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 June 10, 2012.

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

2012 June 10 - Released on the web.

2012 April 6 - Polygon page created.


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Basic Two Dimensional Shapes

Basic 2-D 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

Regular Convex Polygons

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

Regular Star Polygons

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.

Binary Filling of Star Polygons

Time to be MESMERIZED!

Binary Filling of 75 / 32 Gon


Regular Compound Polygons

Regular Compound Polygons

A compound polyhedron is uniform when all of its component polyhedra are uniform and all of its vertices are congruent. Some of them have a continuum of morphs. Below are the uniform compound polyhedra plus the two infinite groups divided up into categories.


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.

Pentadecagon - has 15 sides.

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

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 - has four sides of two edge lengths. A variant of a square. Symbol x y.

Bowtie - has two parallel sides and two criss-crossing sides.

Ditrigon - A variant of a hexagon with triangular symmetry. Symbol xy.

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

Ditetragon - Variant of the octagon. Symbol x'y.

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

Ditetragram - 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. Symbol x^y.

Dipentagram - Looks like a truncated star. Symbol x*y.

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

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

Pentapod - Looks like pentagon with triangles dangling off the corners. 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). The second looks like a 2-D version of quit sissid. The third version (shut) looks like a doubled up star. Symbol x^'y - where x/y < tau (y smaller).


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