, anomalous [[[SCP-6189|artifacts]]] and other archaeological evidence of the Daevites have been dated back to at least 10,000 BCE. At the apex of Daevite society was a matriarchal, theocratic aristocracy of thaumaturgically enhanced //[[[scp-9140|Homo sapiens daeva]]]// (the "Daeva") who are known to have helmed the Empire's lengthy imperial conquests and mandated worship of the seven "Fontbearers": the various goddesses/philosophies that made up the overarching Daevite pantheon. Most of the Daevites' contributions to the ancient world are believed to have been destroyed during slave revolts or outright erased through retrocausal means; however, an increasing number of [[[SCP-9280|hemomantic]]], [http://scp-int.wikidot.com/scp-es-133 arboreal], [[[SCP-3399|horticultural]]], and [[[SCP-9640|temporal]]] anomalies recovered by the Foundation have been identified as having ties to the Daevite Empire.

Meanwhile, though the study of trigonometry has followed humanity for as long as written language has existed, the common theory of complex numbers was birthed much later from the study of polynomial equations and their solutions. The square roots of negative numbers were originally discarded whenever encountered due to their supposedly unreal nature; however, it was discovered in the 16th century that certain formulae for the zeros of certain polynomials would require the manipulation of these roots despite the final answer being real valued. In 1637, René Descartes [[a class="external" target="_blank" href="https://archive.org/details/geometryofrenede00rend"]]derogatorily coined the term[[/a]] "imaginary number" for such quantities:

[[div class="darkbox"]]

...sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.

[[/div]]

It was only in the late 18th and early 19th century that the connection between complex numbers and trigonometry, and their geometric interpretation as points on a two-dimensional plane, became accepted across mathematics. In the [[a class="external" target="_blank" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015073697180&view=1up&seq=283"]] words[[/a]] of Carl Friedrich Gauss:

[[div class="darkbox"]]

If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, -1, [[$\sqrt {-1}$]] positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.

[[/div]]

Our analysis of the book //Peágomthra//, translated as //Geometric Veneration//, indicates that the Daevites avoided this pitfall and embraced complex numbers five thousand years earlier than other civilizations did with their mathematics. Furthermore, in developing an intuitive theory of complex numbers from geometric principles, we find that Daevite pedagogy is arguably superior to many //current// approaches to the subject of trigonometry.

+ Analysis
Notable excerpts from //Geometric Veneration//, originally recovered by Yezhov //et al.// late last year [((bibcite archaeology))], are rewritten below in English and modern mathematical language (e.g. from the Daevites' base seven number system to our base ten system) and are accompanied by translation footnotes (marked with "[[span class="fnnum" style="margin:0px"]].[[/span]][[span class="fncon"]]These translation notes are optional, but may offer additional cultural insights into the Daevites' treatment of mathematics.[[/span]]") from the original [[[Old Daevite Language|Old Daevite script]]] when relevant. As many of the original diagrams were heavily damaged or faded, digital recreations of them are also provided. Comparisons to mathematics in Sumer (an ancient Mesopotamian civilization that coexisted with the Daevites) are used to highlight significant deviations from traditional mathematical pedagogy in antiquity.

++ Foundations

The excerpts below are taken from Chapter 3.

[[div class="blockquote"]]

+++ Rotations

Remind your students of the interpretation of addition
[[math]]
x \to x + a,
[[/math]]
as an act of __translation__. In this view, a quantity [[$x$]] grows or is shifted forward by a variable [[$a$]] (called the addend)[[span class="fnnum"]].[[/span]][[span class="fncon"]]In the original text, variables were represented by thematically appropriate Daevite letters, with circles drawn around them when their values were unknown. Abstract variables without any units were instead denoted by circles (or rings of concentric circles) without anything inside them.[[/span]]; examples include the stockpiling of grain and the distance traversed along a river or road towards the beating Heart of the Empire. Remind the students that when [[$a$]] is a negative quantity, it can be used to represent a decline or backwards shift in [[$x$]]; examples include the culling of livestock or an enemy battalion, the consumption of Blood for a ritual, and the accrual of debt[[span class="fnnum"]].[[/span]][[span class="fncon"]]"Negative number" can translated literally as "retreating numeral"; a positive number would instead be translated literally as an "advancing numeral". Note that the Daevites simply wrote negative numbers/variables upside down instead of using negative signs.[[/span]].

Similarly, the operation of multiplication
[[math]]
x \to m x
[[/math]]
can be interpreted as an act of __scaling__ or __partitioning__ by a multiplier [[$m$]]. The students should construct examples of this operation, such as the doubling of the Empire's population ([[$m = 2$]]) or the division of a fowl pie into four even pieces ([[$m = 0.25$]]).

[[/div]]

**Historical Analysis:** Notice that the text starts off by addressing the reader as a teacher who uses this material to help instruct their students. Unlike the cuneiform clay tablets of the ancient Sumerians, //Geometric Veneration// is a proper book (made with preserved calfskin) written in a script usually associated with the Daevite commoner class [((bibcite archaeology))], suggesting that even the teachers that use this book (alongside their students) could come from a wide variety of backgrounds and socioeconomic classes.

**Mathematical Analysis:** The interpretation of numbers-as-operations over numbers-as-quantities demonstrated here is a crucial construction that we argue enabled many of the Daevites' significant mathematical advancements. Like complex numbers, negative numbers were considered illogical and thus discarded for many centuries, as one cannot concretely possess "minus-five planks of wood" or "negative three barrels of wheat". The interpretation of negative numbers as a representation of debt accrual and backwards movement along a number line began to become commonplace in the 1st century CE through the Chinese book //[[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art"]]The Nine Chapters on the Mathematical Art[[/a]]//—over three millennia after the writing of //Geometric Veneration//. Meanwhile, the interpretation of numbers as something one //does// to the world, rather than just //being// a measure of the world's properties, allowed the Daevites to push past this line of thought and intuitively invent negative numbers far earlier than their peers.

Furthermore, the general Daevite conceptualization of a number is itself notable—for example, while distinct cuneiform [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Metrology"]]metrological[[/a]] symbols such as "one-day" and "one-sheep" were well-established, the Sumerians lacked any such symbol for "one" as an abstract number stripped of its unit of measurement. While concrete quantities were of course essential to the functioning of the Daevite Empire, previous work from Zhang //et al.// [((bibcite linguistics))] suggests that the finer details of units of measurement were generally omitted and left implicitly defined by the specific context the measurements appeared in.

[[div class="blockquote"]]

Next, have the following discussion with your students:

__Question:__ How is one to interpret a negative multiplier?
__Answer:__ Consider for instance the product [[$2 \cdot -1 = -2$]]. This act transforms a quantity of two forward units into two backward units; multiplying again returns the quantity to two forward units. Thus, a multiplier of [[$-1$]] can be interpreted as an __inversion__ or __reflection__ of a quantity's direction over the null/zero point[[span class="fnnum"]].[[/span]][[span class="fncon"]]Translated literally as the "Void point".[[/span]].
__Discussion:__ Multiplication by [[$-1$]] can also be understood as the reflection of a concept across the zero point in order to acquire its opposite and/or complement in search of completion and fulfillment. While this is a useful motif throughout mathematics, inform your students that adhering only to this reflection-based ideology ultimately limits their thinking to one dimension, and that the upcoming pursuit of a multi-dimensional mathematical framework will soon allow us to simultaneously represent all seven schools of thought celebrated by the Empire with ease.

If face-to-face discussion is infeasible, have each student independently draft her response as part of her midday prayers.

[[/div]]

**Historical Analysis:** As suggested by its title, //Geometric Veneration// constructs an active dynamic between the students, the teacher, and religious faith. Education is framed as dialogue between an instructor and their pupils, with prayer and mathematical inquiry being made one and the same in a manner so casual that its inseparability being commonplace amongst Daevite society is the most natural conclusion.

[[div class="blockquote"]]

Crucially, the holy work of High Priestess Velka—the eighth Light of Daevon, who shines the wondrous blessings of Noru-gai upon our radiant Empire—reveals an alternative interpretation of inversion that is more readily apparent when working with more intricate shapes in two dimensions[[span class="fnnum"]].[[/span]][[span class="fncon"]]"Noru-gai" can be translated literally as "Sun-Font"; refers to a member of the Daevite pantheon (see translation discussion below).[[/span]]. When performing this inversion on a triangle, it is equally valid to either reflect each constituent side or to rotate the original by a half turn in order to acquire the desired result, as shown in the diagram below[[span class="fnnum"]].[[/span]][[span class="fncon"]]The Daevites measured angles as fractions of a full turn ([[$360^\circ$]]), rather than using degrees, radians, or similar units; however, we will translate to degrees from here on out for simplicity.[[/span]].

[[include :scp-wiki:component:image-block-base name=Reflection.png
|align=center
|width=90%
|alt=alt|alt-text=A right triangle with a base and height of one on a coordinate plane is reflected across the origin to create one with a base and height of negative one. A semicircular arc connects the two, showing that the new triangle can also be obtained by rotating the original.
|caption=Diagram A: When the base and height of the original triangle are multiplied by [[$-1$]], they are individually reflected to run in the opposite direction. However, the resulting triangle this creates can also be acquired by rotating the original by [[$180^{\circ}$]].
]]

Thus, one can interpret a negative multiplier as carrying with it a __rotation__ by a half-turn angle. However, this schema raises another question: how might one construct operands for rotations by any desired angle, as measured by a level-plumb device[[span class="fnnum"]].[[/span]][[span class="fncon"]]See discussion below.[[/span]]?

To achieve this task, we define [[$\angle \theta$]] to be a __unit phasor__, i.e. an angled line segment of length 1, that is at an angle [[$\theta$]] from the positive horizontal direction[[span class="fnnum"]].[[/span]][[span class="fncon"]]"Unit phasor" can be translated literally as a "directed unit". The term "phasor" is a modern portmanteau of "phase vector".[[/span]]. Two of the most fundamental phasors are the two horizontal directions
[[math]]
\angle 0^{\circ} = 1, \quad \angle 180^\circ = -1,
[[/math]]
such that a positive number [[$x$]] can be written as [[$x \angle 0^{\circ}$]], with its inversion [[$-x$]] taking the form [[$x \angle 180^{\circ}$]].

[[/div]]

**Historical Analysis:** Notice that this geometric insight is referred to as "holy work". More specifically, the figure "Noru-gai" is identified as the second goddess of the Daevite pantheon, who presided over the sun and the harvest, light and retribution, and beauty and fiery wrath [((bibcite pantheon))]; this places the "Light of Daevon" as an influential Daeva (more specifically, a member of the Ruling Council of Mothers) who helmed the worship of Noru-gai across the Empire. Mathematical discovery, and its distribution to the masses, was thus relevant (and perhaps even central) to this priestess' role as a religious leader—and, as we will explore later, lower class Daevites would consider the application of these discoveries to their labor as their own kind of holy work.

**Mathematical Analysis:** The exact nature of the level-plumb device mentioned is unknown, but is believed to be akin to the proto-protractor found in the tomb of ancient Egyptian architect [[a class="external" target="_blank" href="https://arxiv.org/pdf/1107.4946"]]Kha[[/a]] (c. 1,400 BCE): a circle is engraved with protractor-like spokes that are compared to a string held taut by a small bob that defines the vertical direction.

[[div class="blockquote"]]
Since multiplication by [[$-1$]] is a rotation of [[$+180^{\circ}$]], multiplication by an arbitrarily angled number is equivalent to rotating by its angle measure[[span class="fnnum"]].[[/span]][[span class="fncon"]]This is the Daevites' equivalent of the angle addition postulate of Euclidean geometry.[[/span]]:
[[math angleaddition]]
(\angle \alpha) \cdot (\angle \beta) = \angle(\alpha + \beta)
[[/math]]
For instance, the quantity [[$\angle 90^{\circ}$]], known as the __quarter-turn multiplier__, is so often used that [[$i$]] is used as a distinct symbol to more accurately represent its ascendancy[[span class="fnnum"]].[[/span]][[span class="fncon"]]The quarter-turn multiplier was either written as the Daevic symbol for [[$1$]] turned [[$90^{\circ}$]] counterclockwise or denoted by the letter [[$c$]] (see translation discussion below).[[/span]]. Using ([[eref angleaddition]]), its value is understood to obey the relation
[[math]]
\angle 90^{\circ} \cdot \angle 90^{\circ} = \angle (90^{\circ} + 90^{\circ}) = \angle 180^{\circ} = -1,
[[/math]]
i.e.
[[math]]
(\angle 90^{\circ})^2 = i^2 = -1.
[[/math]]
Repeated multiplication by [[$i$]] then yields the cyclic rotational pattern
[[math]]
\angle 0^\circ &= i^0 = +1, \\
\angle 90^\circ &= i^1 = +i, \\
\angle 180^\circ &= i^2 = -1, \\
\angle 270^\circ &= i^3 = -i, \\
\angle 360^\circ &= i^4 = +1,
[[/math]]
which the students are encouraged to derive for herselves[[span class="fnnum"]].[[/span]][[span class="fncon"]]Grammatical gender in the Old Daevite script follows a hierarchal system based on the status of the subject being referred to. Save for the inanimate form that refers to objects and slaves, these genders can most accurately be translated as feminine.[[/span]].

