Isohedra

 

This page is under construction.

 

Here we present convex isohedral dice based on the methodology introduced by Mogensen [4], where we use the following notation

*        V: number of vertices of the die

*        N: number of faces of the die

*        E: number of edges of the die.

Such dice are characterized by identical faces where

*        M: number of sides (and corners) of a face (not of the die!)

It can be shown that for isohedra 2<M<6, hence the faces must thus be triangles, quadrangles, or pentagons.

While all the faces on the dice are equivalent, the same is not true for the vertices. The different types of vertices are labeled with a number i, i=1,2,... up to the number of different vertices, with

*        Ri: number of corners that meet in a vertex (rank)

*        Vi: number a vertex type occurs

*        Mi: number of different corners a vertex is made out of

 

The Vertex degree used by Pegg [5] is similar to Mogensenís Ri, and Peggís edge code is equivalent to Mogensenís vertex type. If a greek letter occurs only once in an edge code, that edge is a singlet, if it occurs twice it is a doublet.

 

An alternative construction of Isohedra is based on isohedral transforms, as introduced by loki3 [9].

 

In brackets we include Mogensenís type and loki3ís isohedral transform, e.g. for the Tetrahedron (D, 4T).

Triangles (M=3)

Equilateral Triangles

A polyhedron with faces that are all congruent equilateral triangles is called a Deltahedron. There are five deltahedra which are isohedra: regular tetrahedron, regular trigonal dipyramid, regular octahedron, regular pentagonal dipyramid, and regular icosahedron.

Regular Tetrahedron (D)

N=4, M=3, R=3, V=4, E=6

[3,3,3],aaa

Only one type of vertex with 3 edges.

Platonic solid

 

?or?

Regular Trigonal Dipyramid (I)

N=6, M=3, M1=2, R1=4, V1=3, M2=1, R2=3, V2=2

[4,4,3],aab

1 doublet plus 1 singlet corner.

The singlet and doublet corners are part of different types of vertices.

This is a special case where the triangles are equilateral. However, there are still two different vertex types (and thus this is not a Platonic solid).

 

Regular Octahedron

(E)

N=8, M=3, R=4, V=6, E=12

[4,4,4],aaa

Only one type of vertex with 4 edges.

Platonic solid, dual of the cube.

 

Regular Pentagonal Dipyramid (I)

N=10, M=3, M1=2, R1=4, V1=5, M2=1, R2=5, V2=2

[4,4,5],aab

1 doublet plus 1 singlet corner.

The singlet and doublet corners are part of different types of vertices.

This is a special case where the triangles are equilateral. However, there are still two different vertex types (and thus this is not a Platonic solid).

 

Regular Icosahedron

(F)

N=20, M=3, R=5, V=12, E=30

[5,5,5],aaa

Only one type of vertex with 5 edges.

Platonic solid, dual of the dodecahedron

 

Isosceles Triangles

Isosceles Tetrahedron / Tetragonal Disphenoid (D)

N=4, M=3, V=4, R=3, E=6

†[3,3,3],aab

1 doublet (a) plus 1 singlet (b) corner.

The singlet and doublet corners are part of the same type of vertex.

The Iscosceles Tetrahedron has only one type of vertex, but in each vertex one singlet and two doublet corners meet, satisfying Mogensenís equation (8).

(Note: Mogensen [7] does not explicitly state that (D) need not be a regular tetrahedron)

Each edge is the diagonal of a cuboid

 

Triangular Dihedron / Double Pyramid / Bi-Pyramid / Di-Pyramid

†(I)

N even (N>=6), M=3, M1=2, R1=4, V1=N/2, M2=1, R2=N/2, V2=2

[4,4,N/2],aab

1 doublet (a) plus 1 singlet (b) corner.

The singlet and doublet corners are part of different types of vertices.

Opposing faces if N/2 even

Add loki3ís d6 - di-pyramid

N=6

Trigonal Dipyramid

N=8

Tetragonal Dipyramid

N=10

Pentagonal Dipyramid

N=12

Hexagonal Dipyramid

 

N=16

Octagonal Dipyramid

N=36

Octadecagonal Dipyramid

 

Triakis Tetrahedron / Tristetrahedron / Trigonal tristetrahedron,

(J)

N=12, V=8,E=18

M=3, M1=2, R1=6, V1=4, M2=1, R2=3, V2=4,

[3,6,6],abb

1 doublet (a) plus 1 singlet (b) corner.

