# Regular Polygons and Other Two Dimensional Shapes

This page will deal with regular polygons, regular compound polygons, 2-D dice, semi-uniform polygons, and any other interesting 2-D shapes. Last updated February 29, 2020.

2020 February 29 - Added some new short names and sections.

2014 April 22 - Updated polygon short names, they sound better now.

2012 June 10 - Released on the web.

2012 April 6 - Polygon page created.

## Basic Two Dimensional Shapes

These two shapes can be considered as the basic 2-D shapes, they are either flat in both dimensions or they join them in a uniform curve. The two shapes are:

Square - which can be generalized as a variety of rectangles when the dimensions are of different length. The square is the square of the dyad (line segment) and the rectangle is a dyad times a dyad (of variant lengths). It could be represented by ||.

Circle - no flat dimensions, both dimensions are curved together. It can roll back and forth on a linear path. It could be represented as (). Sometimes the word "circle" refers to the hollow version where the solid version is called a disk.

## Regular Polygons

A polygon is regular when all of its vertices are congruent and all of its edges are the same length. A uniform polygon is considered to be a regular polygon. There are an infinity of them, they can be either convex or star polygons. All polygons are orientable and all regulars are tame. Vertex figures are simply dyads of variant lengths which can be calculated from the formula 2cos(180*d/n) for an n/d-gon.

Convex Cases

Pictures are above and a listing is below.

Triangle - Also called trigon. Short names include trig (previously tri) and equit (for equalateral triangle). It has 3 vertices and 3 edges. Non regular triangles include isot (isosceles) and scalene. Symbol is ox.

Square - Short name is also square. Also called tetragon or tetrangle. It has 4 vertices and 4 edges. The symbol can be o'x or x x.

Pentagon - Short name is peg (previously pe). It has 5 vertices and 5 edges. Symbol is o^x.

Hexagon - Commonly called hex (not to be confused with the hexadecachoron's short name). My short name for it is hig (previously he). It has 6 vertices and 6 edges. The symbol is either o6x (hexagon symmetry) or xx (trigon symmetry).

Heptagon - Short name is heg (previously ha). It has 7 vertices and 7 edges. Symbol is o7x.

Octagon - Short name is oc. It has 8 vertices and 8 edges. Symbol can be either o8x or x'x depending on which symmetry we consider it under.

Enneagon - Short name is en. It has 9 vertices and 9 edges. Symbol is o9x.

Decagon - Short name is dec (previously de). It has 10 vertices and 10 edges. Symbol can be o10x or x^x depending on symmetry.

Hendecagon - Short name is heng. It has 11 vertices and 11 edges. Symbol is o11x.

Dodecagon - Short name is dog. It has 12 vertices and 12 edges. The symbol is either o12x or x6x.

Tridecagon - Short name is tad. It has 13 vertices and 13 edges. Symbol is o13x.

Tetradecagon - Short name is ted. It has 14 vertices and 14 edges. Symbol can be either o14x or x7x depending on which symmetry we consider it under.

Pentadecagon - Short name is ped. It has 15 vertices and 15 edges. Symbol is o15x.

Hexadecagon - Short name is hed. It has 16 vertices and 16 edges. Symbol can be o16x or x8x depending on symmetry.

n-gon - It has n vertices and n edges. Symbol is onx.

2n-gon - It has 2n vertices and 2n edges. Symbol can be o2nx or xnx depending on symmetry.

Star Cases

Pictures are above and a listing is below.

Pentagram - Commonly called star, and thus will be my short name for it as well. It has 5 vertices and 5 edges. Symbol is o*x. It is the 5/2-gon.

Heptagram - Short name is hag. It has 7 vertices and 7 edges. The symbol is o7/2x. It is the 7/2-gon.

Great Heptagram - Short name is gahg. It has 7 vertices and 7 edges. Symbol is o7/3x. It is the 7/3-gon.

Octagram - Short name is og. It has 8 vertices and 8 edges. Symbol can be either o8/3x or x"x depending on which symmetry we consider it under. It is the 8/3-gon and can be considered as the quasitruncated square.