The Light of Daevon's work then reveals the following: while they are naturally well-suited for rotation via multiplication, the addition of these angled values can be interpreted as __translations in two dimensions__, e.g. movement across a field of grain instead of merely a straight road. As such, if [[$\pm 1$]] are taken to represent a unit of travel forwards or backwards, respectively, [[$\pm i$]] the __imaginary units__ represent one unit of travel forwards or backwards in a perpendicular direction[[span class="fnnum"]].[[/span]][[span class="fncon"]]"Imaginary unit" can be translated literally as "lateral increment" or "vertical increment", depending on context.[[/span]].

Thus, we see that reflections, rotations, and translations are all united by our mathematical pursuit of ascendancy, through which all possibilities enter the grasp of our radiant, Deathless Empire[[span class="fnnum"]].[[/span]][[span class="fncon"]]See linguistic discussion below.[[/span]].

[[include :scp-wiki:component:image-block-base name=Vector%20Addition.png
|align=center
|width=90%
|alt=alt|alt-text=An arrow pointing three units to the right is added head-to-tail with an arrow pointing four units upwards. A third arrow pointing towards the final destination has a length of five and an angle of fifty three point two eight.
|caption=Diagram B: You can visually add numbers such as [[$3$]] and [[$4i$]] by drawing them as directed arrows and placing the tail of one against the head of the other. The length and angle of a third arrow pointing towards the resulting destination defines their sum.
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

**Linguistic Analysis:** The actual letter the Daevites used for [[$i$]] was equivalent to [[$c$]] in the Latin script, which is believed to be a reference to "She Who Weeps Crimson"[[span class="fnnum"]].[[/span]][[span class="fncon"]]Unlike the other six Daevite deities, no proper name (paired with the "-gai" suffix) has been given for this figure in any texts recovered by the Foundation to date.[[/span]]—the seventh and most powerful deity of the Daevite pantheon [((bibcite pantheon))]. Also known as the "Great Weaver" and the "Red Goddess"[[span class="fnnum"]].[[/span]][[span class="fncon"]]The overlap in title with the hypothesized [[[tuftos-proposal|Scarlet King]]]/"Red God" does not seem to be a coincidence—our sources suggest that his historical association with the Daevite Empire is in fact rooted in mistranslations and misunderstandings of myths surrounding She Who Weeps Crimson.[[/span]], this chief goddess was said to preside over life and [[[The Smell of Blood|corruption]]]/ascendancy[[span class="fnnum"]].[[/span]][[span class="fncon"]]The distinction here is unclear, as the Daevite language uses the same word for both concepts.[[/span]], manipulation and schemes, and nature and the stars. Notably, worship of her is continued in the modern day by a thaumaturgy-centric cult known as the "Crimson Weavers" (pending GoI classification). We hypothesize that the ascendancy/celestial theming here is the reason for this choice in symbol—instead of degrading "imaginary" numbers for their supposed unreality, the Daevites held these vertical quantities in high regard and associated them with the most powerful deity of their pantheon.

It is also worth considering that the English word "mathematics" comes from the ancient Greek word //máthēma//; its meaning, "something learned, knowledge", is reflective of the view that math was a means of studying and constructing a model of the world around them. The [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Pythagoreanism"]]Pythagoreans[[/a]] in fact held as religious doctrine that everything in the universe can be reduced to whole numbers and their ratios, so much so that numbers like [[$\pi$]] and [[$\sqrt{2}$]] were considered illogical and blasphemous (hence the origins of the term "irrational"). Meanwhile, the Old Daevite word for mathematics (//puvhiázai//) is best translated as "creation-tongue" or "transform-words", which reflects the Daevic view of mathematics as a means of exerting one's will on the world around them. With this in mind, //Geometric Veneration// using math as a tool to enact internal change via religious devotion should come as no surprise—and, as the Applications section later in this paper will demonstrate, the book espouses the use of math as a tool for reshaping the external world as well.

**Mathematical Analysis:** As previously discussed, mathematicians discovered complex numbers during the Scientific Revolution from the relatively abstract perspective of polynomial equations and their solutions; while the rules for them were algebraically well-defined, their geometric comprehension had been thoroughly rejected for many decades until [[a class="external" target="_blank" href="https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_complex/article/1003/&path_info=F34_Argands_Development_of_Complex_Numbers_2022_4_25.pdf"]]Jean-Robert Argand[[/a]] invented the complex plane visualization. Meanwhile, the Daevites //started// with a key geometric insight—the equivalence between a reflection over the origin and a [[$180^{\circ}$]] rotation around it—and extended the rules of arithmetic in order to algebraically describe and further interpret the "angled numbers" that naturally arise from that insight. This approach is far from uncommon in conventional mathematics, of course, but the fact remains that the Daevites' extensive use of it in //Geometric Veneration// allowed them to discover the complex numbers far earlier than any other civilization on Earth.

An astute reader might also notice that these diagrams take [[$\arctan(4/3) \approx 53.28^{\circ}$]], even though [[$53.13^{\circ}$]] would be a better approximation. This is a matter of rounding: while [[$3 + 4i$]] has an angle equal to about [[$14.75836\%$]] of a full turn, the Daevites rounded this to [[$14.8\%$]] (giving [[$0.148 \cdot 360^{\circ} = 53.28^{\circ}$]]). As we will see throughout the excerpts below, //Geometric Veneration// often rounds numbers to three decimal places at most; this is done during every intermediate steps of a calculation, and thus allows rounding errors to accumulate over time. This is indicative of a larger trend: the Daevites had no interest in finding the //exact// decimal expansions of the numbers they worked with, and thus made no distinction between rational and irrational numbers in any works recovered by the Foundation thus far. Given the usage of math as a tool, this makes sense—if only a few [[[mandatory-lecture-on-significant-figures|significant figures]]] of [[[SCP-4314|precision]]] for a given decimal are necessary for reasonable application, why go further and study its //exact// decimal and fractional forms (e.g. finding continued fraction expansions for irrational numbers)? For instance, the Daevites were seemingly content with the estimation [[$\pi \approx 3$]] for the area of a unit circle, with the rough approximation [[$\pi \approx 25/8 = 3.125$]] (also used by the Babylonians centuries later) being employed for more careful calculations when needed (as we will see later on).

[[div class="blockquote"]]

+++ Rectangular and Polar Form

Point out to the students that in Diagram B, the sum [[$3 + 4i$]] was rewritten as the angled number [[$5 \angle 53.28^{\circ}$]]. The second form—its __polar form__—is written in terms of the length and angle of the number, while the first is called its __rectangular form__ as it resolves the number into the diagonal of a rectangle of width [[$3$]] and height [[$4$]].

__Question:__ How might one convert between polar and rectangular form for any angled number?
__Hint 1:__ Since the length [[$5$]] is simply a scaling factor, we only need consider this matter for unit phasors. For example, how might we derive the equality
[[math]]
\angle 53.28^{\circ} = \frac{3}{5} + \frac{4}{5}i
[[/math]]
starting from either side?
__Hint 2:__ If one allows the angle of a unit phasor to vary between [[$0^{\circ}$]] and [[$360^{\circ}$]], what shape does it sweep out?
__Answer:__ A unit phasor can be drawn as a radius of the unit circle (circle of radius [[$1$]]) drawn at the specified angle. If the angle is known, the horizontal and vertical components can be physically measured from a construction of this circle. If the two components are known, one can add them head-to-tail and draw the resulting radius. Then, use a standard technique to measure the angle precisely—for instance, evenly fill the circle with a collection of rice grains, and compute the fraction of grains whose angle falls between [[$0^{\circ}$]] and the radius in question.

Allow the students to perform three to five of these rice grain calculations before showing them the following alternative. Inspired by the eternal dance of the Sun and Moon, we define two angle-dancing functions[[span class="fnnum"]].[[/span]][[span class="fncon"]]Translated literally as "numeric actions", since the concept of a mathematical function was not fully fleshed out at the time.[[/span]] __cosine__ and __sine__[[span class="fnnum"]].[[/span]][[span class="fncon"]]See translation discussion below.[[/span]] that satisfy the relation
[[math euler]]
\angle \theta = \cos{\theta} + i \sin{\theta}.
[[/math]]
That is, a unit phasor is simply the radius of a unit circle drawn at some arbitrary angle, with the cosine and sine of that angle simply being the horizontal and vertical contributions required to get there from the origin. A standard reference table of outputs for these functions, such as [[$\cos{53.28^{\circ}} = 3/5$]] and [[$\sin{53.28^{\circ}} = 4/5$]], is included in the appendix of this book.

[[include :scp-wiki:component:image-block-base name=Unit%20Circle.png
|align=center
|width=90%
|alt=alt|alt-text=A unit circle with two unit phasors drawn. Each radius has its angle labeled and is resolved into horizontal (cosine) and vertical (sine) components.
|caption=Diagram C: A unit phasor [[$\angle \theta$]] is the same as a radius of the unit circle drawn at an angle of [[$\theta$]] from the positive horizontal direction. Its arrowhead can also be reached by moving [[$\cos{\theta}$]] units horizontally and [[$\sin{\theta}$]] units vertically.
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

**Linguistic Analysis:** The Daevites named cosine and sine "noru" and "voen", meaning sun and moon, respectively. The former is associated with the previously discussed Noru-gai, while the latter is tied to the goddess "Voen-gai"[[span class="fnnum"]].[[/span]][[span class="fncon"]]Translated literally as "Moon-Font".[[/span]]—the third Fontbearer venerated by the Daevites—who presided over the moon and the night, love and loss, and solitude and entropy; the two complementary/opposing deities were said to be locked in a bitter stalemate described as an "eternal dance across the heavens" [((bibcite pantheon))].

The common word for sine traces its origins to the Sanskrit word //[[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81"]]jyā[[/a]]//, meaning bow-string, due to the visual similarity of a bow to a circular arc and a chord connecting its ends. Meanwhile, the text implies that the Daeva named cosine and sine with more symbolic (and religious) intent: the harmonic motion of the cosine and sine functions was likened to the dance between the sun and moon (see Figure 1). The origins of this conceptual connection are unclear, but we hypothesize that it stems from Daevite astronomy: the illuminated portion of the moon's area during the lunar cycle can be approximated by the function [[$(1 + \sin{\phi})/2$]], with the new moon being at an angle of [[$\phi = 0^{\circ}$]] relative to the sun and the full moon being at [[$\phi = 180^{\circ}$]].

[[div class="scp-image-block" style="width: 90%; margin: auto;"]]
[[html]]
<style type="text/css">
video {
  filter: brightness(50%);
}
video:hover {
  filter: brightness(100%);
}
</style>
<video controls autoplay loop class="video" title="An animation of oscillating cosine and sine waves." aria-label="An animation of oscillating cosine and sine waves." style="max-width: 100%;">
  <source type="video/webm" src="https://scp-sandbox-3.wdfiles.com/local--files/badooga%3Adaevite-mathematics-complex-numbers/Sun%20and%20Moon.webm">
</video>
[[/html]]
[[div class="scp-image-caption"]]
Figure 1 (Animated): The oscillation of a sine wave (black) lags behind that of a cosine wave (orange) by a quarter of a cycle; analogously, the Daevites envisioned the moon as lagging behind the sun as the two "dance" across the sky.
[[/div]]
[[/div]]

**Mathematical Analysis:** This example problem provides multiple hints that the instructor can use to provide structured support, with the students being expected to solve the problem using both visual/geometric intuition and tactile means—teaching methods that are underused by and highly desirable for educational institutions (including the Foundation's Project Athena initiative) even to this day.