The singlet and doublet corners are part of different types of vertices.

Triangular pyramids on tetrahedron faces, i.e., cumulation of the tetrahedron.

Catalan solid, dual of the truncated tetrahedron

 

Tetrakis Hexahedron / Tetrahexahedron

(K)

N=24, V=14, E=36

M=3, M1=2, R1=6, V1=8, M2=1, R2=4, V2=6,

[4,6,6], abb

1 doublet (a) plus 1 singlet (b) corner.

The singlet and doublet corners are part of different types of vertices.

†Square pyramids on cube faces, i.e., cumulation of the cube.

Catalan solid, dual of the truncated octahedron.

(Note: Mogensen [7] mentions octahedron instead of cube faces)

 

Triakis Octahedron / Small Triakis Octahedron/ Trisoctahedron / Trigonal Trisoctahedron

(M)

N=24, V=14, E=36

M=3, M1=2, R1=8, V1=6, M2=1, R2=3, V2=8,

[3,8,8], abb

1 doublet (a) plus 1 singlet (b) corner.

The singlet and doublet corners are part of different types of vertices.

Triagonal pyramids on octahedron faces, i.e., cumulation of the octahedron.

Catalan solid, dual of the truncated cube.

(Note: Mogensen [7] mentions cube instead of octahedron faces)

 

Pentakis Dodecahedron

(L)

N=60, V=32, E=90

M=3, M1=2, R1=6, V1=20, M2=1,R2=5, V2=12

[5,6,6],aab

1 doublet (a) plus 1 singlet (b) corner.

†The singlet and doublet corners are part of different types of vertices.

Pentagonal pyramids on dodecahedron faces, i.e., cumulation of the dodecahedron.

Catalan solid, dual of the truncated icosahedron.

 

Triakis Icosahedron

(N)

N=60, V=32, E=90

M=3, M1=2, R1=10, V1=12, M2=1, R2=3, V2=20

[3,10,10],aab

1 doublet (a) plus 1 singlet (b) corner.

The singlet and doublet corners are part of different types of vertices.

Triagonal pyramids on icosahedron faces, i.e., cumulation of the icosahedron.

Catalan solid††††† , dual of the truncated dodecahedron

 

Scalene Triangles

Scalene tetrahedron / Rhombic disphenoid

(D)

N=4, M=3, R=3, V=4, E=6

[3,3,3],abg

3 singlet corners.

The three singlet corners are part of the same type of vertex.

The Scalene Tetrahedron has only one type of vertex, but in each vertex one each of the singlet corners meet, satisfying Mogensenís equation (8).

(Note: Mogensen [7] does not explicitly state that (D) need not be a regular tetrahedron).

Each edge is the diagonal of a cuboid

 

Tetragonal Scalenohedron

N=8, M=3, M1=2, R1=4, V1=4, M2=1, R2=4, V2=2

3 singlet corners, yet only two types of vertices.

Same values as for a tetragonal Dipyramid (I), although composed of scalene triangles.

This is a shape of a family called Triangular Dihedron skewed up/down (D4n) which can be constructed by skewing the equatorial vertices of a dipyramid with 4n faces up and down.

"Scalenohedron" from the Wolfram Demonstrations Project
 
http://demonstrations.wolfram.com/Scalenohedron/

 

Skewed Triangular Dihedron / Double Pyramid / Bi-Pyramid / Di-Pyramid

(O)

N=8,12,16,.., M=3, M1=M2=M3=1, R1=R2=4, V1=V2=N/4, R3=N/2, V3=2

3 singlet corners, each is part of a different type of vertex.

deform variant of the Dipyramid (I)

This is a shape of a family called Triangular Dihedron skewed in/out (D4n) which can be constructed by squeezing the equatorial vertices of a dipyramid with 4n faces in and out from the center of the solid.

