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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.


Convex isohedra are  fair dice,  by symmetry.

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 Hexakis Icosahedron

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Fair Dice  by  Ed Pegg, Jr.   (Math Games, MAA, 2005-05-16).
Alea Kybos' Dice CollectionShapes.
Polyhedral Dice  by  George W. Hart (1999).
What shapes do dice have?  by  Kevin Cook  (largest collection of dice).

Wikipedia :   Dice  |  Isohedra

Video :   Weird but Fair Dice  by Henry Segerman  in "Numberphile" by Brady Haran (2016-04-11).


Isohedral Symmetry

(2013-04-14)   Classification of Convex Isohedra
All the faces of an  isohedron  are  equivalent.

An  isohedron  is a polyhedron whoses  faces  are all  equivalent.  That's to say that every face can be transformed into any other through some spatial isometry  (rotation or reflection)  that maps the polyhedron onto itself.

The dual of an isohedron is an isogonal polyhedron, and vice-versa  (duality  being understood with respect to the sphere inscribed in the isohedron or circumscribed to its dual).

The following complete classification is ordered by the number of faces in the relevant  (primal)  topology.

Our primary tool to reach this classification is just the enumeration of  uniform convex polyhedra  (prisms, antiprisms and 13 Archimedean solids)  published in 1619 by  Johannes Kepler (1571-1630).  We simply put to good use the following facts:

  • Every isogonal polyhedron  is patterned a uniform one, allowing  d  degrees of freedom  (d is the degree of every vextex).
  • An isohedron is the dual of an isogonal polyhedron  and vice versa  (the aforementioned  d  is thus the number of sides of every face).
  • The dual of a convex body is convex.

Still, the transformation by geometric duality of some topologically equivalent isogonal polyhedra reserves a few surprises, including the distinction between  radial  and  axial  amphihedra  (whose respective duals are two different types of non-equilateral isogonal prisms).

The two families with infinitely many topologies  (amphihedra and deltohedra)  are sometimes collectively known as  bipolar dice.  They start with 6-sided isohedra and are therefore listed after the disphenoids  (the 4-sided isihedra).

The classification avoids  overlaps,  So, a few familiar isohedra do not appear as such because they  are  members of larger families:

  • The  regular tetrahedron  is an equilateral  disphenoid.
  • The  cube  is a trigonal deltohedron.
  • The  regular octahedron  is a tetragonal dipyramid.

Thus, only two  Platonic solids appear in their own right, in a classification where only the following prototypes are retained:

  1. Disphenoid  (isogonal tetrahedron; equilateral, isoceles or scalene).
  2. Amphihedron  (axial, radial or dipyramidal).
  3. Deltohedron  (chiral or not).
  4. Pentagonal Dodecahedron  (regular, octahedral or tetrahedral).
  5. Rhombic Dodecahedron  &  Trapezoidal Dodecahedron
  6. Triakis Tetrahedron
  7. Regular Icosahedron
  8. Hexakis Tetrahedron
  9. Tetrakis Hexahedron
  10. Triakis Octahedron
  11. Trapezoidal Icositetrahedron
  12. Pentagonal Icositetrahedron
  13. Dyakis Dodecahedron
  14. Rhombic Triacontahedron
  15. Hexakis Octahedron
  16. Triakis Icosahedron
  17. Pentakis Dodecahedron
  18. Trapezoidal Hexecontahedron
  19. Pentagonal Hexecontahedron
  20. Hexakis Icosahedron  (includes  disdyakis triacontahedron ).

Classification of Convex Isohedra
( * )
Isosceles Dipyramidh,d2n+6
Radial Amphihedronh,d,a4n+8
Axial Amphihedronh,d,a4n+8
(Achiral) Deltohedronh,d2n+6
Chiral Deltohedronh,d,q
Pentagonal Dodecahedronh,d12
Octahedral Dodecahedron
Tetrahedral Dodecahedron
Rhombic Dodecahedron12
Deltoidal Dodecahedron
Triakis TetrahedronTriakis Tetrahedron12
IcosahedronRegular Icosahedrond20
 Come back later, we're
 still working on this one...
( * )   The  parms  column primarily indicate the number of  parameters  needed to fully specify the geometric shape of the isohedron  (including its size).  The suggested names are not essential:  h for height, d for diameter, etc.

Isohedra (MathWorld)   |   Fair Dice  by  Ed Pegg, Jr.
29 Fair Dice  by  Klaus Æ. Mogensen
Fair Dice  &  Classification of Isohedra  by  Scott Shermann (2012).

 Isogonal Tetrahedron (2013-04-19)   Disphenoids   (Monge, 1809)
Tetrahedra where nonadjacent edges are equal.

A tetrahedron is isohedral if and only if it's isogonal.  This  two-parameter  family is often  needlessly  split up into  3  subcategories  (equilateral, isosceles and scalene).  The scalene ones are the simplest  chiral  isohedra.

The  regular tetrahedron  is  a disphenoid.

 Gaspard Monge 
 1746-1818 Disphenoids  were first investigated in 1809, by Gaspard Monge (1746-1818)  who called them  isosceles tetrahedra.  This alternate name, which is still used, shouldn't be confused with the  (fairly useless)  special case of  isosceles disphenoids,  which denotes disphenoids where at least four edges have the same length.

Surprisingly enough, a disphenoid can also be characterized as a tetrahedron for which  either one  of the following two conditions holds:

  • The insphere and circumsphere are concentric.
  • The sum of the face angles at every vertex is  180°.

Wikipedia :   disphenoid.

(2013-04-20)   Amphihedra
An  amphihedron  is an isohedron with the topology of a dypyramid.

