Tuesday, June 7, 2016

Polygons in the order Trianglia:An Introduction

This is actually about math, more precisely triangles, but it's based off of biology papers. Enjoy the biomath!


Abstract

Polygons that possess three separate sides are known as triangles, and are hereby classified into the order Trianglia to avoid confusion. Within the family Trianglia, separate orders, genuses, and species exist, all similar but with differences distinguishing them from other triangles. In this paper, we will discuss these differences.


Classification of Triangles

Domain: Mathematica (Figures and concepts used in Mathematics)
Kingdom: Geometrika (Concepts and figures used in Geometry)
Phylum: Duodimensiva (Two-dimensional figures)
Class: Polygona (Polygons)
Order: Trianglia (Triangles)

Family Nonaginta

Triangles in the family Nonaginta, commonly known as right triangles, are triangles in which a right angle, or an angle of 90 degrees, is found. This is the reason for the classification name Nonaginta, from the Latin word for the number 90. Genuses within the family include Analogia, or right triangles possessing particular proportions. These include the 3-4-5 right triangle (Analogia duodecima), the 30-60-90 right triangle (Analogia triginta), the 5-12-13 right triangle (Analogia tredecima), and the 1-1-√2 right triangle (Analogia irrationalis). The other genus in Family Nonaginta is Genus Nonaginta, or other right triangles. Nonaginta nonaginta is your typical right triangle. It possesses one right angle. There is debate within the field of morphogenetics about whether Nonaginta nonaginta is the common ancestor to all right triangles and that potentially, the family Analogia should not exist with all other right triangles as subspecies of Nonaginta nonaginta. The other species that is currently classified within Genus Nonaginta is Nonaginta aequipa or the isoceles right triangle, in which the bottom two angles are congruent as are their corresponding sides. There is much controversy among triangle hobbyists about whether or not isoceles right triangles, found in their perfectly circular deli cups at geometric pet expos, should be labeled as such if they are human-made integrades between other right triangles, often captive-bred Nonaginta nonaginta and other isoceles triangles, which we will discuss later. 

A captive-bred isoceles right triangle inside a perfectly circular deli cup.
Family Aequipa
The triangles found in Family Aequipa (from aequipes, the Latin word for isoceles), are commonly known as isoceles triangles. In order for a triangle to be classified in Family Aequipa, it must have two congruent sides and angles. Isoceles triangles are very popular amongst triangle hobbyists due to the fact that various integrades and sizes, including isoceles right triangles exist, their eye-catching congruent sides and angles, and the fact that they can very easily be inscribed in circular deli cups. Amongst morphogeneticists, the theory that equilateral triangles evolved from isoceles triangles, which in turn evolved from scalene triangles, often considered the most ancestral form of a triangle is a highly discussed and very controversial issue. If you're a morphogenetics grad student, anything about this theory will probably defend your dissertation fairly well. According to this theory, pyramids evolved from the two-dimensional forms of triangles, with equilateral triangle pyramids being the most recent development.


Scalene and Equilateral Triangles

Scalene and equilateral triangles are classified in the family Trianglia. Technically, all equilateral triangles are isoceles and could hypothetically be classified in Aequipa, but in practice, according to the current morphogenetics theory, equilateral triangles, despite evolving from isoceles triangles, have split off genetically enough that they should be classified in a separate family. Equilateral triangles are triangles in which all 3 angles and sides are congruent. They are prized by triangle hobbyists, and purebred equilaterals often win first place at triangle shows, in many cases being classified as isoceles and being considered "Best of Breed" despite not being truly isoceles. Scalene triangles are considered the most ancestral form of triangle, from which all other triangles split off during geometric evolution. In order to be a scalene triangle, a figure has to have internal angles that add to 180 degrees and have 3 sides that connect. They are also the only kind of triangle commonly found in nature, as unlike other triangles, they do not require precise proportions, which are not commonly found in nature. 

A wild scalene triangle, found in nature.

Blogger is not cooperating with me, so for some reason, it would not let me put my picture of an equilateral triangle anywhere but the top of the post.



Anatomy

In all triangles, the interior angles add to 180 degrees. 

A formal proof exists, but a simplified version, demonstrated on a triangle dead of natural causes, is that, if you detach the other two sides from one and arrange them to connect, perfectly straight, with the other side, a straight line or 180 degree angle is formed. However, some more specific anatomy exists for right triangles.

Right triangles are the only triangles to possess legs, which adds to the theory that they are mobile or potentially, the first triangles to develop legs and crawl onto land. Although, for the kinds of triangles found in nature, they have somehow found their way onto land. Perhaps they have legs but we don't realize it? Anyway, the Pythagorean Theorem relies on the right triangle's legs. You can use it to find any side of a right triangle, given the other two. But, if you only know the length of one side, you can use the field known as trigonometry to study right triangles in more depth. For the purpose of trigonometry, right triangles possess specialized anatomy and this anatomy is used for functions.

Trigonometry and its associated functions can be used to find almost anything about a right triangle, even when you only know the length of one of its sides. The other advantage to trigonometry and the Pythagorean Theorem is that you can measure a right triangle and study it in depth without hurting or dissecting the triangle, which would enrage triangle rights groups like PETT or People for the Ethical Treatment of Triangles. Because, hey, there are advocacy groups for everything else, so why not triangles?