"Scalenohedron" from the Wolfram Demonstrations Project
 
http://demonstrations.wolfram.com/Scalenohedron/

 

missing

Hexakis Tetrahedron / Hextetrahedron

(P)

N=24, M=3, M1=1, R1=4, V1=6, M2=M3=1,R2=R3=6, V2=V3=4, V=14, E=36

[4,6,6],abg

3 singlet corners, each is part of a different type of vertice.

deform variant of (K)

(Note: Mogensen [7] incorrectly states that V1=4)

 

Hexakis Octahedron / Disdyakis Dodecahedron †/ Hexoctahedron/ Octakis Hexahedron

(Q)

N=48, V=26, E=72

M=3, M1=1, R1=4, V1=12, M2=1, R2=6, V2=8, M3=1, R3=8, V3=6

[4,6,8],abg

3 singlet corners, each is part of a different type of vertice.

Rhombic pyramids on rhombic dodecahedron faces.

Catalan solid, dual of the truncated cuboctahedron

 

Hexakis Icosahedron / Disdyakis Triacontahedron

(R)

N=120, V=62, E=180

M=3, M1=1, R1=4, V1=30, M2=1, R2=6, V2=20, M3=1, R3=10, V3=12

[4,6,10],abg

3 singlet corners, each is part of a different type of vertice.

Largest non-bipolar die.

Rhombic pyramids on rhombic triacontahedron faces.

Catalan solid, , dual of the truncated icosidodecahedron.

 

Quadrangles (M=4)

Square

Regular Hexahedron, Cube

(G)

N=6, M=4, R=3, V=8, E=12

[3,3,3,3],aaaa

Only one type of vertex with 3 edges.

Platonic solid, dual of the octahedron.

Rhombus

Rhombic Hexahedron,

Rhombohedron

(S)

N=6, M=4, M1=2, R1=3, V1=4, M2=2, R2=3, V2=4

[3,3,3,3], aaab

2 opposite doublet corners (a,b). Vertex type 2 consists of identical corners (b), vertex type 1 consists of different corners (a,a,b).

Deform variant of the Cube (G), special case of the Trapezohedron (V)

 

Note that it can be proven that there are no trapezohedra made of rhombi other than for N=6 (friz).

††††††† †

Rhombic Dodecahedron

(T)

N=12, V=14, E=24

M=4, M1=2, R1=3, V1=8, M2=2, R2=4, V2=6

[3,4,3,4],abab

2 opposite doublet corners (a,b), each is part of a different type of vertice.

Catalan solid, dual of the cuboctahedron.

 

Rhombic Triacontahedron

(U)

N=30, V=32, E=60

M=4, M1=2, R1=3, V1=20, M2=2, R2=5, V2=12

[3,5,3,5],abab

2 opposite doublet corners (a,b), each is part of a different type of vertice.

Catalan solid, dual of the icosidodecahedron

 

Deltoid (Kite)

Trapezoidal Dihedron / Trapezohedron / Antidipyramids / Deltohedron

Crystallographic examples: Tetragonal Trapezohedron (N=8), Hexagonal Trapezohedron (N=12)

(V)

N even (N>6), V=N+2, E=2N

M=4, M1=3, R1=3, V1=N, M2=1, R2=N/2, V2=2

[3,3,3,N/2],aaab

1 doublet and 2 singlet corners, doublet and one of the singlets are part of the same type of vertex

double cone made from kites

opposing faces if N/2 odd

"Streptohedron and Trapezohedron" from the Wolfram Demonstrations Project
 
http://demonstrations.wolfram.com/StreptohedronAndTrapezohedron/

 

Note that it can be proven that there is no Trigonal Trapezohedron (N=6) composed of deltoids (proof by friz). The faces have to be squares (a cube), rhombi (a rhombic cube) or irregular quadrilaterals.

