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.

Back to the First Dimension . . . Home Page . . . On to the Third Dimension

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.

**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 o^{6}x (hexagon symmetry) or xx (trigon symmetry).

**Heptagon** - Short name is heg (previously ha). It has 7 vertices and 7 edges. Symbol is o^{7}x.

**Octagon** - Short name is oc. It has 8 vertices and 8 edges. Symbol can be either o^{8}x 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 o^{9}x.

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

**Hendecagon** - Short name is heng. It has 11 vertices and 11 edges. Symbol is o^{11}x.

**Dodecagon** - Short name is dog. It has 12 vertices and 12 edges. The symbol is either o^{12}x or x^{6}x.

**Tridecagon** - Short name is tad. It has 13 vertices and 13 edges. Symbol is o^{13}x.

**Tetradecagon** - Short name is ted. It has 14 vertices and 14 edges. Symbol can be either o^{14}x or x^{7}x depending on which symmetry we consider it under.

**Pentadecagon** - Short name is ped. It has 15 vertices and 15 edges. Symbol is o^{15}x.

**Hexadecagon** - Short name is hed. It has 16 vertices and 16 edges. Symbol can be o^{16}x or x^{8}x depending on symmetry.

**n-gon** - It has n vertices and n edges. Symbol is o^{n}x.

**2n-gon** - It has 2n vertices and 2n edges. Symbol can be o^{2n}x or x^{n}x 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 o^{7/2}x. It is the 7/2-gon.

**Great Heptagram** - Short name is gahg. It has 7 vertices and 7 edges. Symbol is o^{7/3}x. It is the 7/3-gon.

**Octagram** - Short name is og. It has 8 vertices and 8 edges. Symbol can be either o^{8/3}x 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 o^{9/2}x. It is the 9/2-gon.

**Great Enneagram** - Short name is geng. It has 9 vertices and 9 edges. Symbol is o^{9/4}x. It is the 9/4-gon.

**Decagram** - Short name is dag. It has 10 vertices and 10 edges. Symbol can be o^{10/3}x 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 o^{11/2}x.

**Hendecagram** - Short name is henge. It has 11 vertices and 11 edges. Symbol is o^{11/3}x.

**Great Hendecagram** - Short name is ghen. It has 11 vertices and 11 edges. Symbol is o^{11/4}x.

**Grand Hendecagram** - Short name is gahn. It has 11 vertices and 11 edges. Symbol is o^{11/5}x.

**Dodecagram** - Short name is dodag. It has 12 vertices and 12 edges. Symbol can be o^{12/5}x or x^{6/5}x depending on symmetry.

**Small Tridecagram** - Short name is sat. It has 13 vertices and 13 edges. Symbol is o^{13/2}x.

**Tridecagram** - Short name is trad. It has 13 vertices and 13 edges. The symbol is o^{13/3}x.

**Medial Tridecagram** - Short name is mat. It has 13 vertices and 13 edges. Symbol is o^{13/4}x.

**Great Tridecagram** - Short name is get. It has 13 vertices and 13 edges. Symbol is o^{13/5}x.

**Grand Tridecagram** - Short name is gat. It has 13 vertices and 13 edges. Symbol is o^{13/6}x.

**Tetradecagram** - Short name is tedag. It has 14 vertices and 14 edges. Symbol can be o^{14/3}x or x^{7/3}x depending on symmetry.

**Great Tetradecagram** - Short name is getag. It has 14 vertices and 14 edges. Symbol can be o^{14/5}x or x^{7/5}x depending on symmetry.

**Small Pentadecagram** - Short name is sped. It has 15 vertices and 15 edges. The symbol is o^{15/2}x.

**Pentadecagram** - Short name is pad. It has 15 vertices and 15 edges. Symbol is o^{15/4}x.

**Great Pentadecagram** - Short name is gepad. It has 15 vertices and 15 edges. Symbol is o^{15/7}x.

**Small Hexadecagram** - Short name is shed. It has 16 vertices and 16 edges. Symbol can be o^{16/3}x or x^{8/3}x depending on symmetry.

**Hexadecagram** - Short name is had. It has 16 vertices and 16 edges. Symbol can be o^{16/5}x or x^{8/5}x depending on symmetry.

**Great Hexadecagram** - Short name is gahd. It has 16 vertices and 16 edges. Symbol can be o^{16/7}x or x^{8/7}x 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 o^{n/d}x.

**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 o^{2n/d}x or x^{n/d}x depending on symmetry.

Below are the same star polygons filled binary style.

**Time to be MESMERIZED!**

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.

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 S_{1} sides where S_{1} represents the circle, contact regions are points.

*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).

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