The use of rice grains for computing the ratio of areas—a considerably practical method that pairs well with the Daevites' lack of computational precision beyond what was deemed strictly necessary—is not terribly out of the ordinary. For instance, in 1599, Galileo Galilei computed the [[a class="external" target="_blank" href="https://galileoandeinstein.phys.virginia.edu/more_stuff/Applets/Cycloid/cycloid.html"]]area under a cycloid[[/a]] not through calculation, but by cutting a cycloid out of a sheet of metal, weighing it, and comparing its weight with that of a cutout of the circle used to generate its shape. Furthermore,  the Daevite method here is akin to the [[a class="external" target="_blank" href="https://www.advanced-ict.info/mathematics/monte_carlo.html"]]Monte Carlo method[[/a]] of approximating [[$\pi$]], which does so by estimating the ratio of a square's area and the area of a circle inscribed within it using the fraction of randomly placed points that fall within the circle. In any case, offering the students multiple hints and having them perform these tedious calculations only three to five times before offering a simpler method suggests that a surprising amount of mercy was extended towards them by their teachers—perhaps, then, the Daeva believed that overly harsh discipline is ineffective in fostering behavioral compliance and intellectual stimulation in an educational setting.

[[div class="blockquote"]]

In addition to direct measurement, some identities for sine and cosine can be derived from the basic properties of the unit phasor. For example, expanding the angle addition formula ([[eref angleaddition]]) gives
[[math type="gather"]]
\cos(\alpha + \beta) + i \sin(\alpha + \beta) \\
= (\cos \alpha + i \sin \alpha)(\cos \beta + i \sin \beta).
[[/math]]
Distributing the product and isolating the horizontal and vertical components then allows the students to extract the individual formulae
[[math angleadditiontrig]]
\cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta, \\
\sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta;
[[/math]]
make it clear to the students that the symmetry of these equations is representative of the Sun and Moon's eternal inseparability. Meanwhile, repeated multiplication [[$(\angle \theta)^n = \angle (n \theta)$]] yields the exponentiation rule
[[math]]
(\cos{\theta} + i \sin{\theta})^n = \cos{n \theta} + i \sin{n \theta}
[[/math]]
for any integer angle scaling factor [[$n$]]. As an exercise, have the students use this formula for [[$n = 3$]] to prove the identities
[[math]]
\cos(3 \theta) &= 4 \cos^3 \theta - 3 \cos \theta, \\
\sin(3 \theta) &= 3 \sin \theta - 4 \sin^3 \theta.
[[/math]]

[[/div]]

**Mathematical Analysis:** In contemporary mathematics, identities such as these were derived independently of complex numbers, with later discoveries such as [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/De_Moivre%27s_formula"]]de Moivre's formula[[/a]] (the exponentiation rule detailed above) unifying them into a cohesive, elegant framework after the fact. Here, the Daevites were able to //start// within this framework, granting them easy access to many trigonometric identities right off the bat.

[[div class="blockquote"]]

+++ Complex Numbers

As previously discussed, addition by an arbitrary length __complex number__ can be interpreted as a two-dimensional displacement[[span class="fnnum"]].[[/span]][[span class="fncon"]]"Complex number" can be translated literally as "angled numeral".[[/span]]. Given two complex numbers [[$z = a \angle \alpha$]] and [[$w = b \angle \beta$]], their multiplication in polar form is simply
[[math]]
z \cdot w = (a \angle \alpha)(b \angle \beta) = ab \angle(\alpha + \beta),
[[/math]]
i.e. simultaneous rotation (the angular addition part) and scaling (the length part), as shown in Diagram D. 

Using ([[eref euler]]), we know that a complex number [[$z = r \angle \theta$]] can be split into a horizontal/__real__ part [[$x$]] and a vertical/__imaginary__ part [[$y$]] such that
[[math]]
z = r \cos{\theta} + i r \sin{\theta} = x + iy.
[[/math]]
This means that in rectangular form, complex multiplication is resolved simply as
[[math]]
(x + iy)(u + iv) = (xu - yv) + i(xv + yu);
[[/math]]
several sample problems for such matters are listed at the end of this chapter.

[[include :scp-wiki:component:image-block-base name=Complex%20Multiplication.png
|align=center
|width=90%
|alt=alt|alt-text=Two complex numbers are drawn on a coordinate plane. The first is three units right plus four units up, and thus has a length of five units and an angle of fifty three point two eight degrees. The second is three halves units left plus one unit up, and thus has a length of one point eight zero units and an angle of one four six point three one degrees. Their product, also drawn, is seventeen halves units left plus three units up, and thus has a length of nine point zero one units and an angle of one nine nine point four four degrees.
|caption=Diagram D: Multiplying two complex numbers adds their angles and multiplies their lengths. Performing this multiplication in either polar form or rectangular form leads to the same result.
]]

To acquire a formula for the length of [[$z$]] from its rectangular form, the Silence of Daevon—who exits her shadowed reclusion only [[[When Daevon Fell|once every decade]]] to bless our serene Empire with Voen-gai's tranquility—chose to start with its __conjugate__[[span class="fnnum"]].[[/span]][[span class="fncon"]]In place of an asterisk, the Daevites used a downward-pointing arrow to represent conjugation's vertical reflection.[[/span]]
[[math conjugate]]
z^{*} = x - iy = r \angle(-\theta);
[[/math]]
just as the transition from day to night and from night to day plunges the blessed Sun and Moon beneath the horizon, conjugation reflects [[$z$]] over the horizontal axis.

In polar form, multiplying by this conjugate yields
[[math]]
z^{*} z = [r \angle (-\theta)][r \angle \theta] = (r \cdot r) \angle 0^{\circ} = r^2,
[[/math]]
while in rectangular form it becomes
[[math]]
z^{*} z = (x - iy)(x + iy) = x^2 - i^2 y^2 = x^2 + y^2.
[[/math]]
Thus, the Silence of Daevon's holy work reveals the __distance formula__
[[math]]
|z|^2 = x^2 + y^2 = r^2.
[[/math]]
(Defer to Chapter 1 if your students need a reminder of what the absolute value [[$|z|$]] means.) When the angled radius is resolved as a right triangle, this relates the length of the hypotenuse to that of the two other sides[[span class="fnnum"]].[[/span]][[span class="fncon"]]This is the Pythagorean theorem.[[/span]].

Meanwhile, the angle [[$\theta$]] can instead be derived from [[$x$]] and [[$y$]] through a new angle-dancing function __tangent__ such that[[span class="fnnum"]].[[/span]][[span class="fncon"]]See translation discussion below.[[/span]]
[[math]]
\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{y}{x}.
[[/math]]
Tangent represents the inclination/slope of the angled radius, as well as the sharp edge between the Sun and Moon that characterizes their battle dance across the sky. A column for tangent values is included in the trigonometric tables in the appendix, which can be cross referenced to recover [[$\theta$]] based on the value of [[$y/x$]].

Notice also that because of this symmetry, we have
[[math trigdefs]]
\cos \theta &= \frac{\angle \theta + \angle (-\theta)}{2}, \\
\sin \theta &= \frac{\angle \theta - \angle (-\theta)}{2i},
[[/math]]
which the students are encouraged to derive for herselves. This formulation can be used to acquire a number of properties for cosine and sine. For instance, under vertical reflections [[$\theta \to -\theta$]], cosine remains the same, while sine picks up a negative sign—in other words, while the Sun shines brightly upon the Empire each and every day, the Moon alternates between scattering Her radiance across the night sky and withdrawing into the somber isolation of the celestial Void as the Lunar cycle progresses.

[[/div]]

**Linguistic Analysis:** The Daevite word for "tangent" is "tiḳápor", which translates literally as "battle-love"; in other words, the sun and moon goddesses Noru-gai and Voen-gai were considered both adversaries and lovers[[span class="fnnum"]].[[/span]][[span class="fncon"]]This conceptual dualism is very prevalent in the Daevite tongue, with the corruption/ascendancy conflation earlier being a more extreme example of it.[[/span]]. Between this, the reference to the Silence of Daevon (the Lunar equivalent of the Light of Daevon), and the horizon analogy with conjugation, we see that //Geometric Veneration// serves as both educational material for mathematics and the active instillment of religious lore within Daevite children.

**Mathematical Analysis:** Notice again the strong emphasis on using complex numbers as transformation instructions. Even in the absence of [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Euler's_formula"]]Euler's formula[[/a]] [[$\angle \theta = e^{i \theta}$]] and tools derived from calculus, we believe that Daevite mathematics were in many ways a precursor of [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Group_theory"]]group theory[[/a]] and its applications towards the study of symmetries (e.g. [[[SCP 1484|gauge symmetry]]]); if the Daevites' mathematical knowledge had not been lost to time, perhaps this field would have been formally developed centuries (if not millennia) earlier.

Furthermore, the case [[$r = 1$]] reduces the distance formula given above to the Pythagorean identity
[[math]]
\cos^2 \theta + \sin^2 \theta = 1,
[[/math]]
which seemingly isn't mentioned anywhere in the text recovered from //Geometric Veneration//. It is unknown whether Daeva mathematicians failed to derive this identity, failed to find an adequate religious or computational reason to merit its inclusion, or waited to cover it until later chapters of //Geometric Veneration// that couldn't be recovered by the Foundation.

++ Applications

The second recoverable portion of //Geometric Veneration// consists of the majority of Chapter 4, which applies these concepts to real-life situations.

[[div class="blockquote"]]

+++ Incentivizing the Students
One cannot learn without practice, and one cannot practice without incentive. In place of rudimentary physical punishment, whenever the students are given a problem in this chapter, you are encouraged to reward the first three students that solve it (a) most accurately, (b) most insightfully, and (c) most swiftly in a manner of your choosing. In order to improve their creativity, vigilance, and wits, we also encourage you to silently authorize sabotage amongst students so long as (a) no direct lies are uttered; (b) no student passes off a peer's work as their own; and (c) no physical harm is inflicted on any parties, nor their associates.

Furthermore, if a student wishes to enroll in the prestigious Fifth Academy within the illustrious City of Daevon, preserve their work here using a standard hemomantic seal to be submitted alongside their application.

[[/div]]

**Historical Analysis:** The "Fifth Academy" is presumed to have been an academic and religious institution in service of the fifth goddess of the Daevite pantheon; as of this paper's publication, no information on this deity has been recovered. Notably, this is the first reference to the "City of Daevon" ever discovered by the Foundation—until now, the city of [[[metaphysician-karpin-proposal|Adytum]]] (sometimes referred to as [[[rounderhouse-bone-proposal|Black Adytum]]]) was assumed to be the sole capital of the Daevite Empire. The previously cited sources discuss this finding and its possibly-retrocausal ramifications in more detail, but for our purposes, there are two important insights to be had here.

First, the Daevites appear to have forwent "rudimentary" [[a class="external" target="_blank" href="https://pmc.ncbi.nlm.nih.gov/articles/PMC3447048/"]]corporal punishment[[/a]] in favor of using competition to keep students motivated and disciplined. In the modern day, intra-Veil studies have generally found structured competition (when used [[a class="external" target="_blank" href="https://www.york.ac.uk/media/economics/documents/discussionpapers/2002/0211.pdf"]]in moderation[[/a]]) to be [[a class="external" target="_blank" href="https://ncspe.tc.columbia.edu/working-papers/files/OP35V2.pdf"]]effective[[/a]] in an academic setting. As such, we posit that this structure helped solidify the Daevite Empire's accelerated academic development compared to its peers. It is also quite revealing of their social stratification: while surprisingly accepting of upward mobility of lower class students through academic merit, Daevite society was highly cutthroat and adversarial, with active sabotage (though without plagiarism) being actively encouraged for students as young as 12 years of age (and possibly younger).

This upward mobility for students appears to have culminated in enrollment at a prestigious institution within one of the Empire's two capital cities. Rural talent was siphoned towards urban population centers before even reaching adolescence, with laborers (willing or otherwise) being utilized for construction projects within these urban centers and agricultural work in the rural expanses beyond their walls.

