# Polygons and Polyhedrons

## Definitions

A polygon is a many-sided 2-dimensional closed shape. All the sides (= edges) must be straight lines. All the angles between edges must be greater than zero (and shouldn't be 180 degrees). A vertex is another name for a corner where the edges of polygons meet. There must be at least 3 polygons at each vertex to make a grid or a solid. If only 2 polygons are involved then they are sharing a bent edge rather than meeting at a true vertex.

A polyhedron is a many-sided 3-dimensional closed shape. All the sides (= faces) must be polygons. Note that the plural term may be polyhedrons or polyhedra.

The lower limit on the number of edges for a polygon is 3. It is impossible to construct a shape that holds area with less than 3 lines. A polygon with 3 edges is called a triangle (because of the three angles between them). A polygon with 4 edges is called a quadrilateral rather than a quadrangle! A polygon with 5 edges is called a pentagon, one with 6 edges is a hexagon and so on (using greek numbering). There is no upper limit on the number of edges but the polygon becomes hard to draw.

The lower limit on the number of faces for a polyhedron is 4. It is impossible to construct a shape that holds volume with less than 4 planes. There is no upper limit on the number of faces but the polyhedron becomes hard to construct.

A convex polygon is one with only interior angles less than 180 degrees. For example, a rectangle is convex but a chevron is not. The centre of gravity of a convex polygon is always within its area. The centre of gravity of a non-convex polygon may be inside or outside its area.

A regular polygon has all sides and all angles the same and therefore must also be convex. The regular 3-sided polygon is called an equilateral triangle. The 4-sided one is a square. There are no special names for the regular polygons with more sides. Simply specify a regular pentagon or hexagon etc.

There are special names for some irregular or semi-regular polygons though. An isosceles triangle has 2 of 3 angles the same and therefore 2 of 3 sides the same. A parallelogram has 4 sides and 4 angles and the opposite ones of each are the same. A rectangle is a parallelogram with all 4 angles the same - ie 90 degrees. Note that both diagonals of a rectangle are the same length but do not cross at 90 degrees. A rhombus is a parallelogram with all 4 sides the same. Note that the diagonals of a rhombus cross at 90 degrees but are not the same length. One special rhombus has one diagonal the same length as its sides but unfortunately has no special name.

A pentangle is a special non-convex polygon. It has 10 equal sides but 2 different angles (36 and 252 degrees) alternating to form a 5-pointed star. This shape is significant because it occurs in several non-convex polyhedrons. These are "stellations" of the regular convex ones.

A central angle is one between the lines (radii) joining 2 adjacent vertices to the centre. With a regular polygon, all central angles are the same. An interior angle is the angle between sides measured inside a vertex. With a regular polygon, all interior angles are the same. An exterior angle is the angle between sides measured outside a vertex when one of the sides is extended. With a regular polygon, all exterior angles are the same as the central angles. The central or exterior angle is 360 degrees divided by the number of sides. The interior angle is 180 degrees minus the central angle.

## Regular Polygons

There are an infinite number of regular polygons (from 3 sides upwards). However, only a few can be used in a regular vertex.

Sides Central Angle Interior Angle
3 120 60
4 90 90
5 72 108
6 60 120
7 51 3/7 128 4/7
8 45 135
9 40 140
10 36 144
12 30 150
15 24 156
18 20 160
20 18 162
24 15 165
42 8 4/7 171 3/7

## Regular 2-D Vertices

The sum of the interior angles of the polygons at a 2-dimensional vertex must be equal to 360 degrees. The following table shows all possible permutations with regular polygons. Not all can be used in practice. Some vertex types can be repeated to fill the plane on their own. Some must be combined with others to fill the plane. Some are just not usable (ie triples with an odd-sided polygon and the others not being a pair).

Vertex Usable Single Multi
3.3.3.3.3.3 Yes Yes Yes
3.3.3.3.6 Yes Yes Yes
3.3.3.4.4 Yes Yes Yes
3.3.4.3.4 Yes Yes Yes
3.3.4.12 Yes No Yes
3.4.3.12 Yes No Yes
3.3.6.6 Yes No Yes
3.6.3.6 Yes Yes Yes
3.4.4.6 Yes No Yes
3.4.6.4 Yes Yes Yes
4.4.4.4 Yes Yes Yes
3.7.42 No No No
3.8.24 No No No
3.9.18 No No No
3.10.15 No No No
3.12.12 Yes Yes Yes
4.5.20 No No No
4.6.12 Yes Yes Yes
4.8.8 Yes Yes No
5.5.10 No No No
6.6.6 Yes Yes Yes

## Irregular 2-D Vertices

Here the error in the sum to 360 degrees is given to show how irregular the polygons must be. The smaller the error (negative or positive) the less irregular the polygons will look. Only convex polygons are considered here.

VertexError
5.5.5-36
4.5.7-33 3/7
4.6.6-30
4.5.8-27
5.5.6-24
4.5.9-22
4.6.7-21 3/7
3.8.10-21
4.5.10-18
5.5.7-15 3/7
VertexError
3.8.12-15
4.6.8-15
4.7.7-12 6/7
3.10.10-12
4.5.12-12
5.6.6-12
3.9.12-10
4.6.9-10
5.5.8-9
4.7.8-6 3/7
VertexError
3.10.12-6
4.6.10-6
5.5.9-4
5.6.7-3 3/7
4.7.9-1 3/7
4.7.10+2 4/7
5.6.8+3
5.7.7+5 1/7
4.8.9+5
5.5.12+6
VertexError
5.6.9+8
4.7.12+8 4/7
6.6.7+8 4/7
4.8.10+9
4.9.9+10
5.7.8+11 4/7
5.6.10+12
4.9.10+14
4.8.12+15
6.6.8+15
VertexError
5.7.9+16 4/7
6.7.7+17 1/7
4.10.10+18
5.6.12+18
4.9.12+20
6.6.9+20
6.7.8+23 4/7
4.10.12+24
6.6.10+24
7.7.7+25 5/7

## Polygon Grids / Lattices

Note that V stands for vertex, F for face, E for Edge and t for type. So Vt is vertex type and Vs is vertices in one unit cell of the grid. For my purposes, a unit cell is the smallest parallelogram that can be repeated in translations to make an infinite 2-dimensional grid. A crystallographer would choose a different (bigger) unit cell in some cases and would specify the symmetry a little differently.