Enneagram - Short name is eng. It has 9 vertices and 9 edges. Symbol is o9/2x. It is the 9/2-gon.

Great Enneagram - Short name is geng. It has 9 vertices and 9 edges. Symbol is o9/4x. It is the 9/4-gon.

Decagram - Short name is dag. It has 10 vertices and 10 edges. Symbol can be o10/3x or x*'x. It is the 10/3-gon and can be considered as the quasitruncated star.

Small Hendecagram - Short name is shen. It has 11 vertices and 11 edges. The symbol is o11/2x.

Hendecagram - Short name is henge. It has 11 vertices and 11 edges. Symbol is o11/3x.

Great Hendecagram - Short name is ghen. It has 11 vertices and 11 edges. Symbol is o11/4x.

Grand Hendecagram - Short name is gahn. It has 11 vertices and 11 edges. Symbol is o11/5x.

Dodecagram - Short name is dodag. It has 12 vertices and 12 edges. Symbol can be o12/5x or x6/5x depending on symmetry.

Small Tridecagram - Short name is sat. It has 13 vertices and 13 edges. Symbol is o13/2x.

Tridecagram - Short name is trad. It has 13 vertices and 13 edges. The symbol is o13/3x.

Medial Tridecagram - Short name is mat. It has 13 vertices and 13 edges. Symbol is o13/4x.

Great Tridecagram - Short name is get. It has 13 vertices and 13 edges. Symbol is o13/5x.

Grand Tridecagram - Short name is gat. It has 13 vertices and 13 edges. Symbol is o13/6x.

Tetradecagram - Short name is tedag. It has 14 vertices and 14 edges. Symbol can be o14/3x or x7/3x depending on symmetry.

Great Tetradecagram - Short name is getag. It has 14 vertices and 14 edges. Symbol can be o14/5x or x7/5x depending on symmetry.

Small Pentadecagram - Short name is sped. It has 15 vertices and 15 edges. The symbol is o15/2x.

Pentadecagram - Short name is pad. It has 15 vertices and 15 edges. Symbol is o15/4x.

Great Pentadecagram - Short name is gepad. It has 15 vertices and 15 edges. Symbol is o15/7x.

Small Hexadecagram - Short name is shed. It has 16 vertices and 16 edges. Symbol can be o16/3x or x8/3x depending on symmetry.

Hexadecagram - Short name is had. It has 16 vertices and 16 edges. Symbol can be o16/5x or x8/5x depending on symmetry.

Great Hexadecagram - Short name is gahd. It has 16 vertices and 16 edges. Symbol can be o16/7x or x8/7x depending on symmetry.

Density d n-Gram - Also called n/d-gon. n and d are relatively prime. It has n vertices and n edges. Symbol is on/dx.

Density d 2n-Gram - Also called 2n/d-gon. 2n and d are relatively prime. It has 2n vertices and 2n edges. Symbol can be o2n/dx or xn/dx depending on symmetry.

Below are the same star polygons filled binary style.

Time to be MESMERIZED!

## Regular Compound Polygons

A compound polygon is regular when all of its component polygons are regular and all of its vertices are congruent. Some of them have a continuum of morphs (the ones that are a compound of an even number of polygons). Below are the regular compound polygons.

Hexagram - short name is Shig for stellated hexagon - also called Star of David. It is the 6/2-gon and is a compound of two triangles.

Stellated Octagon - short name is Soc, it is the 8/2 and is a compound of two squares.

Fissal Enneagram - short name is Fen, it is the 9/3 and is a compound of three triangles.

Stellated Decagon - short name is Sadeg, it is the 10/2 and is a compound of two pentagons.

Stellated Decagram - short name is Sadag, it is the 10/4 and is a compound of two stars.

Stellated Dodecagon - short name is Sedog, it is the 12/2 and is a compound of two hexagons.

Trisquare - short name is the same, it is the 12/3 and is a compound of three squares.