This revelation offers considerable recontextualization of the ancient diary (c. 2,500 BCE) recovered during Yezhov //et al.//'s expedition nearly a decade ago [((bibcite sabotage))]. The relevant entry, penned by a Daevic child estimated to be between 12 and 14 years of age, pertaining to academic sabotage is reprinted below:

[[div class="darkbox"]]

Once the embers of dusk had faded, I knew it was time to strike. Using the preplaced ladder, I slithered up the bluff and through Vida's window. When my feet landed on the floor, the wood creaked so loud. My Blood went cold! But I knew I had to press on. Before she could fully awaken, I put the rope around Vida's mouth and limbs so that she would be bound to her bed. Though I took a few glancing blows to the face, it was a success! The Great Weaver truly loves me.

I didn't have long before Vida would worm her way out of her bindings and call for help. She's such a worm! One of her mothers also possesses an axe, and I __really__ didn't want to be chopped into bits by it. So I moved quickly. It didn't take much of the bucket of Blood to drench her parchment beyond repair. I poured so much that it spilled off the desk and onto the floor. It was so cool! And more importantly, she'll __never__ get to be Teacher's favorite again.

That's when Vida freed her mouth and screamed and hurt my ears real badly. If I didn't do something fast, I'd be axe food! So I poured the rest of the bucket on her face and into her mouth to shut her up. I felt a bit ashamed of it, but then I remembered how that scar on the back of my calf is her fault, so instead I started smiling like crazy. When it got in her throat, she started coughing a bunch, and I started hearing footsteps approach from down the hall. I left the bucket behind and dove out the window! Fortunately the hay at the bottom caught me. Then I ran into the night and didn't stop until I made it home.

When I went into class the next day, Vida got yelled at by Teacher for not doing her work, and I got perfect marks. I also left a few faint Blood stains on my arms just so Vida would know it was me. I think she got the message, because she glared at me a bunch and refused to talk to me for the rest of the day. I think the divines would be proud of me. Now I have to make sure she doesn't take revenge on me anytime soon.

[[/div]]

This story was originally thought to be an isolated incident spurned by a feud between two students. However, given the new context offered by //Geometric Veneration//, perhaps such incidents were far more commonplace than previously thought.

[[div class="blockquote"]]

+++ Solving Triangles

A triangle can be formally defined as a shape with three straight side lengths [[$a, b, c$]] that are each connected. Without loss of generality, we can rotate the triangle to place the side [[$c$]] along the horizontal axis. If the arrows for [[$a,b$]] are at angles [[$\alpha$]] above the horizontal and [[$-\beta$]] below the horizontal, respectively, this gives
[[math triangledef]]
a \angle \alpha + b \angle (-\beta) = c.
[[/math]]
If one were to instead consider this triangle to be a loop being traversed, the walker would start facing forward and end in the same location facing backwards. Thus, the interior angles add up to a half turn ([[$180^{\circ}$]]).

[[include :scp-wiki:component:image-block-base name=Triangle.png
|align=center
|width=90%
|alt=alt|alt-text=A triangle is drawn on a coordinate plane. The horizontal base has a length of c, the leftmost side has a length of a and an angle of alpha from the horizontal, and the rightmost side has a length of b and an angle of negative beta from the horizontal.
|caption=Diagram E: Any triangle can be drawn as follows; its three sides with lengths [[$a, b, c$]] can be described by the complex numbers [[$a \angle \alpha$]], [[$b \angle(-\beta)$]], and [[$c \angle 0^{\circ}$]], respectively. Note that a clever student can determine that the triangle's interior angles sum up to [[$180^{\circ}$]] by identifying congruent angles at the top and seeing that they form a semicircular half-turn.
]]

By converting ([[eref triangledef]]) to rectangular form and isolating its real and imaginary parts, we acquire two separate relations
[[math]]
a \cos{\alpha} + b \cos{\beta} &= c, \\
a \sin{\alpha} - b \sin{\beta} &= 0.
[[/math]]
First, we consider the second equation, which we rearrange as
[[math lawofsines]]
a \sin \alpha = b \sin \beta;
[[/math]]
this is called the __Law of Lunar Equivalency__. This tells us that a vertical deviation [[$a \sin \alpha$]] from the path must be met with an equivalent deviation [[$b \sin \beta$]] in the opposite direction in order to reach the destination [[$c \angle 0^{\circ}$]].

This Law also reveals to us the true nature of ascension: for one to truly rise above the rest, she must be willing to be humbled in equal measure, lest she plummet towards the Earth and bleed out amidst the smouldering crater of her own ego. It was none other than Voen-gai who first taught She Who Weeps Crimson this lesson when She crashed down upon this world—the very same lesson that later spurned Our Lady Of Crimson Stardust to reject the falsity of the heavens in favor of the serenity of Daevon. When discussing this history with your students, have them share stories of past humblings and how recognizing the Law of Lunar Equivalency may have saved them from embarrassment and shame.

[[/div]]

**Mathematical Analysis:** If we let the third angle of the triangle (purple) be denoted [[$\gamma$]], the "Law of Lunar Equivalency" is really just the law of sines
[[math]]
\frac{\sin \beta}{a} = \frac{\sin \alpha}{b} = \frac{\sin \gamma}{c}.
[[/math]]
Notice how the Daevite formulation neglects the third side and angle altogether—despite its utility, more emphasis was placed on the mythology of the Fontbearers and the morals/ideals that can be extracted from the equation in relation to them. Also notice that this religious material is framed as a discussion topic between teachers and students—one that uses mathematics as a vehicle of delivery, or perhaps as a lens through which one might explore the metaphor.

[[div class="blockquote"]]

Similarly, multiplying ([[eref triangledef]]) by its conjugate gives us the expression
[[math]]
c^2 &= [a \angle \alpha + b \angle (-\beta)] [a \angle (-\alpha) + b \angle \beta] \\
&= \dots \\
&= a^2 + ab [\angle(\alpha + \beta) + \angle(-[\alpha + \beta])] + b^2
[[/math]]
Using ([[eref trigdefs]]), we know that
[[math]]
\angle \theta + \angle(-\theta) = 2 \cos \theta,
[[/math]]
so this becomes the elegant equation
[[math lawofcosines]]
c^2 = a^2 + b^2 + 2ab \cos(\alpha + \beta).
[[/math]]
This equation, known as the __Law of Sunlit Paths__, allows the distance to a destination to be computed from the constituent distances and angles of the path taken to get there. In other words, the destination [[$c$]] cannot be reached through some combination of [[$a$]] and [[$b$]] alone: their angles [[$\alpha, \beta$]] must dance together through the cosine function in order to make up the missing difference. It is Noru-gai's beautiful rhythm that first enabled the Red Dancer to rise from the ruins of Her former astral life: in embracing the Wrathful Sun's radiance and pride, She Who Weaves The Stars was born anew. Point out to your students how this Law isn't the opposite of the Law of Lunar Equivalency, but its complement—only by embracing both can a child of Daevon ever hope to prosper.

[[/div]]

**Mathematical Analysis:** Since [[$\alpha + \beta = 180^{\circ} - \gamma$]] using the previous definition of [[$\gamma$]], the "Law of Sunlit Paths" is really the law of cosines:
[[math]]
c^2 = a^2 + b^2 - 2ab \cos \gamma.
[[/math]]
Of course, despite being simpler and more compact, formulating it as subtraction would interfere with the "making up the missing difference" analogy that constitutes the bulk of the cooperation-based metaphor. Given the emphasis on beauty in the context of Noru-gai and her philosophy, this should come at no surprise—perhaps, then, Daevite standards for [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Mathematical_beauty"]]mathematical beauty[[/a]] were more rooted in symbolic beauty than aesthetic beauty.

The law of cosines and law of sines as they appear in modern mathematics were developed in the time of Hellenistic Greece; without complex numbers, they are generally proved by drawing additional altitudes that divide the triangle into smaller triangles that are then manipulated further. Here, both are derived rather straightforwardly through the purely algebraic manipulation of the simple, one-line equation ([[eref triangledef]]) that defines a triangle in its entirety—an approach that is arguably more intuitive and tends to bring about less confusion.

**Historical Analysis:** It must be emphasized that the Daevites did not conceive of the heavens as a supernatural or extradimensional space, nor as a place where the gods dwelled, nor as some sort of afterlife (as the Daevites distinctly do not conceive of existence after death); rather, the Fontbearers were considered to be one and the same with the objects, elements, and philosophies they embodied [((bibcite pantheon))]. This means that the prior references to She Who Weeps Crimson being from the heavens and having an "astral life" before her time on Earth might be literal—if she does tangibly exist in baseline reality, it is possible that she has extrasolar origins that are worthy of further exploration by the Foundation's Department of Xenohistory.

[[div class="blockquote"]]

+++ Architecture
Have the students complete the problems below.

__Question 1:__ Those who construct roofs that have inclinations less than [[$25^{\circ}$]] in areas of heavy snow are fools deserving of the gruesome Fate in store for them. If a trading post is [[$1.6 \text{ ulv}$]] wide[[span class="fnnum"]].[[/span]][[span class="fncon"]]Approximately [[$16$]] meters.[[/span]] and is built with a roof inclination of [[$30^{\circ}$]], how long are the rafters?
__Answer:__ The roof is an isosceles triangle[[span class="fnnum"]].[[/span]][[span class="fncon"]]Translated literally as "twilit triangle", possibly in reference to the balance between day and night during sunrise/sunset.[[/span]] with base angles [[$30^{\circ}$]]. Using ([[eref triangledef]]) and ([[eref trigdefs]]), we find
[[math]]
1.6 = L \angle 30^{\circ} + L \angle (-30^{\circ}) = 2L \cos{30^{\circ}};
[[/math]]
using the table of values in the appendix, this becomes
[[math]]
L = \frac{1.6}{2 \cos{30^{\circ}}} = \frac{1.6}{2 \cdot 0.866} = 0.924 \text{ ulv.}
[[/math]]

[[include :scp-wiki:component:image-block-base name=Architecture%201.png
|align=center
|width=90%
|alt=alt|alt-text=A roof is drawn as an isosceles triangle with base angles of thirty degrees. The triangle's base is one point six units wide, and the two other side lengths are denoted L. The triangle is characterized by a base width of one point six and a height of L times the sine of the thirty degree angle.
|caption=Diagram F: A diagram of the roof described in Question 1.
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

The harsh phrasing at the start of the question might seem unwarranted at first—but remember that Siberian winters are long and harsh, with temperatures dropping as low as [[$-50^{\circ}$]] or [[$-60^{\circ} \text{ C}$]] ([[$-58^{\circ}$]] or [[$-76^{\circ} \text{ F}$]]) in certain regions. Too much snow buildup can lead to a roof's collapse, bringing with it [[a class="external" target="_blank" href="https://www.aljazeera.com/video/newsfeed/2026/1/21/deadly-roof-collapse-at-russian-shopping-mall-after-heavy-snowfall"]]lethal[[/a]] consequences that warrant treating the precise angle and sloping of a roof as a matter of life or death.