There are only 3 regular and 8 semi-regular grids. Using vertex type as the primary characteristic, there are 18 demi-semi-regular grids. An objective measure of grid complexity might be the total number of edges in a unit cell divided by the motif repeats its symmetry requires. The lists of grids with arbitrary numbers of regular or irregular polygons can never be complete. However, note that for all grids: Vs + Fs = Es.

Regular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
R4444 (square) 4m 1 1 1 1 1 2 0.25 1[4444] 1[4]
R666 (hexagon) 6m 1 1 1 2 1 3 0.25 2[666] 1[6]
R666 Er (triangle) 6m 1 1 1 1 2 3 0.25 1[333333] 2[3]
Semi-Regular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
S6363 (R6 E) 6m 1 2 1 3 3 6 0.5 3[6363] 1[6], 2[3]
S884 (R4 Er+Vr) 4m 1 2 2 4 2 6 0.75 4[884] 1[8], 1[4]
S123 (R6 Er+Vr) 6m 1 2 2 6 3 9 0.75 6[CC3] 1[12], 2[3]
S43433 (R4 E twist) 4g 1 2 2 4 6 10 1.25 4[43433] 2[4], 4[3]
S44333 (R4/R3 slide 1) 2x 1 2 3 2 3 5 1.25 2[44333] 1[4], 2[3]
S6434 (R6 V+Er) 6m 1 3 2 6 6 12 1.0 6[6434] 1[6], 2[3], 3[4]
S63333 (R6 E twist) 6 1 3 3 6 9 15 2.5 6[63333] 1[6], 2[3], 6[3]
S1264 (R6 V+Vr+Er) 6m 1 3 3 12 6 18 1.5 12[C64] 1[C], 2[6], 3[4]

Demi-Semi-Regular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
T3636 (R6 slide 1) 2m 2 2 3 3 3 6 1.5 1[6363], 2[6633] 1[6], 2[3]
T663 (R3 div) 6m 2 2 3 7 8 15 1.25 1[333333], 6[6633] 6[3], 2[6]
- (R3 slide 2) 2x 2 2 4 3 4 7 1.75 1[4444], 2[44333] 2[4], 2[3]
T3366 (R6 slide 2) 2x 2 2 4 4 5 9 2.25 2[63333], 2[6633] 1[6], 4[3]
T1243 (R123 slide 1) 4m 2 3 3 8 6 14 1.75 4[CC3], 4[C343] 1[C], 1[4], 4[3]
T3434 (S6434 div 6) 6m 2 3 3 7 11 18 1.5 1[333333], 6[43433] 6[3], 2[3], 3[4]
T633 (R6 div V') 6m 2 3 3 8 13 21 1.75 2[333333], 6[63333] 1[6], 6[3], 6[3]
- (R4 slide 2) 2m 2 3 4 3 5 8 2.0 1[333333], 2[44333] 1[4], 2[3], 2[3]
- (S6363 slide 2a) 2x 2 3 4 5 5 10 2.5 1[6363], 4[6443] 1[6], 2[3], 2[4]
- (R3 slide 3) 2x 2 3 5 4 5 9 2.25 2[4444], 2[44333] 1[4], 2[4], 2[3]
T4436 (S6363 slide 2b) 2m 2 4 4 5 5 10 2.5 1[6363], 4[6443] 1[6], 1[4], 1[4], 2[3]
T6433 (S123 div 12a) 6m 2 4 4 12 15 27 2.25 6[6434], 6[43433] 1[6], 2[3], 6[3], 6[4]
T6343 (S123 div 12b) 6m 2 4 4 12 15 27 2.25 6[6434], 6[44333] 1[6], 2[3], 6[4], 6[3]
T12643 (R6 E+Er+Vr) 6m 2 4 4 18 12 30 2.5 6[6434], 12[C64] 1[C], 2[3], 3[6], 6[4]
T1234 (S1264 div 6) 6m 2 4 4 14 16 30 2.5 2[333333], 12[C433] 1[C], 3[4], 6[3], 6[3]
- (R4 slide 3) 2x 2 4 5 4 7 11 2.75 2[44333], 2[333333] 1[4], 2[3], 2[3], 2[3]
- (S43433 double) 4g 2 4 5 12 18 30 3.75 4[44333], 8[43433] 2[4], 4[4], 4[3], 8[3]
- (S1264 div 12) 6m 2 5 5 18 18 36 3.0 6[6434], 12[6443] 1+2[6], 3+6[4], 6[3]