An  amphihedron  is defined as an isohedron which has the topology of an  N-gonal  dipyramid  (or "bipyramid").  The amphihedra corresponding to  odd  values of  N  are necessarily straight symmetrical dipyramid  (as listed first).  For  even  values of  N  however, there are two other kinds of amphihedra  (listed last)  whose numbers of faces are divisible by 4.  Triangular 

  • N-gonal symmetrical dipyramids  (featuring an N-hedral axis of symmetry perpendicular to a plane of symmetry).   Octahedron This family depends on a discrete parameter  (N)  and   a continuous one  (the aspect ratio ).   Pentagonal 
 Dipyramid The  regular octahedron  (D8)  shown at left belongs to this family  (it's a tetragonal dipyramid).
  • Axial amphihedra  each obtained from a 2N-gonal symmetrical dipyramid by moving equatorial points alternately up or down by the same nonzero distance in the direction of the axis of symmetry.  This family depends on one discrete parameter and two continuous  (positive)  ones.
  • Radial amphihedra  each obtained from a 2N-gonal symmetrical dipyramid by moving every other equatorial point radially outward by a nonzero distance  (small enough to maintain convexity).  This family depends on one discrete parameter and two continuous  (positive)  ones.

A priori, one might have expected that the two-parameter  axial  and  radial  families of amphihedra could be combined into a single three-parameter family by allowing at once both a radial and axial component of the displacements involved involved.  This ain't so  (HINT: .../... ).

(2013-04-20)   Deltohedra  (a.k.a  trapezohedra )
There are  chiral  and  achiral  isohedral deltahedra.

  Pentagonal Deltohedron An N-gonal deltohedra  is obtained as the intersection of two congruent opposing coaxial  rotationally misaligned  N-gonal symmetrical pyramids of unbounded extend  (if those two were aligned, a dipyramid would be obtained instead).  Such polyhedra are  chiral  unless the rotational misalignment is at the midway point.   Cube

This family of isohedra depends on one discrete parameter and two continuous ones.  The cube,  by far the most commonly used die,  belongs to this family  (it's just an achiral trigonal deltohedron with a special aspect ratio).  So do the 10-sided dice  (D10)  that are standard in role-playing games   (they are achiral  pentagonal deltohedra).

 Hexakis Icosahedron
(2013-04-16)   Hexakis Icosahedra
Fair dice with  120  faces.

Hexakis icosahedra  form a two-parameter family of isohedra featuring:

  • 120 triangular faces.
  • 180 edges  (in three classes).
  • 62 vertices  (in three classes).

Every such isohedron can be obtained from a  regular icosahedron  by creating a new vertex above the center of every face and above the center of every edge.  Both displacements should be small enough to preserve convexity.

The isohedral  disdyakis triacontahedra  are special cases of isohedral  hexakis icosahedra  obtained from isohedral rhombic triacontahedra by pasting a tetragonal pyramid on every face.  This entails only one degree of freedom  (the common height of the 30 added pyramid)  which goes to show that not all isohedral hexakis icosahedra can be constructed this way  (since the whole family has  two  degrees of freedom).  Great Rhombicosidodecahedron

Among those, only  one  shape is canonical:  It's a  Catalan solid  best described as the  geometric dual  of the  uniform great rhombicosidodecahedron  shown at right  (one of the  13  Archimedean solids).

From a distance, all  isohedral hexakis icosahedra  look the same but they're not created equal  (don't believe the Wikipedia editors who don't bother to make any distinctions at the outset).

(2013-02-02)   Chiral isohedra with  achiral  faces :
Two  chiral isohedra  have mirror-symmetric pentagonal faces.

There are only two  chiral  Archimedean solids, pictured below:  The  snub cube  (at left)  and the  snub dodecahedron  (at right).

 Snub Cube  Snub Dodecahedron  

Their respective duals are the  pentagonal icositetrahedron  (24 faces, 60 edges, 38 vertices)  and  pentagonal hexacontahedron  (60 faces, 150 edges, 92 nodes).  The French term is  hexacontaèdre;  the corrupted spelling  "hexecontahedron"  is dominant in modern English.

 Pentagonal Icositetrahedron    Pentagonal Hexecontahedron
Pentagonal icositetrahedron
Wikipedia   |   MathWorld
Pentagonal hexecontahedron
Wikipedia   |   MathWorld

Those two are  chiral  isohedra  whith pentagonal faces.  Chirality  means that they don't have any mirror symmetry.  Isohedrality  means that no face can be distinguished from any other.

There are other types of chiral isohedra  (scalene disphenoids being the simplest example)  but they feature  chiral faces.  Those two don't.

(2013-02-23)   Non-Isohedral Fair Dice ?
Is the  pseudo deltoidal icositetrahedron  a fair die?

The requirement that a fair die should be both orthohedral and equispherical is enough to pronounce as unfair almost all nonisohedral dice.

However, even a die whose faces are all congruent isn't guaranteed to be fair unless it's isohedral.  The simplest such example to investigate is the dual of the J37 Johnson solid  (the  elongated square gyrobicupola ).  Notoriously, J37 is an equiradial polyhedron which isn't isogonal, although every vertex is surrounded by the same configuration of four faces.

Although the  24  faces of  Q24  are congruent to the same quadrilateral, this  quasifair  die isn't an isohedron.  Specifically, its faces are divided into two distinct equivalence classes:  8  polar  faces and  16  equatorial  faces.  A polar face is  not  equivalent to an equatorial one.

Strombic Icositetrahedron Pair  by  Robert Webb  (author of Stella).   |   J37 Dual (Polytope Explorer)
Wikipedia :   Pseudo-uniform polyhedra   |   Pseudo-deltoidal icositetrahedron   |   Dual polyhedron

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