[[div class="blockquote"]]

__Question 3:__ The blessed Sun reaches its lowest Solar angle—[[$50^{\circ}$]] from the horizontal—during the summer. The roof of a temple to Voen-gai is designed with an overhang such that the bottom of the window beneath it is [[$0.2 \text{ ulv}$]] away. How long should the overhang be at minimum to provide the Lunar temple full shade from the Sun year-round?
__Answer:__ In order to block the Sun at all times of year, the arrow pointing from the bottom of the window towards the Sun should be at an angle of [[$50^{\circ}$]] or less. This allows us to set up the equation
[[math]]
r \angle 50^{\circ} = x + iy = L + 0.2i.
[[/math]]
This means that
[[math]]
\tan 50^{\circ} = \frac{y}{x} = \frac{0.2}{L};
[[/math]]
the table of values for tangent thus gives us a minimum overhang length of
[[math]]
L = \frac{0.2}{\tan 50^{\circ}} = \frac{0.2}{1.19} = 0.17 \text{ ulv}.
[[/math]]

[[include :scp-wiki:component:image-block-base name=Architecture%202.png
|align=center
|width=90%
|alt=alt|alt-text=A roof overhangs the side of a house by some length L. The sun's rays hit the roof and side at an angle of fifty degrees from the horizontal, thus forming a right triangle with a width of L. The height of that right triangle is zero point two units.
|caption=Diagram G: A diagram of the roof overhang described in Question 3. Notice that the red and blue guiding lines below and to the right of the angle are equal in length to those above and to the left.
]]

__Question 4:__ In order to estimate the height of an enemy watchtower, an invisible scout stands some distance from it at ground level. From this point, she finds the angle of elevation from the ground to the top of the tower to be [[$30^{\circ}$]]. She then moves [[$2 \text{ ulv}$]] closer to the tower and finds the angle of elevation to now be [[$60^{\circ}$]]. What is the height of the tower?
__Answer:__ For unknown distances [[$d_1, d_2$]], the first measurement is an arrow [[$d_1 \angle 30^{\circ}$]] pointing from the scout to the top of the tower, while the second is a steeper arrow [[$d_2 \angle 60^{\circ}$]] offset by [[$3.5$]] units horizontally. Thus,
[[math triangulation]]
d_1 \angle 30^{\circ} = 3.5 + d_2 \angle 60^{\circ}.
[[/math]]
We could then straightforwardly find the height of the tower by solving the pair of real/imaginary equations for one of the distances and multiplying it by the sine of the appropriate angle:
[[math]]
d_1 \cos 30^{\circ} &= 3.5 + d_2 \cos 60^{\circ}, \\
d_1 \sin 30^{\circ} &= d_2 \sin 60^{\circ}
[[/math]]
However, since we only wish to know the vertical component [[$y$]], these two distances are irrelevant for the final answer, so a clever student (and a clever scout, once she returns to her camp) would instead write the tangent relations
[[math]]
\tan 30^{\circ} &= \frac{y}{x}, \\
\tan 60^{\circ} &= \frac{y}{x - 3.5}.
[[/math]]
Using substitution and tedious algebraic manipulation, this yields the solution
[[math]]
y &= \frac{3.5 \tan{60^{\circ}} \tan{30^{\circ}}}{\tan{60^{\circ}} - \tan{30^{\circ}}} \\
&= \frac{3.5 (1.732) (0.577)}{1.732 - 0.577} \\
&= 3.028 \text{ ulv}.
[[/math]]
__Discussion:__ A clever student could use this work to conclude that the general formula for estimating some height [[$y$]] from ground level is
[[math]]
y = \frac{d \tan{\theta_2} \tan{\theta_1}}{\tan{\theta_2} - \tan{\theta_1}},
[[/math]]
where [[$d$]] is the distance traversed between the two angle measurements. This can be used for a variety of purposes, e.g. estimating the height of a mountain or the depth of a valley using only a level-plumb device. For best results, have the students use this formula for nearby geographical landmarks (assuming relatively flat ground) and compare their answers to see who attains the most accurate one. (A clever student will walk farther between angle measurements than the rest to improve their precision.)

[[include :scp-wiki:component:image-block-base name=Architecture%203.png
|align=center
|width=90%
|alt=alt|alt-text=A tower of height y is drawn alongside two points on the ground at both x and x minus two units to the left of it. Two dashed lines are drawn from those points to the top corner of the tower. The first line has an angle of thirty degrees and a length of d subscript one units, while the second line has an angle of sixty degrees and a length of d subscript two units.
|caption=Diagram H: A diagram of the tower and measurement points in Question 4. Notice that the equality ([[eref triangulation]]) is shown visually in the form of a triangle with side lengths [[$3.5, d_1, d_2$]].
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

**Mathematical Analysis:** For Question 4, it is worth noting that the general height estimation formula can be simplified to
[[math]]
y = \frac{d}{\cot \theta_1 - \cot \theta_2}
[[/math]]
However, the Daevites seemingly did not develop the reciprocal functions secant, cosecant, and cotangent (let alone tables of values for them) in spite of their notational efficiency. Regardless, notice the reoccurrence of the competitive element—particularly in the hands-on context of "go outside and take an actual measurement" that takes learning outside of the classroom. For Foundation personnel who wish to experience this exemplary "learn by doing" pedagogy for themselves, we suggest estimating the height of your secure facility with this measuring process (use an angle measuring app on your mobile phone) and comparing the results to those of your coworkers.

The dozens of problems thereafter follow a similar pattern. For our purposes, we will focus only on the final problem: a challenge question that applies what the student has learned to a more advanced subject.

[[div class="blockquote"]]

__Question 49:__ A natural spring is [[$4900 \text{ ulv}$]] away from the City of Daevon by foot; its elevation is [[$10 \text{ ulv}$]] higher than that of the city[[span class="fnnum"]].[[/span]][[span class="fncon"]]About [[$49$]] kilometers and [[$100$]] meters, respectively.[[/span]]. The Empire has its construction-slaves[[span class="fnnum"]].[[/span]][[span class="fncon"]]See translation discussion below.[[/span]] build an aqueduct that transports [[$7 \text{ sigai}$]] of water and [[$10 \text{ logai}$]] of life energy per citizen per day from the spring to the City, providing its beloved citizens with splendor beyond description[[span class="fnnum"]].[[/span]][[span class="fncon"]]These quantities are equivalent to about [[$10$]] gallons ([[$37.85$]] liters) of water and [[$5$]] Caspers of Aspect Radiation (see below), respectively. Notice that both units are derived from the word "gai".[[/span]]. (Naturally, the City has nine more aqueducts just like it.)

A Daeva uses holy magic to raise the spring's elevation, increasing the aqueduct's angle of inclination by [[$0.05^{\circ}$]]. How much water and life energy does the spring now provide per person per day? Use the proportionality
[[math]]
V \propto \sqrt{ S },
[[/math]]
where [[$V$]] is the amount of water/energy received and [[$S$]] is the water's stream slope.
__Answer:__ If the City of Daevon is the origin, the spring is originally at the coordinates
[[math]]
d \angle \theta = F = 4900 + 10i
[[/math]]
To rotate by [[$0.05^{\circ}$]], we first use ([[eref euler]]) and our table of trigonometric values to calculate the unit phasor associated with it:
[[math]]
\angle 0.05^{\circ} = 1 + 0.000873i.
[[/math]]
Multiplication then gives
[[math]]
F \cdot \angle 0.05^{\circ} &= (4900 + 10i)(1 + 0.000873i) \\
&= \dots \\
&= 4899.99 + 14.2777i.
[[/math]]
The slope [[$S$]] can be identified with the inclination ratio [[$\tan \theta$]]. This means that the water and energy carried by the aqueduct are both multiplied by
[[math]]
k &= \sqrt{\frac{\tan{\theta'}}{\tan{\theta}}} \\
&= \sqrt{\frac{14.2777/4899.99}{10/4900}} \\
&= \sqrt{1.42777} \\
&= 1.195.
[[/math]]
Thus, the modified aqueduct can supply about [[$8.365 \text{ sigai}$]] of water and [[$11.95 \text{ logai}$]] of life energy per person per day.
__Discussion:__ While the students could have solved this problem by finding the original angle of inclination using their tangent table, adding [[$0.05^{\circ}$]] to the angle directly, and taking tangent again, notice that this method computed the same result directly in rectangular coordinates: once the rotation phasor was computed, no angles entered the picture again! Further, point out that if this problem were given again for a different aqueduct, the phasor [[$\angle 0.05^{\circ}$]] could be reused for it, which improves efficiency compared to the manual method.

Next, point out how a small angle increase of [[$0.05^{\circ}$]] improved the aqueduct's flow by nearly [[$20\%$]]. Of course, a student could argue that raising a spring by [[$4.2777 \text{ ulv}$]] is no small feat, and thus that it is no surprise that doing so yields significant gains from it. This is the power of angles: over long distances, even small angle changes will translate to large shifts in either coordinate.

[[include :scp-wiki:component:image-block-base name=Aqueduct.png
|align=center
|width=90%
|alt=alt|alt-text=A right triangle is drawn out from a depiction of a city with a base of forty nine hundred units and a height of ten units. The hypotenuse of that triangle is then rotated by an angle of zero point zero five degrees, which raises its height by some amount.
|caption=Diagram I: A diagram of the geometry for the aqueduct from Question 49. Of course, this diagram isn't to scale—in reality, the slope is so gentle that the horizontal displacement of the spring upon the rotation of [[$0.05^{\circ}$]] is entirely negligible.
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

**Historical Analysis:** In addition to being a wonderful culmination of the pedagogy we've seen so far, this question indicates that aqueducts were used to transport water to urban population centers in the Daevite Empire in a manner not dissimilar to the [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Roman_aqueduct"]]Roman Empire[[/a]]—although the parallel transportation of [[[goc-supplemental-arad|Elan-Vital Energy]]] (EVE) certainly makes for a significant deviation from them. In some sense, these aqueducts could be described as artificial leylines that transported EVE for utilization by the Daeva upper class and even ordinary citizens in order to perform thaumaturgical feats of some kind. Perhaps, then, EVE (alongside with the water carrying it) was a rudimentary equivalent of electricity used to, say, provide artificial lighting at night or power lumber mills.

Further speculation is, of course, outside the scope of this paper—and until archaeological evidence of these aqueducts' existence can be found, speculation is all it shall remain. For now, consider how these dual-purpose aqueducts heavily strengthen the Daevite metaphor of life energy springing forth from a "Font" of water: water is an excellent carrier of EVE, and it's likely that the EVE itself was used to keep the water from freezing during even the harshest of Siberian winters. From here, the mechanics of essential utilities like these being introduced in a core mathematics textbook for 12 year old Daevite students makes it clear that a strong workforce of engineers was actively cultivated by the Daeva in order to strengthen and reinforce the Empire on the domestic front (and perhaps on its various military fronts as well).

As discussed more thoroughly by Bakkal //et al.// [((bibcite government))], Daevite slavery was divided into two state-enforced categories that when combined made up about 30% to 40% of the Empire's population. As demonstrated in this problem, the first was called "construction-slavery", in which slaves were utilized within population centers to gather and process raw materials like stone and ores into refined goods, build and maintain structures like bridges, houses, and monuments (and, as indicated here, aqueducts), and support other infrastructure projects. The other is aptly called "destruction-slavery", and is believed to be similar to the military slavery of the [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Auxilia"]]Romans[[/a]], [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Janissary"]]Ottomans[[/a]], and [[a class="external" target="_blank" href="https://archive.nytimes.com/opinionator.blogs.nytimes.com/2011/07/27/slavery-and-freedom-at-bull-run/?_r=0"]]Confederates[[/a]], in which slaves bolstered secondary labor efforts for military legions and/or directly fought on the front lines in times of need. Intriguingly, the evidence gathered by Bakkal //et al.// suggests that slaves were expressly prohibited from interacting with Daevite agricultural production and anything food-related, with anomalous means of labor being utilized in their place (as demonstrated in the Agriculture section below).

**Mathematical Analysis:** The Discussion section of this problem further illuminates the previously discussed dynamic between a teacher and their students. Active debate was acceptable and actively encouraged, with the teacher taking the role of a supportive mentor over a harsh superior. Perhaps Daevite math classes were akin to Greek [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Socratic_method#Socratic_seminar"]]Socratic seminars[[/a]]—although, given the prior authorization of inter-student sabotage, this setting would likely be more akin to a war room of scheming commanders and politicians vying for superiority.

[[div class="blockquote"]]

+++ Triangle Areas
Previous mathematical study has (hopefully) revealed to the students that the area of a rectangle with width [[$b$]] and height [[$h$]] is equal to [[$bh$]]. If the rectangle's angles are tilted into a parallelogram, the overall area remains constant, and so the same formula applies. If we then cut the parallelogram along one of its diagonals, the resulting triangle has half the area, i.e.
[[math]]
A = \frac{1}{2} bh.
[[/math]]
We can do better than this. If the bottom left corner is taken as the origin, a triangle is defined by two complex numbers [[$z$]] and [[$w$]] which extend out from that corner, as shown in the diagram below.

[[include :scp-wiki:component:image-block-base name=Agriculture%200.png
|align=center
|width=90%
|alt=alt|alt-text=A triangle is drawn on a coordinate plane with one corner at the origin. Its two other vertices are denoted z and w, with a line towards z forming the base of the triangle with length b and a dashed line from w to that base forming its altitude/height. The base z is rotated by an angle of theta compared to the positive horizontal axis. A second triangle with thinner lines is drawn; it is congruent to the first, but is rotated so that its base lies along the positive horizontal axis. The height of this rotated triangle, denoted h, can be identified as the vertical position of the rotated point for w, which thus serves as the height of the original triangle as well.
|caption=Diagram J: A triangle (red) can be uniquely defined by the origin point [[$0$]] and the two complex numbers [[$z$]], [[$w$]] that form the other vertices. If the triangle is rotated (blue) such that [[$z$]] lies along the positive real axis, the height of both triangles (purpled, dashed) can be identified as the imaginary part of the rotated version of [[$w$]].
]]

The base is the length of [[$z$]], so we can simply write
[[math]]
b = |z| \implies z = b \angle \theta.
[[/math]]
Were the triangle rotated by an angle [[$-\theta$]] such that [[$z$]] is purely horizontal, the height [[$h$]] would simply be the imaginary/vertical component of [[$w$]]. Thus, in general,
[[math]]
h = \begin{vmatrix} \Im\{w \cdot \angle (-\theta)\}  \end{vmatrix},
[[/math]]
where [[$\Im$]] means "the imaginary part of"[[span class="fnnum"]].[[/span]][[span class="fncon"]]Translated literally as "the ascended part of". Note that the Daevite equivalent of a capital C was used in the original text (in reference to She Who Weeps Crimson) instead of J.[[/span]]. Note that an absolute value is taken to ensure that the height is always positive.