Less Regular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
- (R6 Vr div) 6m 3 2 3 9 9 18 1.5 1[333333], 2[666], 6[6633] 6[3], 3[6]
- (S43433 div 2a) 4g 3 3 4 14 14 28 3.5 2[4444], 4[6434], 8[6443] 8[4], 2[6], 4[3]
- (R6 reduce) 6m 3 3 4 11 19 30 2.5 2[333333], 3[333333], 6[63333] 1[6], 12[3], 6[3]
- (S44333 div) 2m 3 3 5 4 6 10 2.5 1[4444], 1[333333], 2[44333] 2[4], 2[3], 2[3]
- (R6 slide 3) 2m 3 3 5 5 7 12 3.0 1[333333], 2[63333], 2[6633] 1[6], 2[3], 4[3]
- (S6363 slide 4a) 2x 3 4 5 7 7 14 3.5 1[6363], 2[4444], 4[6443] 1[6], 2[3], 2[4], 2[4]
T3433 (S123 div 12a+) 6m 3 4 5 13 20 33 2.75 1[333333], 6[43433], 6[43433] 6[3], 2[3], 6[3], 6[4]
- (S123 div 12b+) 6m 3 4 5 13 20 33 2.75 1[333333], 6[44333], 6[43433] 6[3], 2[3], 6[4], 6[3]
- (S6363 slide 4b) 2m 3 4 6 7 7 14 3.5 1[6363], 2[4444], 4[6443] 1[6], 2[4], 2[4], 2[3]
- (R6 slide twice) 2x 3 4 6 7 11 18 4.5 1[333333], 2[63333], 4[63333] 1[6], 2[3], 4[3], 4[3]
- (S6363 slide 1) 2 3 4 7 5 7 12 6.0 1[6363], 2[63333], 2[63333] 1[6], 2[3], 2[3], 2[3]
- (S123 slide 2) 2x 3 4 7 10 9 19 4.75 2[CC3], 4[C343], 4[C433] 1[C], 2[4], 2[3], 4[3]
- (S6363 div 2a) 6m 3 5 5 15 27 42 3.5 3[333333], 6[63333], 6[333333] 1[6], 2+6+12+6[3]
- (S6363 slide 3a) 2x 3 5 6 7 11 18 4.5 1[6363], 2[333333], 4[63333] 1[6], 2+2+2+4[3]
- (S6434 x3 div 1) 6m 3 5 6 19 23 42 3.5 1[333333], 6[43433], 12[6434] 6+6[3], 2[6], 3+6[4]
- (S1264 div 12+) 6m 3 5 6 19 23 42 3.5 1[333333], 6[43433], 12[6443] 6+6[3], 2[6], 3+6[4]
- (S123 x3 div 1a) 6m 3 5 6 24 21 45 3.75 6[CC3], 6[6434], 12[C343] 1[6], 2[C], 6+6[3], 6[4]
- (S123 x3 div 1b) 6m 3 5 6 24 21 45 3.75 6[CC3], 6[6434], 12[C433] 1[6], 2[C], 6+6[3], 6[4]
- (S6434 dbl+div 2a) 6m 3 5 6 18 27 45 3.75 6[44333], 6[43433], 6[63333] 1[6], 2+6+12[3], 6[4]
- (T12643 div 6) 6m 3 5 6 21 27 48 4.0 3[333333], 6[43433], 12[C433] 1[C], 2+6+12[3], 6[4]
- (T12643 div 12) 6m 3 6 6 24 24 48 4.0 6[6434], 6[6434], 12[6443] 1+3[6], 2+6[3], 6+6[4]
- (S6434 x3 div 2) 6m 3 6 6 20 28 48 4.0 2[333333], 6[6434], 12[43433] 1[6], 6+6+6[3], 3+6[4]
- (S1264 div 6+12) 6m 3 6 6 20 28 48 4.0 2[333333], 6[6434], 12[44333] 1[6], 6+6+6[3], 3+6[4]
- (S123 x3 div 2b) 6m 3 6 8 30 33 63 5.25 6[44333], 12[6434], 12[C433] 1[C], 2[6], 6+6[4], 6+12[3]

Less Regular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
- (S6363 div 2b) 3m 4 4 6 13 17 30 5.0 1[666], 3[6633], 3+6[63333] 3[6], 2[3], 6[3], 6[3]
- (S43433 div 2b) 4g 4 4 6 16 24 40 5.0 2[4444], 2[333333], 4[43433], 8[44333] 8[4], 4[3], 4[3], 8[3]
- (S6363 slide 3b) 2m 4 5 6 7 11 18 4.5 1[6363], 1+1[333333], 4[63333] 1[6], 2+2+2+4[3]
T12443 (S1243 slide) 4m 4 5 6 17 19 36 4.5 1[4444], 4[44333], 4[C343], 8[C433] 4+2[4], 1[C], 4+8[3]
- (S6434 dbl+div 2+) 6m 4 5 7 19 32 51 4.25 1+6[333333], 6[44333], 6[43433] 6+2+6+12[3], 6[4]
- (T12643 div 12+) 6m 4 6 7 25 29 54 4.5 1[333333], 6[6434], 6[43433], 12[6443] 6+2+6[3], 3[6], 6+6[4]
- (S6434 x4 div 1) 6m 4 6 7 25 29 54 4.5 1[333333], 6[6434], 6[43433], 12[6434] 6+2+6[3], 3[6], 6+6[4]
- (S1264 div 6+12+) 6m 4 6 7 21 33 54 4.5 1+2[333333], 6[43433], 12[44333] 6+6+6+6[3], 3+6[4]
- (S123 x3 div 2a) 6m 4 6 7 30 33 63 5.25 6+6[6434], 6[43433], 12[C343] 1[C], 2[6], 6+6+6[3], 12[4]
T4346 (S6434 slide) 2m 4 6 8 10 12 22 5.5 2+2[43433], 2[6434], 4[6434] 1[6], 1+4[4], 2+2+2[3]
- (S123 slide 3) 2m 4 6 9 12 12 24 6.0 2[44333], 2[CC3], 4[C343], 4[C433] 1+2[4], 1[C], 2+2+4[3]
T41233 (S1243 twist) 4 4 6 9 16 18 34 8.5 4[43433], 4[C343], 4[C433], 4[C433] 1+4[4], 1[C], 4+4+4[3]
- (S123 slide 4) 2x 4 6 10 14 15 29 7.25 2[CC3], 4[C343], 4[C433], 4[44333] 1[C], 2+4+4[3], 2+2[4]
- (S6434 x4 div 3) 6m 4 7 8 27 39 66 5.5 3[333333], 6[43433], 6[6434], 12[43433] 1[6], 2+6+6+12[3], 6+6[4]
- (T12643 div 6+12) 6m 4 7 8 27 39 66 5.5 3[333333], 6[43433], 6[6434], 12[44333] 1[6], 2+6+6+12[3], 6+6[4]
T46343 (T1243X div) 4g 4 7 9 28 36 64 8.0 4+8[6434], 8+8[43433] 2+4+8[4], 2[6], 4+8+8[3]
- (S63333 slide) 2 4 7 12 8 13 21 10.5 2+2+2[63333], 2[333333] 1[6], 2+2+2+2+2+2[3]
- (S123 x3 div 2b 1a) 6m 4 8 10 36 45 81 6.75 6[44333], 6+12[6434], 12[43433] 1+2[6], 6+6+6[4], 6+6+12[3]