Tetratriangle - short name is Tetri, it is the 12/4 and is a compound of four triangles.

Stellated Tetradecagon - short name is Sted, it is the 14/2 and is a compound of two heptagons.

Great Stellated Tetradecagon - short name is Gosted, it is the 14/4 and is a compound of two heptagrams.

Spinostellated Tetradecagon - short name is Nisted, it is the 14/6 and is a compound of two great heptagrams.

Tripentagon - short name is Tripen, it is the 15/3 and is a compound of three pentagons.

Pentatriangle - short name is Pentri, it is the 15/5 and is a compound of five triangles.

Tripentagram - short name is Tristar, it is the 15/6 and is a compound of three stars.

Stellated Hexadecagon - short name is Shed, it is the 16/2 and is a compound of two octagons.

Tetrasquare - short name is Tesq, it is the 16/4 and is a compound of four squares.

Dioctagram - short name is Diog, it is the 16/6 and is a compound of two octagrams.

M/N-gon - where M and N are not relatively prime and their greatest common divisor is D, they are compounds of D (M/D)/(N/D)-gons.

## Two Dimensional Dice

In a geometric sense, dice are normally defined as convex polytopes with congruent sides. We could allow curved objects into the mix by changing the definition so that a die is any convex shape with congruent "contact regions". Contact regions can be thought of as the part that can contact a surface when the object is setting on one. The contact region of a square is a dyad (one of its edges), vertices don't count as contact regions. The contact region of a circle is a point. Two dimensional dice are as follows:

Triangle - has 3 sides.

Square - has 4 sides, can be morphed into a rhombus and still be a die.

Pentagon - has 5 sides.

Hexagon - has 6 sides, can be morphed into a triambus and still be a die.

Heptagon - has 7 sides.

Octagon - has 8 sides, can be morphed into a tetrambus and still be a die.

Enneagon - has 9 sides.

Decagon - has 10 sides, can be morphed into a pentambus and still be a die.

Hendecagon - has 11 sides.

Dodecagon - has 12 sides, can be morphed into a hexambus and still be a die.

Tridecagon - has 13 sides.

Tetradecagon - has 14 sides, can be morphed into a heptambus and still be a die.

Hexadecagon - has 16 sides, can be morphed into a octambus and still be a die.

Regular Polygons - has any number of sides, even numbered dice can be morphed. Contact regions of all polygons are edges.

Circle - has one curved continuous side which can be refered to as S1 sides where S1 represents the circle, contact regions are points.

## Semi-Uniform Polygons

A semi-uniform polygon is one with congruent vertices, but could have one or two edge lengths. All regular polygons are included, here are the other polygons to include:

Rectangle - Short name is rect. It has four sides of two edge lengths. A variant of a square. Symbol x y.

Bowtie - Short name is also bowtie. It has two parallel sides and two criss-crossing sides.

Ditrigon - Short name is dit. A variant of a hexagon with triangular symmetry. Symbol xy.

Tripod - Short name is also tripod. A six sided star figure with trigon symmetry. There are two versions, the propeller version has a dangling appearance. Symbol x,y.

Ditetragon - Short name is diteg. Variant of the octagon. Symbol x'y.

Tetrapod - The inflected square, with the short name 'tepod', looks like a square with triangles dangling off the corners. Symbol x"y - where x/y > sq2 (if y is the smaller value).

Ditetragram - Ditag for short. Variant of the octagram, there are two version (the ditetragram and the octagram). Symbol x"y - where x/y < sq2 (y being the smaller value).

Dipentagon - Variant of the decagon, dipeg for short. Symbol x^y.

Dipentagram - Distar for short, it looks like a truncated star. Symbol x*y.

Stellapod - It has two variants, one looks like a star with triangles dangling off the points (propeller version with short name prost), the other looks like five star points fused together with their points inward, short name starpod. Symbol x*'y - where x/y > tau (when y is smaller).

