(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).
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:
Disphenoid (isogonal tetrahedron; equilateral, isoceles or scalene).
Amphihedron (axial, radial or dipyramidal).
Deltohedron (chiral or not).
Pentagonal Dodecahedron (regular, octahedral or tetrahedral).
( * ) 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.
(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.
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°.
(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.
N-gonal symmetrical dipyramids
(featuring an N-hedral axis of symmetry perpendicular to a plane of symmetry).
This family depends on a discrete parameter (N) and
a continuous one (the aspect ratio ).
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.
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.
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).
(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).
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 chiralArchimedean solids,
pictured below:
The snub cube (at left) and the snub dodecahedron (at right).
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.
Those two are chiralisohedra
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.
(2013-04-19) Disphenoids (Monge, 1809)
This family depends on a discrete parameter (N) and
a continuous one (the 
The regular octahedron (D8) shown at left belongs to this family
(it's a tetragonal dipyramid).