Less Regular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
- (S6434 dbl+div 2b) 3x 5 6 8 16 17 33 5.5 1[666], 3[6633], 3[43433], 3[6434], 6[6443] 3[6], 1+1+6[3], 3+3[4]
- (S123 slide 5) 2m 5 6 9 14 12 26 6.5 2[6434], 2[43433], 2[CC3], 4[C433], 4[C64] 1[C], 1[6], 2+2+2[3], 4[4]
T4343 (S6434 slide+div) 2m 5 7 10 11 17 28 7.0 1[333333], 2+2+2+4[43433] 1+4[4], 2+2+2+2+4[3]
- (T12643 slide) 2m 5 7 10 18 12 30 7.5 2+4[6434], 4+4+4[C64] 1[C], 1+2[6], 1+1+4[4], 2[3]
- (R6 div3/4) 2m 5 7 10 11 19 30 7.5 1+2+2[333333], 2+4[63333] 1[6], 2+2+2+4+4+4[3]
- (S123 x3 div 2a 1b) 6m 5 8 9 36 45 81 6.75 6+6+6[6434], 6+12[43433] 1+2[6], 6+6+6+6[3], 6+12[4]
- (T1243 div 12) 2m 5 8 11 14 18 32 8.0 2+4[6434], 2+4[43433], 2[44333] 1+2+4[4], 1[6], 2+2+2+4[3]
T43343 (T1243X div12+) 4g 5 8 11 30 46 76 9.5 2[333333], 4+8+8+8[43433] 2+4+8[4], 4+4+8+8+8[3]
- (S123 sl.3 div 12a) 2m 6 10 14 18 24 42 10.5 2+4[6434], 2+2[44333], 4+4[43433] 1+2+2+4[4], 1[6], 2+2+2+4+4[3]
- (S123 sl.3 div 12b) 2m 6 10 14 18 24 42 10.5 2+4[6434], 2+4[43433], 2+4[44333] 1+2+2+4[4], 1[6], 2+2+2+4+4[3]

Irregular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
- (R4 as rhombus) 2x 1 1 1 1 1 2 0.5 1[4444] 1[4]
S6363 Er 6m 2 1 1 3 3 6 0.5 1[444444], 2[444] 3[4]
S43433 Er 4g 2 1 2 6 4 10 1.25 2[5555], 4[555] 4[5]
U53 3m 2 2 2 5 4 9 1.5 2[555], 3[5553] 1[3], 3[5]
U52 (with 4s) 2x 2 2 3 4 3 7 1.75 2[554], 2[5554] 2[5], 1[4]
U47 (S43433) 4g 2 2 3 12 6 18 2.25 4[777], 8[774] 2[4], 4[7]
U52 (with 3s) 2x 2 2 4 4 4 8 2.0 2[5533], 2[5553] 2[5], 2[3]
U1056 Er2-10r 2x 2 2 4 6 3 9 2.25 2[A44], 4[AA4] 1[A], 2[4]
U693 6m 2 3 3 12 9 21 1.75 6[993], 6[9363] 1[6], 2[9], 6[3]
U1293 6m 2 3 3 15 9 24 2.0 3[9393], 12[C93] 1[C], 2[9], 6[3]
U12865 Er2-12r-8r 4m 2 3 4 12 6 18 2.25 4[C44], 8[C84] 1[C], 1[8], 4[4]
U1683 (S884) 4m 2 3 4 12 6 18 2.25 4[GG3], 8[G83] 1[G], 1[8], 4[3]
U1294 (U1293) 6m 2 3 4 18 9 27 2.25 6[944], 12[C94] 1[C], 2[9], 6[4]
U534 (U53) 3m 2 4 4 9 9 18 3.0 3[3454], 6[3545] 1[3], 2[3], 3[5], 3[4]
U12684 (S6434) 6m 2 4 5 24 12 36 3.0 12[C84], 12[684] 1[12], 2[6], 3[8], 6[4]