Distellagram - Variant of decagram, there are two variants, the distellagram (distag for short) and the decagram. Symbol x*'y - where x/y < tau (y smaller).

Pentapod - Looks like pentagon with triangles dangling off the corners. Short name is pepod. Can be called inflected pentagon. Symbol x^'y - where x/y > tau (y smaller).

Distellagon - There are three versions, one looks like a small pentagon with large triangles dangling off the corners but fusing together (the open version - odist for short). The second looks like a 2-D version of quit sissid - dist for short. The third version (shut) looks like a doubled up star - shudst for short. Symbol x^'y - where x/y < tau (y smaller).

## Types of Triangles and Tetragons

Types are determined by shape type and symmetry. The triangle has three possible symmetries with only one shape type (convex). The symmetries are trigonic, mirrored, and none. Tetragons have three shape types - convex, concave, and crossed. They also have the following symmetry types: square, rectangular, mirrored, "S" symmetry (symmetry of the letter S), and none. Below are the types of trigons and tetragons with short names.

Equit - the equalateral triangle with trigon symmetry. Edges and angles have one type each: AaAaAa (edges are capital, angles lower case)

Isot - the isoscelese triangle has mirror symmetry, can be tall or squashed. Edges and angles come in two types: AbBaBb.

Scalene - the scalene triangle has no symmetry. Edges and angles come in three types each: AcBaCb.

Square - it has square symmetry and has the convex type. Edges and angles come in one type each: AaAaAaAa.

Rect - the rectangle has rectangle symmetry with the convex type. Edges come in two types, angles in one type: AcBcAcBc.

Rhomb - the rhombus has rectangle symmetry with the convex type. Edges come in one type, angles in two: CaCbCaCb.

Bowtie - the bowtie has rectangle symmetry with crossed type. It can be wide or long. Edges come in two types, inner and outer. The angles have one type but with different orientations - IpOpIqOq.

Traz - the trapezoid has mirror symmetry and convex type. Edges come in three types, angles in two. AbCaBaCb.

Crotraz - the crossed trapezoid has mirror symmetry and crossed type. Edges come in three types where the crossing edges are represented as C. The angles has two type and are orientated differently. AbCaBaCb.

Butter - the butterfly has mirror symmetry with crossed type. Edges and angles come in two types, both angle types come in one orientated one way and the other the other way. ObIpOdIq - where b and d are one angle type and p and q are the other.

Kite - it has mirror symmetry with convex type. Edges come in two types and angles in three: BcAbAcBa.

Dart - it has mirror symmetry with concave type. Edges come in two types and the angles in three. The angle b is the inner one. BcAbAcBa.

Par - the parallelogram has S symmetry and the convex type. Edges and angles come in two types. AcBdAcBd.

Cropar - the crossed parallelogram has S symmetry and the crossed type. Edges come in three types, two types cross each other while the outer edges (C) come in parallel. The angles come in two types, each with two orientations - CbXdCpYq.

Irteg - the irregular tetragon has no symmetry and has convex type, it has four edge types and four angle types.

Irdart - the irregular dart has no symmetry and has concave type. It also has four angle and edge types.

Irtie - the irregular bowtie has no symmetry and has crossed type. It has four angle and edge types.

## The Other Types of Polygons

In one dimension, there is the dyad and it's antidyad counter part (ad). Here there are four types: Normal, anti, antidyadic, and anti-antidyadic. Below are the 'other types' of squares.

Asq - the antisquare, it has four dyad edges, but the inside is actually the outside and vise versa. Think of a plane with a square cut out of it. Asq and ye shall receive

Adesq - the antidyadic square, it has four ads for edges. It looks like a plus with an infinite sized arms.

Adasq - the antidyadic antisquare, it also has four ads for edges. It looks like the interior and exterior of adesq swaped. It is in four pieces. Think of two infinite hallways that intersect between four infinite solid buildings in front and behind, both left and right. Adasq is its floorplan.

```Page created by Jonathan Bowers, 2012
e-mail = hedrondude at suddenlink dot net ```