 

N=8

Tetragonal Trapezohedron

N=10

Pentagonal Trapezohedron

N=12

Hexagonal Trapezohedron

N=14

Heptagonal Trapezohedron

N=18

Enneagonal Trapezohedron

 

N=26

Trisdecagonal Trapezohedron

N=34

Heptadecagonal Trapezohedron

N=50

Icosikaipentagonal Trapezohedron

 

 

Deltoidal Dodecahedron / Trapezoidal Dodecahedron / Trapezohedral Tristetrahedron

(W)

N=12, M=4, M1=2, R1=4, V1=6, M2=M3=1, R2=R3=3, V2=V3=4, V=14, E=24

[3,4,3,4],abgb

1 doublet (a) and 2 singlet (b,c) corners, doublet and both singlets all form their own vertices.

deform variant of the rhombic dodecahedron (T)

 

Deltoidal Icositetrahedron / Trapezoidal Icositetrahedron / Strombic Icositetrahedron / Trapezohedral Trisoctahedron / Trapezohedron,

(X)

N=24, V=26, E=48

M=4, M1=2, R1=4, V1=12, M2=1, R2=3, V2=8, M3=1, R3=4, V3=6

[3,4,4,4],abgb

1 doublet and 2 singlet corners, doublet and both singlets all form their own vertices.

Can be made by replacing each face of a cube with four kites.

Catalan solid, dual of the rhombicuboctahedron

 

Deltoidal Hexecontahedron / Trapezoidal Hexecontahedron / Strombic Hexecontahedron

(Y)

N=60, V=62, E=120

M=4, M1=2, R1=4, V1=30, M2=1, R2=3, V2=20, M3=1, R3=5, V3=12

[3,4,5,4],abgb

1 doublet and 2 singlet corners, doublet and both singlets all form their own vertices.

Can be made by replacing each face of a regular dodecahedron †with five kites.

Catalan solid, dual of the rhombicosidodecahedron.

 

Trapezoid

2 adjacent duplets???

 

Irregular Quadrilateral

Skewed Trapezoidal Dihedron / Trapezohedron / Antidipyramids / Deltohedron

deform variant of the Trapezohedron (V)

N even (N>=6), M=4, M1=3, R1=3, V1=N, M2=1, R2=N/2, V2=2

4 singlet corners, but only 2 types of vertices. Vertex type 2 consists of identical corners (a), vertex type 1 consists of different corners (b,c,d).

 

Examples:

Skewed Trigonal Trapezohedron (N=6)

Lengths a-d and a-b are identical.

 

Skewed Hexagonal Trapezohedron (N=12)

Lengths a-d and a-b are identical.

 

Skewed Rhombic Dodecahedron

N=12, M=4, R1=4, V1=6, R2=R3=3, V2=V3=4

deform variant of (W)

4 singlet corners, but only 3 types of vertices. Vertex type 1 consists of identical corners (a), vertex type 2 consists of corners (b,b,c), vertex type 3 consists of corners (c,d,d).

Length b-c and c-d are identical

 

??? missing in loki3ís table???

 

 

Dyakis Dodecahedron / Didodecahedron / Diploid

N=24, M=4, M1=2, R1=4, V1=12, M2=1, R2=3, V2=8, M3=1, R3=4, V3=6, V=26, E=48

[3,4,4,4],aabg

Two adjacent sides are equal.

deform variant of the Deltoidal Icositetrahedron (X)

 

missing

Skewed Deltoidal Hexecontahedron

N=60, M=4, R1=4, V1=30, R2=3, V2=20, R3=5, V3=12

deform variant of (Y)

??? missing in loki3ís table???

 

Pentagons (M=5)

Regular Pentagon

Regular (Pentagonal) Dodecahedron

(H)

N=12, M=5, R=3, V=20, E=30

[3,3,3,3,3],aaaaa

Only one type of vertex with 3 edges.

Platonic solid, dual of the icosahedron

 

 

Irregular Pentagon

Octahedral Pentagonal Dodecahedron /

Pentagonal Dodecahedron /

Pyritohedron

(Z)

N=12, M=5, M1=3, R1=3, V1=12, M2=2, R2=3, V2=8, V=20, E=30

[3,3,3,3,3],abbbb

2 doublets and 1 singlet, one doublet and the singlet are part of the same type of vertex.

A deform dodecahedron which has a symmetry that mirrors the octahedron.

Note that 4 edges of the pentagon are of equal length with bilateral symmetry. The vertices of Type 2 form a cube.