More Irregular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
U54 (S884) 4m 3 2 3 7 5 12 1.5 1[5555], 2[5555], 4[455] 1[4], 4[5]
U855 (S884) 4m 3 2 3 9 5 14 1.75 1[5555], 4[855], 4[855] 1[8], 4[5]
T3434 (S6434) 6m 3 2 3 11 7 18 1.5 2[555], 3[5555], 6[655] 1[6], 6[5]
U125 (S123) 6m 3 2 3 14 7 21 1.75 2[555], 6[C55], 6[C55] 1[C], 6[5]
U75 2x 3 2 5 8 4 12 3.0 2[775], 2[755], 4[757] 2[7], 2[5]
U522 (with 4s) 2m 3 3 3 5 4 9 2.25 1[5454], 2[5454], 2[554] 2[5], 1[4], 1[4]
U956 3x 3 3 4 10 5 15 2.5 1[555], 3[955], 6[965] 1[9], 1[6], 3[5]
U846 (S884) 4m 3 3 4 12 6 18 2.25 4[866], 4[866], 4[664] 1[8], 1[4], 4[6]
U856 (S884) 4m 3 3 4 13 7 20 2.5 1[5555], 4[655], 8[865] 1[8], 2[6], 4[5]
U1254 4m 3 3 4 13 7 20 2.5 1[5555], 4[C55], 8[C54] 1[C], 4[5], 2[4]
U695 (U693) 6m 3 3 4 18 9 27 2.25 6[995], 6[955], 6[655] 1[6], 2[9], 6[5]
U1256 (R6) 6m 3 3 4 20 10 30 2.5 2[555], 6[655], 12[C65] 1[C], 6[5], 3[6]
U522 (with 3s) 2m 3 3 5 5 6 11 2.75 1[5353], 2[53533], 2[5533] 2[5], 2[3], 2[3]
U586 2m 3 3 5 8 4 12 3.0 2[885], 2[655], 4[865] 1[8], 1[6], 2[5]
U865 (S43433) 4g 3 3 5 20 10 30 3.75 4[665], 8[865], 8[865] 2[8], 4[6], 4[5]
(U1256) 6m 3 3 5 20 13 33 2.75 2[555], 6[5544], 12[C54] 1[C], 6[5], 6[4]
- (T1243 cut) 2m 3 4 5 7 6 13 3.25 1[A3A3], 2[AA3], 4[A343] 1[A], 1[4], 2[3], 2[3]
U4866 (S884) 4m 3 4 5 16 8 24 3.0 4[466], 4[666], 8[866] 1[8], 1[4], 2[6], 4[6]
U6946 6m 3 4 5 24 12 36 3.0 6[666], 6[669], 12[694] 1[6], 2[9], 3[4], 6[6]

More Irregular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
U674 (S6434) 6m 4 3 4 20 10 30 2.5 2[777], 6+6[774], 6[776] 1[6], 6[7], 3[4]
U765 (U53) 3m 4 3 5 16 8 24 4.0 1[777], 3[775], 6+6[765] 2[6], 3[7], 3[5]
U1255 (T1243) 4m 4 3 5 17 9 26 3.25 1[5555], 4[555], 4+8[C55] 1[C], 4[5], 4[5]
T3433 Er 6m 4 3 5 20 13 33 2.75 2+6[555], 6[5555], 6[655] 1[6], 6[5], 6[5]
U568 2m 4 3 6 10 5 15 3.75 2[855], 2[866], 2[665], 4[865] 1[8], 2[6], 2[5]
U1056 2x 4 3 6 12 6 18 4.5 2+4[A55], 2[655], 4[A65] 1[A], 1[6], 4[5]
U6956 6m 4 4 6 30 15 45 3.75 6[955], 6[665], 6[666], 12[965] 1[6], 2[9], 6[5], 6[6]
U69553 (U6956) 6m 4 5 6 30 21 51 4.25 6+12[955], 6[5553], 6[6353] 1[6], 2[9], 6+6[5], 6[3]
U66856 (U6586) 6m 4 5 7 36 18 54 4.5 6[666], 6[665], 12+12[865] 1+2+6[6], 3[8], 6[5]
U8856 (U855) 4m 4 5 8 28 14 42 5.25 4+8[665], 8[865], 8[866] 1+1[8], 4[5], 4+4[6]

More Irregular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
U1265 (S1264) 6m 5 3 5 27 15 42 3.5 3[5555], 6+6[655], 6+6[C55] 1[C], 2[6], 12[5]
U965 3m 5 3 6 20 10 30 5.0 2[666], 3[966], 3+6[665], 6[965] 1[9], 6[6], 3[5]
U567 (U47) 4g 5 3 7 30 16 46 5.75 2[5555], 4[766], 8+8[765], 8[755] 8[5], 4[6], 4[7]
U12865 4m 5 4 6 24 12 36 4.5 4[655], 4+4[855], 4[C55], 8[C65] 1[C], 1[8], 2[6], 8[5]
U6586 6m 5 4 6 32 16 48 4.0 2[555], 6[855], 6[866], 6[666], 12[865] 1[6], 6[5], 3[8], 6[6]
U12965 (U1293) 6m 5 4 6 36 18 54 4.5 6[655], 6+6[C55], 6[955], 12[965] 1[C], 2[9], 3[6], 12[5]
U4657 (S43433) 4g 5 4 7 32 16 48 6.0 4[776], 4+8[775], 8[774], 8[765] 2[4], 2[6], 4[5], 8[7]
U8675 (S43433) 4g 5 4 7 32 16 48 6.0 4[655], 4[755], 8+8[875], 8[765] 2[8], 2[6], 4[7], 8[5]
U1275 6m 5 4 7 38 19 57 4.75 2[777], 6+6[775], 12[755], 12[C55] 1[C], 6[7], 6[5], 6[5]
U7665 2m 5 4 8 14 7 21 5.25 2[776], 2[766], 2[665], 4+4[765] 1[6], 2[7], 2[6], 2[5]
U48675 (Altair1) 4m 5 5 7 28 14 42 5.25 4[766], 4[774], 4[775], 8[765], 8[865] 1[8], 1[4], 4[7], 4[6], 4[5]
U66675 (Altair2) 6m 5 5 8 42 21 63 5.25 6[766], 6[776], 6[775], 12[765], 12[665] 1+2+6[6], 6[7], 6[5]
U69864 (Altair4) 6m 5 5 8 42 24 66 5.5 6[666], 6[6644], 6[944], 12[984], 12[864] 1+6[6], 2[9], 3[8], 12[4]