Putting this all together, we get
[[math trianglearea]]
A &= \frac{1}{2} bh \\
&= \frac{1}{2} |z| \cdot \begin{vmatrix} \Im\{w \cdot \angle (-\theta)\} \end{vmatrix} \\
&= \frac{1}{2} \begin{vmatrix} \Im\{|z| \angle(-\theta) \cdot w\} \end{vmatrix} \\
&= \frac{1}{2} \begin{vmatrix} \Im\{z^{*} w\} \end{vmatrix}.
[[/math]]
The third step uses the basic properties of scaling to move [[$|z|$]] inside the imaginary part, and the fourth step uses the definition of the conjugate/vertical reflection ([[eref conjugate]]).

Crucially, this formula can be used in both rectangular and polar coordinates. In the former case, you simply multiply the second corner's position by the conjugate of the first and isolate the imaginary component; in the latter case, the sine function will need to be used instead. Furthermore, only the portion of [[$w$]] that is perpendicular to [[$z$]] will contribute to this area, as multiplication of the parallel parts by [[$z^{*}$]] will always yield a real number that is consequentially discarded. This also means that [[$w$]] needn't be the raw position of the second vertex—it can simply be the arrow pointing from [[$z$]] to it.

Since all polygons can be divided into triangles, we have thus devised a method of finding the area of any such shape, e.g. in order to survey a field of farmland or compute how much paint is needed to decorate the side of a temple.

[[/div]]

**Mathematical Analysis:** In the text above, the origin of the coordinate system and the orientation of the triangle are considered malleable, which considerably simplifies the derivation of the area formula—an operation-centric mathematical motif we have seen several times throughout the representative excerpts shown so far.

It is also worth noting that the formula given here for the area of a triangle is equivalent to the [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Area_of_a_triangle#Using_vectors"]]cross/wedge product formula[[/a]]
[[math]]
A = \frac{1}{2} \begin{Vmatrix} \vec{z} \wedge \vec{w} \end{Vmatrix} \to \frac{1}{2} \begin{Vmatrix} \vec{z} \times \vec{w} \end{Vmatrix}
[[/math]]
in Euclidean space (with the cross product formula working only in three dimensions). No preserved material regarding the Daevite's treatment of higher-dimensional objects has been recovered to date, but we suspect that such an extension would have led to a similar theory to that of [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Quaternion"]]quaternions[[/a]].

For the students, the representation-free approach (i.e. without converting to rectangular or polar form in particular) taken here eliminates the potential for conceptual confusion stemming from implementation-based technicalities—they are given the freedom to convert a given problem to the form they find most natural, and are capable of comprehending the underlying concepts and equations even if a particular expression of them gives them trouble. If you have any doubts as to the benefits of this coordinate-free approach, ask "What is a tensor?" to a mathematician and to a physicist and witness the stark contrast in higher-level comprehension for yourself.

[[div class="blockquote"]]

+++ Agriculture
Have the students complete the problems below.

__Question 1:__ A holy temple maintains a garden dedicated to the Seven Nurturers in the shape of a regular heptagon that is [[$15 \text{ ulv}$]] on each side[[span class="fnnum"]].[[/span]][[span class="fncon"]]"Regular heptagon" can be translated literally as "Crimson Glyph"; see further discussion below.[[/span]]. What is its area?
__Answer:__ The regular heptagon can be divided into seven identical isosceles triangles with apex angles of [[$360^{\circ}/7 = 51.429^{\circ}$]]. If we rotate this triangle such that the length [[$15$]] is horizontal, the triangle satisfies
[[math]]
r \angle \theta + r \angle (-\theta) = 15
[[/math]]
where
[[math]]
51.429^{\circ} + 2 \theta = 180^{\circ} \implies 2 \theta = 128.571^{\circ}.
[[/math]]
Using ([[eref trianglearea]]), the area of the full shape is
[[math]]
A &= 7 \begin{vmatrix} \frac{1}{2} \Im\{ [r \angle \theta]^{*} [r \angle (-\theta)] \} \end{vmatrix} \\
&= \frac{7}{2} \begin{vmatrix} \Im\{ r^2 \angle(-2 \theta) \} \end{vmatrix} \\
&= \frac{7}{2} r^2 \sin 128.571^{\circ}.
[[/math]]
The task is then to compute [[$r$]] in terms of the given length of 15. Using the Law of Sunlit Paths ([[eref lawofcosines]]), we get
[[math]]
15^2 &= r^2 + r^2 + 2 r^2 \cos(\theta + \theta) \\
&= 2r^2 (1 + \cos 128.571^{\circ}),
[[/math]]
which solves as
[[math]]
r^2 = \frac{15^2}{2 (1 + \cos 128.571^{\circ})}.
[[/math]]
Therefore
[[math]]
A &= \frac{7}{4} (15^2) \frac{\sin{51.429^{\circ}}}{1 + \cos 128.571^{\circ}} \\
&= 15^2 \frac{7}{4} \frac{0.782}{1 - 0.623} = 509.789,
[[/math]]
which is just shy of [[$816.744$]] square ulv[[span class="fnnum"]].[[/span]][[span class="fncon"]]A little over [[$20$]] acres ([[$80937$]] square meters).[[/span]].
__Discussion:__ From this work, a clever student can conclude that the area of a regular [[$n$]]-gon with side length [[$s$]] is
[[math]]
A_n &= \frac{ns^2}{4} \frac{\sin(2 \theta)}{1 + \cos(2 \theta)}, \\
2 \theta &= \frac{n - 2}{n} 180^{\circ},
[[/math]]
and a cleverer student could use trigonometric identities (see Chapter 7) to reduce this further to
[[math]]
A_n = \frac{n s^2}{4 \tan (180^{\circ}/n) }
[[/math]]
Of course, a holy garden with anything other than [[$n = 7$]] would be utterly blasphemous and summarily burnt to ash.

[[include :scp-wiki:component:image-block-base name=Agriculture%201.png
|align=center
|width=90%
|alt=alt|alt-text=A regular seven-sided polygon with a side length of fifteen is drawn; its internal angles are fifty one point four two nine degrees each, and each of its seven sectors are shaded with a different color: white, orange, black, yellow, blue, green, red.
|caption=Diagram K: A diagram of the garden in Question 1. Make sure the students draw their gardens neatly and precisely—if a glyph's drawn angles are uneven to the naked eye, the symbol is blasphemous and must be redone from scratch.
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

**Historical Analysis:** The colors chosen for the heptagonal garden are approximations based on trace pigments recovered from the preserved textbook following chemical analysis [((bibcite archaeology))]. In addition to a robust color palette based on those of the seven Fontbearers, the Daevites demonstrated an ancient form of [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Numerology"]]numerology[[/a]] that manifests most overtly in cases involving the number seven. The ascription of divine or occult power to numbers is [[[SCP-627|far from uncommon]]], but the Daevite Empire enforced it quite strongly—numbers were counted in base seven, heptagons were considered expressions of faith, Daeva imperial power was divided amongst seven branches of government [((bibcite government))], and so on and so forth. Of course, the Daevite Empire was not the first ancient, anomaly-wielding civilization to ascribe occult meaning to numbers, or even to the number seven—the [[[second-hytoth-hub|Ortothan Kingdom]]] (c. 11,000 BCE) worshipped seven "Koru-teusa" (deities that serve as supposed universal guardians against threats from outside reality), with a motif pertaining to the number seven pervading Ortothan belief in a manner not dissimilar to the Daevites. Verification of a deeper connection between these belief systems and their respective numerologies is pending further research.

Notice also that this diagram assigns the Fontbearers seven equal sectors of the heptagon. This implies some degree of equality between these deities and the respective philosophies they embody—though, with other texts describing She Who Weeps Crimson as the leader and possible manipulator or subjugator of her fellow goddesses [((bibcite pantheon))], perhaps this supposed equality was superficial in nature or otherwise more complex than a simple polygon in a children's textbook could convey. Whether this hierarchy (or lack thereof) also extended to the politics of the Daeva upper class is currently unknown [((bibcite government))].

[[div class="blockquote"]]

__Question 13:__ It is said that when She Who Weeps Crimson was taught the secret ways of water by the Dweller of Depths, She chose to pass down this knowledge to the people of Daevon so that the thirst of Her children would be quenched forevermore. One such secret is the method of planting trees across Daevite farmland: in addition to providing crops and livestock alike with shade and protection from harsh weather, simple incantations can be used to modify a tree's branches such that she[[span class="fnnum"]].[[/span]][[span class="fncon"]]As trees and plants are alive/animate, they are gendered as female by the Old Daevite script.[[/span]] acts as sprinklers for irrigation purposes (provided that water is fed to the tree's roots by a standard system of canals).

The branches of one such pine tree—enhanced in size through a simple incantation—uniformly sprinkle water onto a field of wheat below her at a rate of [[$105 \text{ sigai}$]] of water per day. The branches each extend about [[$2.5 \text{ ulv}$]] vertically and are at an angle of [[$20^{\circ}$]] from the trunk/vertical direction. What is the precipitation rate for the wheat in [[$\text{ulv/day}$]]?
__Answer:__ Since the water consumption is given as a rate of volume per day, we simply need to divide it by the horizontal area it covers. In a given horizontal slice of the tree, a branch's tip is at the position
[[math]]
L \angle 70^{\circ} = w + 2.5i,
[[/math]]
so using tangent, the length of wheat [[$w$]] that the branch extends over satisfies
[[math]]
\tan 70^{\circ} = \frac{2.5}{w}
[[/math]]
and thus equals
[[math]]
w = \frac{2.5}{\tan 70^{\circ}} = 0.910.
[[/math]]
Using our work in Chapter 2, the circular area of wheat that the tree covers is
[[math]]
A = \pi w^2 = 3.125(0.910)^2 = 2.588 \text{ ulv}^2.
[[/math]]
We can then convert the rate of water usage to[[span class="fnnum"]].[[/span]][[span class="fncon"]]Since the Daevites used base [[$7$]] for their mathematics, this also carries over to their version of scientific notation.[[/span]]
[[math]]
R = \frac{105 \text{ sigai}}{1 \text{ day}} \cdot \frac{1 \text{ ulv}^3}{77 \cdot 7^4 \text{ sigai}} = 1.364 \cdot 7^{-4} \text{ ulv}^3/\text{day},
[[/math]]
thus giving an answer of
[[math]]
d = \frac{R}{A} = 0.527 \cdot 7^{-4} \text{ ulv}/\text{day}.
[[/math]]
__Discussion:__ In colder climates, wheat uses up to about [[$0.5 \cdot 7^{-4} \text{ ulv}$]] each day[[span class="fnnum"]].[[/span]][[span class="fncon"]]Approximately [[$0.08$]] inches each day.[[/span]], which makes this tree fully sufficient for the wheat's irrigation needs. The tree's roots absorb any excess ground water (e.g. from snow) and store it within her mass, which minimizes waste water compared to methods like flood irrigation used by lesser civilizations.