 

include pix w/ corners

Tetragonal Pentagonal Dodecahedron / Tetartoid

(∆)

N=12, M=5, M1=3, R1=3, V1=12, M2=M3=1, R2=R3=3, V2=V3=4, V=20, E=30

[3,3,3,3,3],abbgg

All singlet corner types, one type of vertex with three corner types, and two vertices with one corner type.

Skewed dodecahedron which looks similar to the Tetrahedron.

 

There are no parallel faces. The design by loki3 consists of a wireframe with embedded parallel sides.

 

Pentagonal Icositetrahedron / Pentagon Trioctahedron / †Gyroid

(ō)

N=24, M=5, M1=3, R1=3, V1=24, M2=1, R2=3, V2=8, M3=1, R3=4, V3=6, V=38, E=60

[3,3,3,3,4],abbgg

Two pairs of equal adjacent sides

All singlet corner types, one type of vertex with three corner types, and two vertices with one corner type.

Can be made by placing sets of four pentagons on each face of a cube, turned a bit to make the corners interlace.

No parallel faces.

Catalan solid, dual of the snub cube

Both enantiomorphous forms, known as laevo (left) and dextro (right) have the same edge code since they are mirror since they are mirror image versions of each other (thanks to loki3 for the hint)

 

 

Pentagonal Hexecontahedron

(Ň)

N=60, M=5, M1=3, R1=3, V1=60, M2=1, R2=3, V=20, M3=1, R3=5, V3=12, V=92, E=150

[3,3,3,3,5],abbgg

All singlet corner types, one type of vertex with three corner types, and two vertices with one corner type.

Can be made by placing sets of five pentagons on each face of a dodecahedron, turned a bit to make the corners interlace.

No parallel faces.

Catalan solid, , dual of the snub dodecahedron.

Both enantiomorphous forms, known as laevo (left) and dextro (right) have the same edge code since they are mirror image versions of each other (thanks to loki3 for the hint)

 

References

*         [1] Huge collection of shapes, including some paper models and prototypes not commercially available: http://www.dicecollector.com/diceinfo_how_many_shapes.html

*         [2] 3D pictures of isohera: http://mathworld.wolfram.com/Isohedron.html

*         [3] Wikipedia http://wikipedia.org/

*         [4] Mathematical derivation of all fair dice: http://web.archive.org/web/20080501074022/http://hjem.get2net.dk/Klaudius/Dice.htm (was: http://hjem.get2net.dk/Klaudius/Dice.htm ). Click here for a local copy in PDF format.

*         [5] Another mathematical derivation of all fair dice: http://dicephysics.info/thesis7.doc. Click here for a local copy in PDF format.

*         [6] Detailed description of all fair dice: http://www.mathpuzzle.com/Fairdice.htm

*         [7] International Bone Rollersí Guild, Mitchel D Klink http://web.archive.org/web/20071118155413/http://members.aol.com/dicetalk/ (was http://members.aol.com/dicetalk/polymore.htm )

*         [8] Polygon Names: http://www.math.com/tables/geometry/polygons.htm

*         [9] Isohedra (loki3): http://loki3.com/poly/isohedra.html

*         [10] Crystallographic Polyhedra, Steffen Weber http://jcrystal.com/steffenweber/

 

To Do

Add loki3ís type code.

 

Pix & corner codes of cube, regular octahedron, regular D10 dipyramid, regular dodecahedron, regular icosahedron, scalene tetrahedron, Pyritohedron, Tetartoid, Pentagonal Icositetrahedron, Pentagonal Hexecontahedron

 

Trapezoid with 2 adjacent duplets???

 

Check Skewed Rhombic Dodecahedron & Skewed Deltoidal Hexecontahedron

 

My wish list:

 

Composed of isosceles triangles:

D6 Trigonal Dipyramid with isosceles triangles

 

Composed of scalene triangles:

D24 Hexakis Tetrahedron: no parallel faces ŗ numbering???

 

Composed of irregular quadrilateral:

D12 Skewed Rhombic Dodecahedron ???

D60 Skewed Deltoidal Hexecontahedron ???

 

Check:

Pyritohedron M1=4?