More Irregular Sym Vt Ft Et Vs Fs Es E/S Vertices Faces
U6765 3x 6 4 7 20 10 30 5.0 1[777], 1[666], 3[776], 3[665], 6+6[765] 1[6], 3[7], 3[6], 3[5]
U16855 (U1683) 4m 6 4 7 28 14 42 5.25 4[555], 4+4[855], 4+4+8[G55] 1[G], 1[8], 4[5], 8[5]
U12565 (T12643) 6m 6 4 7 38 22 60 5.0 2[555], 6+12[655], 6+6[C55], 6[5555] 1[C], 6[5], 3[6], 12[5]
U10556 2m 6 4 8 15 8 23 5.75 1[5555], 2[555], 2[655], 2+4[A55], 4[A65] 1[A], 1[6], 2[5], 4[5]
U88665 2m 6 5 9 18 9 27 6.75 2[866], 2[855], 2[655], 4+4+4[865] 1+1[8], 1+2[6], 4[5]
U85765 4m 6 5 9 33 17 50 6.25 1[5555], 4[557], 4[775], 8[865], 8+8[765] 1[8], 4+4[5], 4[7], 4[6]
T4343 Er 2m 7 5 10 17 11 28 7.0 1+4[5555], 2+2+2[555], 2+4[655] 1[6], 2+2+2+4[5]
U8865 (U856) 4m 7 6 10 40 20 60 7.5 4+4[866], 4+8[666], 4+8[665], 8[865] 1+1[8], 2+4+8[6], 4[5]
U10865 2m 8 5 10 21 11 32 8.0 1[5555], 2[655], 2+2+4[855], 2+4[A55], 4[A65] 1[A], 1[8], 1[6], 4+4[5]
T43343 Er 4g 8 5 11 46 30 76 9.5 2+4+8[5555], 4+8[655], 4+8+8[555] 8+4+8+8[5], 2[6]
U8875 4m 8 6 11 44 22 66 8.25 4+8+8[775], 4+4[755], 4+4[855], 8[875] 1+1[8], 4+4[7], 4+8[5]
U12105 2m 9 6 12 26 13 39 9.75 2[555], 2[655], 2+2+4+4[C55], 2+4[A55], 4[A65] 1[C], 1[A], 1[6], 2+4+4[5]

There are relationships between grids. Some are shown in the following table. Note:
V = original polygon grid (ie join original vertices).
truncate = cut off vertices (ie double polygon edges and leave space for a new polygon).
E = join centres of edges of polygon faces (ie expand / truncate vertices to halfway).
Er = join centres of adjacent polygons (ie the inverse grid from full truncation).
Vr = join vertices to centres of faces (ie expand / shave edges).
snub = reduce and twist polygons to leave space for more polygons (triangles).
* = some readjustment of edge length is required.

V trunc. E Er Vr snub
[4444] square [884] [4444] square square [43433]
[666] hexagon [CC3] [6363] trianglerhombus[63333]
[333333] triangle[666] [6363] hexagon rhombus
[6363] hex.+tri.*[C64]*[6434] rhombus kite
[43433] sq.+tri.*[865] *[435] pentagonkite + rhombus

Note the square is its own inverse whereas the other grids are paired as inverses. As a result, its edges lead to a regular rhombus - squares again! Note that every rhombus is an inverse of a rectangle. The other special rhombus in the table above is the 60 degree one (like 2 equilateral triangles). It ought to have been given a special name too, eg a trombus (concatenating triangle and rhombus). The pentagon is irregular with two 90 degree and three 120 degree angles. The kite has two opposite 90 degree angles and a 60 degree angle opposite a 120 degree one.

## 2-D Symmetry Types

There are only 17 types of 2-dimensional symmetry. They are sometimes called plane groups. There are various standard nomenclatures for them but I use a minimalist classification system. The 2-D symmetry operations are rotation, mirror reflection and glide reflection. Translation should be reserved for repeating the unit cell to fill the plane but standard nomenclatures tend to include some. I disapprove of their "centred" cell as it arises from a prejudice that right angles are best.

Mirror reflection should be obvious. The motif is flipped over to repeat or match itself. The mirror surface is a plane that intersects our plane along a line. Glide reflection is rather strange. The motif is flipped over to repeat itself half a unit cell away. The "glide" is this bit of translation before finishing the reflection.

The unit cell of a lattice is always some sort of parallelogram (opposite pairs of sides and angles being the same). Translation along its 2 axes repeats the motif to fill the plane. However, some parallelograms are special and are required by some symmetry groups. A rectangle has all 4 angles the same (right angles) and diagonals the same. This is required for mirrors parallel to cell walls. A rhombus has all 4 sides the same and diagonals crossing at right angles. This is required for mirrors diagonal to cell walls. The standard nomenclatures use the diagonals as cell walls, doubling the size of a unit cell to make a centred cell. So "p" is used for a primitive cell and "c" for a centred one. A centred cell has twice the number of motifs it should have. A square has all 4 sides & angles the same and diagonals the same & crossing at right angles. This is required for tetrads. A "trombus" (60/120 degree rhombus) has one diagonal the same length as its sides. This is required for triads and hexads. Bizarrely, the standard nomenclatures don't insist on a centred rectangular cell here!

In the following table of plane groups, I list both my classification and some standard ones. The symmetry defines the shape of the smallest possible unit cell and the number of motif repeats (M) within it. I also list the ratio of sides (R) and skew angle (S) used in my pattern software.