__Question 14:__ The pine trees described in Question 13 are planted [[$2 \text{ ulv}$]] on center in a triangular formation within a square patch of farmland that is [[$60 \text{ ulv}$]] on a side[[span class="fnnum"]].[[/span]][[span class="fncon"]]About [[$89$]] acres ([[$0.36$]] square kilometers) of area.[[/span]]. How many trees can be planted on this land at most?
__Answer:__ Since each tree is [[$2 \text{ ulv}$]] apart, a set of three neighboring trees will form an equilateral triangle with base [[$2$]] and sides [[$2 \angle (\pm 60^{\circ})$]]. This means that the width of a given alley is
[[math]]
\Im\{ 2 \angle 60^{\circ} \} = 2 \sin 60^{\circ} = 2 (0.866) = 1.732,
[[/math]]
so the farm can contain
[[math]]
\frac{60}{1.732} = 34.642 \to 34
[[/math]]
alleys of trees. Since [[$60 / 3 = 20$]], each alley will alternate between [[$20$]] and [[$19$]] trees on a side. Thus, we take the average and multiply:
[[math]]
19.5 \cdot 34 = 663 \text{ trees}.
[[/math]]

[[include :scp-wiki:component:image-block-base name=Agriculture%202.png
|align=center
|width=90%
|alt=alt|alt-text=First, a green line representing a tree trunk is drawn in a coordinate plane. A branch then protrudes from it at an angle of twenty degrees from the trunk and spans a vertical space that is two point five units long. The branch sprinkles water down on a horizontal portion of land beneath it with width w. Next, a rectangular area contains a triangular grid of green dots representing trees. One triangle is highlighted as having lengths of two units on each side and interior angles of sixty degrees.
|caption=Diagram L: A diagram of a singular tree and the farm of these trees in Questions 13 and 14, respectively. Of course, the two are not to scale with each other. 
]]

[[div style="margin-top: 15px;"]]

[[/div]]

[[/div]]

**Historical Analysis:** Questions 13 and 14 detail an incredibly revolutionary agricultural tool: a thaumaturgical combination of [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Agroforestry"]]agroforestry[[/a]] and [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Center-pivot_irrigation"]]center-pivot irrigation[[/a]]. The former integrates trees with crops and livestock on the same land, carrying with it benefits such as:
# Supporting greater biodiversity (e.g. avian life) in the local area
# Enhancing and [[a class="external" target="_blank" href="https://openknowledge.fao.org/server/api/core/bitstreams/8cd17c6f-051b-4105-866b-dcd344d4ba78/content"]]restoring[[/a]] soil fertility, water availability, and resistance to erosion
# Sheltering crops and livestock from harsh weather (e.g. [[a class="external" target="_blank" href="https://www.researchgate.net/publication/257510212_Effects_of_shelterbelts_on_snow_distribution_and_sublimation"]]snow[[/a]] or drought)
# Providing food (e.g. vitamin-rich nuts), [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Fodder"]]fodder[[/a]], timber, and medicine to rural communities

Meanwhile, center-pivot irrigation is a modern irrigation system where a long pipeline rotates about a central pivot point and delivers water through a mechanized sprinkler system. Center pivots deliver water uniformly on a large scale, significantly improving the efficiency of water usage, minimizing soil erosion and compaction, and optimizing crop yields.

Both techniques are nonanomalous and thus ubiquitous in intra-Veil communities—the former has existed for millennia and is used on [[a class="external" target="_blank" href="https://uniteddiversity.coop/2009/08/28/agroforestry-found-on-nearly-half-the-worlds-farms/"]]nearly half[[/a]] the world's farms, while the latter is used to irrigate [[a class="external" target="_blank" href="https://www.canr.msu.edu/beginning-farmer/uploads/files/BFRDMG_Plants_Water_Pivot%20Irrigation.pdf"]]about 76%[[/a]] of American farmland as of 2022. Here, however, both techniques are combined through anomalous means: trees were regularly planted across Daevite farmland, with thaumaturgy transforming them into sprinkler systems that irrigate the crops using an external supply of water (the tree needn't rotate about a center pivot, as its branches are assumed to cover the area uniformly). Other benefits, such as the absorption of excess water mentioned in the problem, can be easily extrapolated from there.

[[include :scp-wiki:component:image-block-base name=Agroforestry.png
|align=center
|width=90%
|alt=alt|alt-text=A field of cotton crops with pecan trees planted in rows along it. The picture is attributed to the National Agroforestry Center.
|caption=Figure 2: Agroforestry is used regularly to bolster nonanomalous agriculture, such as with pecan and cotton [[a class="external" target="_blank" href="https://www.fs.usda.gov/nac/practices/alley-cropping.php"]]alley cropping[[/a]].
]]

The scientific basis of this technique is sound: trees already maintain a significant, nonanomalous role in the water cycle, with tiny pores called [[a class="external" target="_blank" href="https://auf.isa-arbor.com/content/16/12/331"]]stomata[[/a]] facilitating the [[a class="external" target="_blank" href="https://www.sciencelearn.org.nz/resources/2632-trees-and-natural-cycles"]]transpiration[[/a]] of water vapor into the atmosphere. Given the text of the question and the Daevites' highly developed arboriculture and thaumaturgy, we conjecture that this technique was ubiquitous across the Empire, granting it a highly diverse and bountiful food supply that could support its large domestic, military, and slave populations. Given our own advancements in [[[a-brief-introduction-to-thaumaturgical-engineering|thaumaturgical engineering]]], we sincerely hope that the Foundation can replicate this technique for its own benefit within the next few years.

On the other hand, the factual basis of the Daevite religious lore surrounding the origins of this technique is shaky at best. The "Dweller of Depths" is //not// given any details in previous sources [((bibcite pantheon))], and the concrete existence of She Who Weeps Crimson as a pluripotent entity (let alone one that directly communicates with its followers) has not yet been confirmed or disproved by the [[[tactical-theology-hub|Department of Tactical Theology]]]. However, the fifth section of the garden from the question before this one is given a blue, water-like color in the original text, which could suggest that the Dweller of Depths is in fact the missing fifth Fontbearer—the very same deity identified with the previously discussed "Fifth Academy", which could have been the location of this technique's genesis. Speculation aside, it is likely that this irrigation technique was discovered through Daeva scientific study, laymen experimentation, or other [[[SCP-9320|mundane explorations]]] and attributed to the Fontbearer pantheon after the fact.

Regardless, this "para-arboreal irrigation" technique could not exist without the mathematics to support it. A tree's application and absorption rates of water must be carefully calibrated to prevent over- or under-watering, which in turn requires the necessary area covered by the tree's branches to be computed (as demonstrated in this problem) and adjusted according to the water availability of the surrounding area. In the absence of solid mathematical theory (and mathematics education that is sufficiently widespread enough to propagate that theory to rural areas), this para-arboreal irrigation would simply be unsustainable.

[[div class="blockquote"]]

__Question 49:__ When the sky still bled[[span class="fnnum"]].[[/span]][[span class="fncon"]]A phrase from Daevite folklore used at the start of legends and other tales; equivalent to "once upon a time" in modern stories.[[/span]], the Fourth Legion of old—eternally fearless and indomitable, for the Wrath of Daevon personally blessed the path they carve in the name of the Matron of War—conquered numerous parcels of farmland used to grow millet. To survey one such prize, esteemed champion Commander Rakshal begins at a corner stone next to the road; thence north [[$10^{\circ}$]] east along the road, [[$32 \text{ ulv}$]] to a stone; thence east [[$15^{\circ}$]] north, [[$24 \text{ ulv}$]] to a corner stone next to a different road; thence south [[$10^{\circ}$]] east along that second road to a pike; thence west [[$30^{\circ}$]] south to the place of beginning.

The commander then directs one of her [[[SCP-3140|wooden behemoths]]] to use its animated roots to till the land; the behemoth can cover approximately [[$40 \text{ ulv}^2$]] of farmland for every 16-hour day it works[[span class="fnnum"]].[[/span]][[span class="fncon"]]This is about one acre ([[$4047$]] square meters) per 16-hour workday; see quantitative discussion below.[[/span]]. How many days will it take the behemoth to till the whole field?
__Answer:__ Since north/south is [[$\pm i = \angle (\pm 90^{\circ})$]] respectively, the four directed quantities that represent the commander's path are
[[math]]
d_1 &= 32 i \angle (-10^{\circ}) = 32 \angle 80^{\circ}, \\
d_2 &= 24 \angle 15^{\circ}, \\
d_3 &= -ia \angle 10^{\circ} = a \angle(-80^{\circ}), \\
d_4 &= -b \angle 30^{\circ},
[[/math]]
where the unknowns [[$a, b$]] must be computed before the area can be found. As this path forms a loop, it must obey the equation
[[math]]
0 &= d_1 + d_2 + d_3 + d_4 \\
&= 32 \angle 80^{\circ} + 24 \angle 15^{\circ} - ia \angle 10^{\circ} - b \angle 30^{\circ}.
[[/math]]

[[include :scp-wiki:component:image-block-base name=Behemoth.png
|align=center
|width=90%
|alt=alt|alt-text=A quadrilateral on a coordinate plane is bordered by four arrows that meet head to tail: the first is thirty two units long at an angle of eighty degrees from the horizontal, the second is twenty four units away from the previous point at an angle of fifteen degrees from the horizontal, the third is at an angle of negative eighty degrees from the horizontal with a distance denoted by the variable a, and the fourth is the origin (zero). The third arrow also leads to a point that is at an angle of thirty degrees from the horizontal and has a distance from the origin that is denoted by the variable b. The quadrilateral is divided into two halves along the diagonal between the origin and the furthest corner; the top half has a shaded green area of A subscript one,  and the bottom half has an area of A subscript two.
|caption=Diagram M: The four movements/arrows [[$d_1, d_2, d_3, d_4$]] lead the commander to the positions [[$p_1, p_2, p_3, 0$]] in sequence. The farmland enclosed within these positions can be split into two triangles along the dashed line between the origin and [[$p_2$]].
]]

For simplicity, we multiply both sides of the equation by [[$\angle (-10^{\circ})$]], which is equivalent to rotating the quadrilateral by [[$10^{\circ}$]] clockwise such that its left edge lies parallel to the northern axis. This reduces our equation to
[[math]]
0 = 32 \angle 70^{\circ} + 24 \angle 5^{\circ} - ia - b \angle 20^{\circ},
[[/math]]
which can be rearranged as
[[math]]
ia = 32 \angle 70^{\circ} + 24 \angle 5^{\circ} - b \angle 20^{\circ}.
[[/math]]
Taking the real and imaginary parts separately then yields the system of equations
[[math]]
0 &= 32 \cos 70^{\circ} + 24 \cos 5^{\circ} - b \cos 20^{\circ}, \\
a &= 32 \sin 70^{\circ} + 24 \sin 5^{\circ} - b \sin 20^{\circ}.
[[/math]]
The first equation immediately reveals the value of [[$b$]]:
[[math]]
b &= \frac{32 \cos 70^{\circ} + 24 \cos 5^{\circ}}{\cos 20^{\circ}} \\
&= \frac{ 32(0.342) + 24(0.996)}{0.940} \\
&= 37.072.
[[/math]]
We can then directly compute [[$a$]] using the second equation:
[[math]]
a &= 32 \sin 70^{\circ} + 24 \sin 5^{\circ} - 37.072 \sin 20^{\circ} \\
&= 32 (0.940) + 24 (0.087) - 37.072 (0.342) \\
&= 19.489.
[[/math]]
With this, the quadrilateral is fully solved. To find its area, we can cut it into two triangles defined by the directions [[$d_1, d_2$]] and [[$d_3, d_4$]]. The first has an area of
[[math]]
A_1 &= \frac{1}{2} \begin{vmatrix} \Im\{ d_1^{*} d_2 \} \end{vmatrix} \\
&= \frac{1}{2} \begin{vmatrix} \Im\{ 32 \angle (-80^{\circ}) \cdot 24 \angle 15^{\circ} \} \end{vmatrix} \\
&= 384 \begin{vmatrix} \sin(-65^{\circ}) \end{vmatrix} \\
&= 384 \begin{vmatrix} -0.906 \end{vmatrix} \\
&= 347.904,
[[/math]]
while the second's area is
[[math]]
A_2 &= \frac{1}{2} \begin{vmatrix} \Im\{ d_3^{*} d_4 \} \end{vmatrix} \\
&= \frac{1}{2} \begin{vmatrix} \Im\{ (19.489 \angle 80^{\circ}) \cdot (-37.072 \angle 30^{\circ}) \} \end{vmatrix} \\
&= 361.248 \begin{vmatrix} - \sin 110^{\circ} \end{vmatrix} \\
&= 361.248 \begin{vmatrix} - 0.940 \end{vmatrix} \\
&= 339.573.
[[/math]]
Thus, the total area of the farm is [[$A_1 + A_2 = 687.477 \text{ ulv}^2$]]. Dividing by [[$40$]] then gives a total tilling time of [[$17.187$]] days.
__Discussion:__ Students who do not multiply both sides of the loop equation by [[$\angle (-10^{\circ})$]] will have more difficulty computing [[$a, b$]] due to [[$a$]]'s presence in both—they will have to divide one by the other in order to eliminate [[$a$]], which overcomplicates the process of finding [[$b$]]. Most students should be practiced enough in the numerous simplification-based rotations used throughout this chapter's problem sets to identify it immediately—those who do not ought to redouble their studies and prayers until their performance is acceptable.

