Verse and Dimensions Wikia
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Verse and Dimensions Wikia

A triangle is the 2 simplex. The study of triangles is called trigonometry (plus the study of angles). Its Bowers acronym is "trig". It is also known as a pyrogon under the elemental naming scheme.

Types of Triangle[]

By Sides[]

Equilateral[]

When all of the angles and edges of a triangle are equal a triangle is equilateral. Each angle must have a measure of exactly 60 degrees.

Isosceles[]

When a triangle has two equal sides, it is isosceles. It must also have two equal angles.

Scalene[]

A triangle with three unique side lengths is known as a scalene triangle.

By Angles[]

Acute[]

A triangle with all three angles smaller than 90 degrees is called an acute triangle. All equilateral triangles are also acute triangles.

Right-Angled[]

A triangle with one right angle is a right-angled triangle. Their side lengths follow the equation

, where c is the edge length of the side opposite the right angle.

Obtuse[]

A triangle where at least one angle is greater than 90 degrees is called an obtuse triangle.

Special Cases[]

A triangle with more than one right angle is usually degenerate but can appear on the surface of a sphere. The same applies to a triangle with angles that sum to greater than 180 degrees, and to triangles that have an angle equal to 180 degrees.

Symbols[]

The triangle has three Dynkin type symbols:

  • x3o (regular)
  • ox&#x (isosceles)
  • ooo&#x (scalene)

Structure and Sections[]

Sections[]

As seen from a vertex, a triangle starts as a point that expands into a line segment.

Hypervolumes[]

  • surface area =

Formulas for a general triangle[]

If we let a, b, c be the sides of the triangle, and h be the higher perpendicular to side a, then we can obtain these formulae:

  • Edge length =
  • Surface area =

Subfacets[]

Radii[]

  • Vertex radius:
  • Edge radius:

Angles[]

  • Vertex angle: 60º

Vertex coordinates[]

The simplest way to obtain vertex coordinates for an equilateral triangle is as a face of a 3D octahedron, thus yielding the coordinates as

  • (1,0,0)
  • (0,1,0)
  • (0,0,1)

thus giving a triangle of size

.

To obtain a triangle in the 2D plane with side 2, the coordinates are

  • (±1,-√3/3)
  • (0,2√3/3)

Notations[]

  • Tapertopic notation:

Related shapes[]

See Also[]

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

Null polytope


Point


Line segment


Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Dodecadakon

Tridecahendon

Tetradecadokon

Pentadecatradakon

Hexadecatedakon

Heptadecapedakon

... Omegasimplex
Cross

Square

Octahedron

Hexadecachoron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

Tetradekacross

Pentadekacross

Hexadekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

Tetradekeract

Pentadekeract

Hexadekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

Tetradekorb

Pentadekorb

Hexadekorb

... Omegaball

...
Zerogon Monogon Digon Triangle Square Pentagon Pentagram Hexagon Heptagon Heptagram Great heptagram Octagon Octagram Enneagon Enneagram Great enneagram Decagon Decagram Hendecagon Small hendecagram Hendecagram Great hendecagram Grand hendecagram Dodecagon Dodecagram Tridecagon Small tridecagram Tridecagram Medial tridecagram Great tridecagram Grand tridecagram Tetradecagon Tetradecagram Great tetradecagram Pentadecagon Small pentadecagram Pentadecagram Great pentadecagram Hexadecagon Small hexadecagram Hexadecagram Great hexadecagram Heptadecagon Tiny heptadecagram Small heptadecagram Heptadecagram Medial heptadecagram Great heptadecagram Giant heptadecagram Grand heptadecagram ... Apeirogon Failed star polygon (-gon) Pseudogon (-gon)
Regular
Rectified
Truncated
Triangle Triangle Hexagon
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