1 p1 //ogram 1--tt'tt' (2), tt't"tt't" (3)
1g pg rectangle2-0gtgt (2), gg'tg'gt (3), ggg'g' (2), ggtg'g't (3)
1m pm rectangle2-0mtm't (2), mtmt (2)
1x cm rhombus 21-mgg (2), gggg (2), mgm'g (2), mggmgg (3), mtggt (4), ggtggt (3)
2 p2 //ogram 2--22'2" (2), 2t2't2" (4), 2t2't (2), 22t'2"2"'t (3), 22'2"2"' (2)
2g p2gg, pggrectangle4-02gg (2), 2gg'gg' (3), gg'gg' (2), 2tggt (3), 22'tggt (3), 22'gg (2), 2g2'g (2), 2g2'g'gg' (3)
2j p2mg, pgmrectangle4-0m22' (2), 222'2' (2), mt2t (2), 2t2t (2), mt22't (3), 22't2'2t (3), m2m'2' (2), m22m2'2' (3)
2m p2mm, pmmrectangle4-0mm'm"m"' (2), mm'mm" (2), mm'mm' (2)
2x c2mm, cmmrhombus 41-mm'2 (2), m22 (2), 2222 (2), mm'm"2 (2), m2m'2 (2), m22m22 (3)
3 p3 trombus 311/3333'3' (3), 333'3'3"3" (3)
3m p31m trombus 611/3m33 (3), 3333 (3), mm'33 (3), mm3333 (3), 333333 (3)
3x p3m1 trombus 611/3mm'm" (2), mmm'm' (2)
4 p4 square 410442 (2), 444'4'2 (3), 444'4' (2)
4g p4gm, p4gsquare 810m44 (2), 4444 (2), mm'44 (2), m4444 (3), 4444 (2)
4m p4mm, p4msquare 810mm'm" (2), mmm' (2), mmm'm' (2)
6 p6 trombus 611/36633 (3), 66332 (3), 332 (3), 662 (2)
6m p6mm, p6mtrombus 1211/3mm'm" (2), mmm' (3), mmm' (2), mmm'm' (3)

Symmetry occurs in nature but it is approximate rather than exact. Gravity has caused organisms to evolve with vertical mirror symmetry. As a result humans are prejudiced in favour of patterns that have vertical mirrors. Psychologically, vertical symmetry is restful while horizontal is restless. When no particular direction of travel is important, rotational symmetry occurs. For example: starfish and most flowers. Glide reflection combined with size change is seen in ferns. Rotation combined with size change or translation produces spirals and coils. For example: fir cones, pineapples and vines.

Islamic art tends to favour higher orders of rotation and as many mirror reflections as possible. This maximised the repetitiveness of their designs. They were only permitted to use abstract forms and the straight edges forced by mirrors were no problem. So most islamic designs are 6m or 4m. In contrast, M.C.Escher tended to avoid too many rotations and mirror reflections. They restrict the shapes in a design and Escher preferred recognisable animal and plant forms to abstract shapes. So most Escher designs are 1, 1g, 2 or 2g.

## Regular 3-D Vertices

The sum of the interior angles of the polygons at a 3-dimensional polyhedron vertex must be less than 360 degrees. Not all possible permutations can be used on their own to make a solid.

Vertex Single
3.3.3.3.3 Icosahedron
3.3.3.3.4 Yes (snub/twisted cubocta.)
3.3.3.3.5 Yes (snub/twisted dodicosa.)
3.3.3.3 Octahedron
3.3.3.X Yes (twisted prism)
3.3.4.X<12 No
3.4.3.4 Yes (cuboctahedron)
3.4.3.X<12 No
3.3.5.X<8 No
3.5.3.5 Yes (dodicosahedron)
3.5.3.X<8 No
3.4.4.4 Yes (rhomb-cubocta.)
3.4.4.5 No
3.4.5.4 Yes (rhomb-dodicosa.)
3.3.3 Tetrahedron
3.3.X No
3.4.4 Yes (triangular prism)
3.4.X No
3.5.X No
3.6.6 Yes (truncated tetra.)
3.6.X No
3.7.X<42 No
3.8.8 Yes (truncated cube)
3.8.X<24 No
3.9.X<18 No
3.10.10 Yes (truncated dodeca.)
3.10.X<15 No
3.11.X<14 No
4.4.4 Hexahedron = Cube
4.4.X Yes (X-sided prism)
4.5.X<20 No
4.6.6 Yes (truncated octa.)
4.6.7 No
4.6.8 Yes (truncated cubocta.)
4.6.9 No
4.6.10 Yes (truncated dodicosa.)
4.6.11 No
4.7.X<10 No
5.5.5 Dodecahedron
5.5.X<10 No
5.6.6 Yes (truncated icosa.)
5.6.7 No

## Polygon Solids / Polyhedrons / Polyhedra

Note that V stands for vertex, F for face, E for Edge and t for type. So Vt is vertex type and Vs is vertices in the whole solid. Truncation refers to polygons of the inverse solid being added by shaving vertices. Shaving edges of regular polyhedrons adds polygons from the faces of rhombus solids. These can be stretched into regular squares or be broken into triangles to make twisted edge forms. Snubbing refers to the reduction and twisting of polygon faces, leaving space for these extra triangle faces.

There are only 5 regular solids (Platonic). There are 13 unique semi-regular solids (Archimedean) and 2 infinite series (prisms and twisted prisms). Using vertex type as the primary characteristic, the next simplest solids are the 2 bi-pyramids. The lists of solids with arbitrary numbers of regular or irregular polygons can never be complete. However, note that for all polyhedrons: Vs + Fs = Es + 2.