[[/div]]

**Mathematical Analysis:** To keep the students engaged and inspired (and propagandized), a legendary war hero (see below) was written into an otherwise unrelated mathematics problem for Daevite students; as an analogy, imagine an American mathematics textbook including a problem where George Washington crosses the Delaware river in order to keep students interested and incite nationalist sentiments. Of course, for all we know, Commander Rakshal //did// personally survey conquered farmland and perform trigonometric calculations throughout her conquests, which would carry with it its own set of interesting implications; given Question 4 in the Architecture section earlier (estimating the height of an enemy watchtower), this possibility is surprisingly plausible.

**Historical Analysis:** Owing to difficulties induced by the harsh climate, indigenous Siberians only joined the Neolithic Revolution in the second half of the 3rd millenium BCE, and only caught up to the rest of the world's agricultural output during the Bronze Age. For instance, indigenous Siberians in antiquity made use of mattocks (digging sticks) to loosen/turn soil before sowing seeds, and only made use of ploughs starting in the 8th century CE.

In place of ploughs, the Daevites seemingly used [[[SCP-3140]]] instances (animated arboreal entities) to gently till soil using their roots (whether through mechanical perturbation or by drawing nutrients to the surface using the roots' nutrient uptake mechanisms). With the numbers given, an SCP-3140 instance could till soil at a rate about half that of a yoke (pair) of oxen. Unlike oxen, however, SCP-3140 instances are fully autonomous and controllable by verbal commands, allowing human labor to be concentrated elsewhere—farm owners could focus their efforts on other farmwork, academic studies, or simple leisure, and slave populations could be allocated purely to construction and military projects as discussed previously. Furthermore, with SCP-3140 serving as both an agricultural tool and an instrument of war, the Daevite military's tactic of targeting enemy farming communities is given new meaning—in addition to starving their enemies, they would claim those farms for themselves and use thaumaturgical tools like SCP-3140 to manage them and reap their spoils.

Of course, agricultural logistics of both the domestic and militaristic nature (not that the Daevites necessarily distinguished between the two) require a strong mathematical foundation. The rate at which an SCP-3140 instance could till land, the size and resulting output of a parcel of farmland, the quantities of food needed to maintain civilian populations and soldier supply lines (and the amount that must be stolen to collapse the enemy's)—the Daevites wielded mathematical mastery of all of these logistical details, and effectively exercised it in the pursuit of territorial expansion and conquest.

Furthermore, the Matron of War referenced here is also known as "Moukar-gai"[[span class="fnnum"]].[[/span]][[span class="fncon"]]Translated literally as "Battle-Font".[[/span]]: the fourth Fontbearer of the Daevite pantheon, who presided over war and strife, craft and wisdom, and conquest and brutality [((bibcite pantheon))]. Similar to the Light of Daevon with Noru-gai and the Silence of Daevon with Voen-gai, the "Wrath of Daevon" was then the Matron's most prominent priestess and Matriarch "representative" on the Ruling Council of Mothers that helmed the Daevite Empire. More important to us, however, is how she maintained a romantic relationship with Commander Rakshal—a romance so important to Daevic history that it is discussed in Chapter 79 of [[[SCP-140]]]:

[[div class="darkbox"]]

Romance between Daevic women is generally considered to be richer and more fulfilling than relationships between other sexes, with the strongest threads of devotion and Blood between two or more women being personally weaved by the Red Goddess. Even so, these threads can serve as two-ended blades—great care must be taken to prevent them from becoming a weakness that the enemy can exploit.

The most famous story of Daevic romance is the tragedy of Commander Durvta Rakshal, who helmed the lengthy extermination of the Batruvians[[span class="fnnum"]].[[/span]][[span class="fncon"]]An ancient, anomalous civilization based in what is now the [[a class="external" target="_blank" href="https://en.wikipedia.org/wiki/Far_North_(Russia)"]]Extreme North[[/a]] of Russia. Besides their destruction at the Daevites' hands c. 5,000 BCE, information on them is sparse.[[/span]]. Her love for the Wrath of Daevon knew no bounds, and was often considered the deciding factor in the Fourth Legion's victories during that millenium. Yet this letter to the Gentle Wrath of Daevon given below—the last known communication from Commander Rakshal to ever be recorded—hints at the nature of her undoing:

[[div class="blockquote" style="font-style:italic"]]

My heart,

By the time you receive this missive, I will have carved through thousands of Batruvians in the name of our great Empire. Yet even as the saccharine Blood of the fallen seeps through my skin and worms its way into my bones, even as I lean into the Matron's cold, guiding grip on my shoulder as she marvels at my handiwork, I am rendered hollow and unbecoming by your crushing absence.

These calloused hands of mine had not known gentleness before you. Even as my Blood runs hot from battle after battle, I think only of lightly tracing the slight divots of your hips with my sword hand, of cupping your golden cheeks in my scar-riddled palm, of feeling the rise and fall of your chest as you drift into slumber within the comfort of my arms. Pools of shed Blood contort into visages of your perfectly structured face. The screams of the fallen blur into your soft, bubbly laughter. The smell of icy rot drifts into a mockery of your enthralling, crimson-soaked scent. There is no escape—no matter how far I stray from you, you shall always be my tranquility amidst the endless slaughter out here in the frozen wastes.

Sometimes, your beckoning call tempts me with fantasies of a peaceful life that rejects the waging of war in your name—yet clarity returns to my mind as quickly as it is clouded. The truth is a simple thing: to return to your side without the spoils of victory would be to deem myself the wretch I have always known myself to be. No, I mustn't falter now—the Great Weaver has revealed to me what I know in my heart to be the only path forward. In three days from the time of this writing, the final push into the Batruvians' stronghold will commence... and three weeks following that, I shall return to your palace with the heads of those who have wronged us in tow.

If I do not survive the battle, do not disgrace the annals of history with my remembrance.

Yours always,
Durvta

[[/div]]

Survivors reported that the Batruvians tricked Commander Rakshal into storming the front gates of their stronghold at a critical moment during the battle; blinded by her fervent love for the Wrath of Daevon, Rakshal did not recognize the obvious trap laid for her. Logic and reason had lapsed in favor of passion and blind devotion; by the time this had been identified, the Commander was already lost. Upon receiving this terrible news, the mournful Wrath of Daevon went against Rakshal's self-loathsome wishes and decreed that her memory shall be immortalized throughout Daevic history, religion, and arts—a decree that has been successfully sustained throughout the Empire's eternal reign.

[[/div]]

Perhaps, then, the inclusion of Commander Rakshal in this math problem within //Geometric Veneration// could be an execution of this immortalization decree. Daevite mathematics could not be divorced from Daevite culture and religion: the two were, by all accounts, one and the same.

+ Conclusions

//Geometric Veneration// is an exemplary demonstration of a novel hypothesis: that mathematical study and application formed the backbone of Daevite society. Math was both a foundational tool for architecture, agriculture, and other domestic endeavors and an essential expression of religious/numerological belief, and a non-insignificant portion of the Daevite Empire's success can be attributed to its usage. Through the study of this math and their fair-yet-competitive pedagogical approach to it, we have identified key elements of Daevite cultural standards, socioeconomic structures, and thaumaturgical practices—only some of which was previously known to the Foundation. While the vast majority of Daevic advancements (mathematical, technological, or otherwise) seemingly did not propagate to other civilizations or survive to the modern day, //Geometric Veneration// still provides us with a glimpse of how ancient academia was structured—and, in select cases, how elements of that structure could be incorporated into modern pedagogy both within the Foundation and beyond it.

We acknowledge that pure speculation is limited in predictive power, and sincerely hope that future studies can yield independent corroboration and verification of the possibilities presented by //Geometric Veneration//. In particular, we posit the following three questions for future researchers:

# What other nonanomalous mathematical and scientific achievements enabled the Daevites' accelerated development as a civilization?
# Were any such accelerated mathematical advancements also achieved by other anomalous civilizations, such as the Ortothan Kingdom or the [[[sarkicism-hub|Sarkic people]]]?
# Did the Daevites also possess knowledge of [[[Math Department Hub|anomalous mathematics]]]? If so, how did this paramathematics affect their societal development?

@@ @@

@@ @@

In Foundation documentation and research, cite this paper as:

[[div class="blockquote" style="text-indent:1.4em hanging"]]

P. Buyantu ([[[math-department-hub|Site-Σ]]]), T. Zhang ([[[secure-facility-dossier-site-12|Site-12]]]). 2023: "[[[daevite-mathematics-complex-numbers|Daevite Mathematics: Complex Numbers]]]", //Foundation Journal of Anthropological Research//, vol. 298., pp. 24-68. [[[Expanded Security Clearance Codes|Clearance]]] G3-2/DOTT/HIST/MATH. {{doi:scp.hist/500219}}

[[/div]]

[[collapsible show="Show Bibliography" hide="Hide Bibliography"]]
[[bibliography]]
: archaeology : A. Yezhov, //et al.// ([http://scp-int.wikidot.com/secure-facilities-locations-cn#toc20 Site-CN-91]). 2022: "Ancient Education: Recovering a Daevic Math Textbook", //Journal of Cryptoarchaeological Sciences//, vol. 285, pp. 166-212. Clearance G2-2/ANY. {{doi:scp.hist/981947}}
: linguistics : T. Zhang ([[[secure-facility-dossier-site-12|Site-12]]]). //et al//. 2023: "Old Daevite Linguistics: Qualitative and Quantitative Insights", //Foundation Journal of Anthropological Research//, vol. 291, pp. 43-73. Clearance G3-2/HIST/LING. {{doi:scp.ling/387419}}
: pantheon : E. Marrak ([[[SCP-6716|Site-89]]]). 2019: "The Daevic Pantheon: Philosophies and Ancestral Worship", //[[[a-once-and-future-king|Dokein]]]: A Journal for [[[Tactical Theology Hub|Tactical Theology]]]//, vol. 86, pp. 24-61. Clearance G4-3/HIST-2/DOTT-2. {{doi:scp.dott/283954}}
: sabotage : A. Yezhov, //et al.// (Site-CN-91). 2014: "Daevic Diaries: did the children of Daevon dream of crimson sheep? A novel recovery", //Journal of Cryptoarchaeological Sciences//, vol. 186, pp. 22-46. Clearance G2-1/HIST. {{doi:scp.hist/570301}}
: government : H. Bakkal (Site-12). 2017: "The Seven Arms of Daeva Hegemony", //Foundation Journal of Anthropological Research//, vol. 225, pp. 93-121. Clearance G3-2/HIST. {{doi:scp.hist/118321}}
[[/bibliography]]
[[/collapsible]]

-----

[[=]]
[[div class="bottom-rating-module"]]
[[module Rate]]
[[/div]]

[[=]]
[[div class="footer-wikiwalk-nav"]]
<< [[[The Smell of Blood]]] | Daevite Mathematics: Complex Numbers | [[[When Daevon Fell]]] >>
[[/div]]
[[/=]]

[[include :scp-wiki:component:license-box | author=badooga]]
=====
> **Filename:** Math Department Logo - Penumbra.png
> **Author:** [[*user BlueEyedFox_]]
> **License:** CC BY-SA 3.0
> **Source Link:** [https://scp-wiki.wdfiles.com/local--files/math-department-hub/Math%20Department%20Logo%20-%20Penumbra.png SCP Wiki]
=====
> **Name:** Agroforestry
> **Author:** Jim Robinson, USDA-NRCS
> **License:** CC BY 2.0
> **Source Link:** [https://www.flickr.com/photos/139938511@N02/26219448802/in/album-72157666621959242 Flickr]
=====
> **Name:** Desmos Graphs
> **Author:** [[*user badooga]]
> **License:** [https://www.desmos.com/terms#use CC BY-SA 4.0]
> **Source Link:** [https://www.desmos.com/ Desmos Graphing Calculator]
=====
[[include :scp-wiki:component:license-box-end]]
[[/=]]

Daevite Mathematics: Complex Numbers

Module Styles

Inline Styles

Includes

CSS Classes

Page Source