Regular Vt Ft Et Vs Fs Es Vertices Faces Model File
tetrahedron 1 1 1 4 4 6 4[333] 4[3] R666 Er
hexahedron/cube 1 1 1 8 6 12 8[444] 6[4] R4444
octahedron 1 1 1 6 8 12 6[3333] 8[3] R666 Er
dodecahedron1 1 1 20 12 30 20[555] 12[5] U52
icosahedron 1 1 1 12 20 30 12[33333] 20[3] R666 Er
Semi-Regular Vt Ft Et Vs Fs Es Vertices Faces Model File
cubocta. 1 2 1 12 14 24 12[4343] 6[4], 8[3] S43433
dodicosa. 1 2 1 30 32 60 30[5353] 12[5], 20[3] U53 (5363)
X-sided prism1 2 2 2X X+2 3X 2X[X44] 2[X], X[4] S44333
twisted prism1 2 2 2X 2X+2 4X 2X[X333] 2[X], 2X[3] S44333
trunc. tetra. 1 2 2 12 8 18 12[663] 4[6], 4[3] R666
trunc. cube 1 2 2 24 14 36 24[883] 6[8], 8[3] S884
trunc. octa. 1 2 2 24 14 36 24[664] 8[6], 6[4] R666
trunc. dodeca.1 2 2 60 32 90 60[AA3] 12[A], 20[3] U1056
trunc. icosa. 1 2 2 60 32 90 60[665] 20[6], 12[5] R666
rhomb-cubocta.1 3 2 24 26 48 24[3444] 6[4], 8[3], 12[4] S6434
rhomb-dodicosa.1 3 2 60 62 120 60[3454] 12[5], 20[3], 30[4] S6434
snub cubocta.1 3 3 24 38 60 24[43333] 6[4], 8[3], 24[3] S63333
snub dodicosa.1 3 3 60 92 150 60[53333] 12[5], 20[3], 60[3] S63333
trunc. cubocta.1 3 3 48 26 72 48[864] 6[8], 8[6], 12[4] S1264
trunc. dodicosa.1 3 3 120 62 180 120[A64] 12[A], 20[6], 30[4] S1264

Less Regular Vt Ft Et Vs Fs Es Vertices Faces Model File
bi-pyramid 3 2 1 2 5 6 9 2[333], 3[3333] 6[3] R666 Er
bi-pyramid 5 2 1 2 7 10 15 2[33333], 5[3333] 10[3] R666 Er

Irregular Vt Ft Et Vs Fs Es Vertices Faces
rhombus 12 2 1 1 14 12 24 6[4444], 8[444] 12[4]
rhombus 30 2 1 1 32 30 60 12[44444], 20[444] 30[4]
kite 24 3 1 2 26 24 48 6[4444], 8[444], 12[4444] 24[4]
kite 60 3 1 2 62 60 120 12[44444], 20[444], 30[4444] 60[4]
triangle 483 1 3 26 48 72 6[38], 8[36], 12[34] 48[3]
triangle 1203 1 3 62 120 180 12[310], 20[36], 30[34] 120[3]

There are relationships between polyhedrons. Some are shown in the following table and its pictorial equivalent. Note:
V = original polyhedron (ie join original vertices).
truncate = cut off vertices (ie double polygon edges and leave space for a new polygon).
E = join centres of edges of polygon faces (ie shave vertices / truncate to halfway).
Er = join centres of adjacent polygons (ie the inverse solid from full truncation).
Vr = join vertices to external centres of faces (ie expand edges to planes)
* = some readjustment of edge length is required.

V trunc. E Er Vr
[333] tetra. [663] [3333] octa. tetra. cube (rhombus 6)
[444] cube [883] [3434] cubocta. octa. rhombus 12
[3333] octa. [664] [3434] cubocta. cube rhombus 12
[555] dodeca. [AA3] [3535] dodicosa.icosa. rhombus 30
[33333] icosa. [665] [3535] dodicosa.dodeca.rhombus 30
[3434] cubocta. *[864]*[3444] rhomb-cubocta. rhombus 12 kite 24
[3535] dodicosa.*[A64]*[3454] rhomb-dodicosa.rhombus 30 kite 60

Note the tetrahedron is its own inverse whereas the other regular solids are paired as inverses. As a result, its edges lead to a regular rhombus solid - the squares of a cube! The halfway truncation of a tetrahedron is also regular - the octahedron. Performing the same edge operations on this as on the other inverse pairs is interesting. The icosahedron can be considered to be a snub/twisted octahedron, while the cuboctahedron is also a rhomb-octahedron.

In addition to convex solids there are also non-convex "stellations" of the regular polyhedra. These can be derived by more than one method. A list of so-called "uniform" polyhedra (convex and non-convex) has been proven complete. However this only includes those that meet certain restrictions. Some non-convex examples follow.

Non-Convex Vt Ft Et Vs Fs Es Vertices Faces
stellated octa. (cube Vx)2 1 2 14 24 36 8[333], 6[38] 24[3]
stellated dodeca. 2 1 2 32 60 90 12[33333], 20[36] 60[3]
stellated icosa. 2 1 2 32 60 90 20[333], 12[310] 60[3]
(dodeca. V*) 2 2 2 80 72 15020[*3*3*3], 60[*33] 12[star], 60[3]
hollow cubocta. 1 2 2 2 18 32 4812[333333], 6[3333] 8[3], 24[3]
hollow cubocta. 2 2 2 2 20 30 4812[433433], 8[333] 6[4], 24[3]

## Polyhedron Grids / Lattices

A unit cell here is the smallest shape that can be repeated in translations to make an infinite 3-dimensional grid or lattice. A rhomboid is a 3-dimensional parallelogram (3 different edge lengths and 3 different angles between them). A cuboid is a 3-dimensional rectangle (3 different edge lengths but 90 degree angles between them).

Of the 5 regular polyhedrons, only 1 can make a regular grid on its own. This is the cube - 1 per unit cell! 8 cubes meet at each vertex (4 meet at an edge). The octahedron and tetrahedron together can make a semi-regular grid. 6 octahedrons and 8 tetrahedrons meet at each vertex (2 of each meet at an edge, alternating). The triangular prism can make a grid on its own. 12 prisms meet at each vertex (6 meet at each edge between squares and 4 meet at each triangle/square edge).