Newton Polyhedra & Polynomial Equations, by Nahum Zobin |
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Polyhedra (3D), Polychora (4D), Polytopes (nD)

(Jerry of Nashville, TN.
2000-11-18)
What [polyhedron] has six faces?

A polyhedron with 6 faces is a hexahedron.
The cube
is the best-known hexahedron, but it's not the only one:
Disregarding geometrical distortions and considering only
the underlying topology, there are 7 distinct hexahedra:

Name of Hexahedron

Edges

Nodes

Triangular Dipyramid

9

5

Pentagonal Pyramid

10

6

Tetragonal Antiwedge

10

6

Hemiobelisk

11

7

Hemicube

11

7

Cube

12

8

Pentagonal Wedge

12

8

The above triangular dipyramid
has 5 vertices and 9 edges. It's the
dual of a triangular prism,
and looks like two tetrahedra "glued" on a common face.

The pentagonal pyramid
has 6 vertices and 10 edges; it's a pyramid whose base is a pentagon.
Like all pyramids, the pentagonal pyramid is self-dual.

The above three hexahedra are the only ones which exist in a version where all 6 faces
are regular polygons.

The tetragonal antiwedge is the least symmetrical of all hexahedra;
its only possible symmetry is a 180° rotation.
This skewed hexahedron has
the same number of edges and vertices as the pentagonal pyramid.
Its faces consist of 4 triangles and 2 quadrilaterals.
Such a solid is obtained from two quadrilaterals
that share an edge [the hinge]
but do not form a triangular prism.
After adding two edges to complete the two triangles whose sides are adjacent to
the hinge, we are left with a nonplanar quadrilateral
and must choose one of its 2 diagonals as the last edge of the polyhedron.
Only one choice gives a convex polyhedron.

Loosely speaking, there are two types of
tetragonal antiwedges which are mirror images of each other;
each is called an enantiomer, or enantiomorph of the other.
The tetragonal antiwedge is the simplest example of a chiral
polyhedron.
Any other hexahedron can be distorted into a shape
which is its own mirror image, and the tetragonal antiwedge
may thus unambiguously be called thechiral hexahedron.
Each enantiomer is self-dual;
a tetragonal antiwedge and its dual have the same chirality.

The other types of hexahedra are more symmetrical and simpler to visualize.
One of them may be constructed by cutting off
one of the 4 base corners of a square pyramid to create a new triangular face.
This hexahedron has 7 vertices and 11 edges.
Its faces include 3 triangles, 2 quadrilaterals and 1 pentagon.
It could also be obtained by cutting an elongated square pyramid
(the technical name for an obelisk)
along a bisecting plane through the apex of the pyramid and the
diagonal of the base prism, as pictured at right.
For lack of a better term, we may therefore call this hexahedron an
hemiobelisk.

Also with 7 vertices and 11 edges, there's a solid which we may call
a hemicube (or square hemiprism), obtained
by cutting a cube in half using a plane going through two opposite corners and
the midpoints of two edges. Its 6 faces include 2 triangles and
4 quadrilaterals.

The cube
(possibly distorted into some kind of irregular prism or truncated tetragonal pyramid)
isn't the only hexahedron with 8 vertices and 12 edges:
Consider a tetrahedron, truncate two of its corners and you have a
pentagonal wedge.
It has as many vertices, edges and faces
as a cube, but its faces consist of 2 triangles,
2 quadrilaterals and 2 pentagons.

We can build a pentagonal wedge with 2 regular pentagons and
2 equilateral triangles, so that all edges but one are equal.
The one "exceptional" edge is the longest side in the two trapezoidal faces.
What's its length? Well, look at the
wedge "from the side" (so both pentagons project into a line) you see two similar
isosceles triangles. The base of the smaller is a regular edge seen perpendicularly
(and therefore at its real size), whereas the base of the larger triangle is the
length we're after.
The ratio of similitude is simply the ratio of the height of a regular pentagon
to the distance from a side to an adjacent vertex, namely
1+sin(p/5)/sin(2p/5) =
(1+Ö5)/2.
A number known as the golden ratio,
which happens to be the ratio of the diagonal to the
side in a regular pentagon. The longest edge in our solid is thus
1.6180339887498948482... times the length of any other.
In other words, both trapezoidal faces are congruent to the diagonal
section of a pentagonal face (pictured at right).

(2007-09-19)
The fattest tetragonal antiwedge isn't a proper one !
Looking for the chiral hexahedron of unit volume and least surface area.

Let's consider convex chiral hexahedra
(see above)
possessing the only allowed symmetry
(namely a 180° rotation about an axis perpendicular
to the hinge where the two tetragonal faces meet).
If the hinge AB has a length of 2 (arbitrary) units,
the shape of such a symmetrical tetragonal antiwedge is
determined by 5 positive parameters (u,v,x,y,t
with y>v for convexity)
giving the coordinates of the 6 vertices in a suitable cartesian system:

A =

-a 0 0

B =

a 0 0

C =

u v -vt

D =

x y yt

E =

-x y -yt

F =

-u v vt

If we call 2q the dihedral
angle between the two tetragonal faces, then:

t = tan q
and
1 + t^{ 2} = 1 / cos^{2 }q

The volume V
is the sum of 2
pairs of tetrahedra (each having O as a vertex).

V = ^{1}/_{3} [ det (B,C,D) + det (D,E,F) ]
= ^{2}/_{3} yt ( av + uy + vx )

The above computation can be done mentally, using the preceeding layout
which features the relevant triplets of columns adjacent to each other
(and in the right order). The first
determinant (2avyt) is trivial.
The second one is quite simple too, when computed as det (D,D+E,F).

Expressing the areas of triangles and quadrilaterals
as cross-products
(and using pairwise equalities)
we obtain the total surface area S of the hexahedron:

S =

|| BC ´ BD ||
+
|| DE ´ DF ||
+
|| AD ´ BF ||

Where :

|| BC ´ BD ||^{ 2}
=

(1 + t^{ 2 }) [ (u-a) y + (a-x) v ]^{ 2} + 4 (vyt)^{ 2}

The maximum volume for a given surface (or, equivalently, the minimum
surface for a given volume) is obtained when the
differential forms dV
amd dS are proportional
(the coefficient of proportionality is the Lagrange
multiplier associated to whichever quantity is considered a
constraint under which the other is to be optimized).

Thus, the distance between A and B must vanish.
This is to say that the fattest symmetrical tetragonal
antiwedge actually degenerates into a triangular dipyramid.
(There must be an easier way to reach that conclusion.
A tantalizing conjecture is that no chiral polyhedron can be
the fattest of its own kind.)

(2000-11-28) Polyhedral Duality
The faces of a polyhedron correspond to the vertices of its dual.

The duality of polyhedra is an involutive
relationship (the dual of the dual is the original polyhedron)
which can be defined either in abstract terms
(topologically) or in more concrete geometrical terms.
When discussing two polyhedra that are duals of each other, it's
convenient to identify one as the primal
and the other as the dual,
but the two rôles could be switched:

Topological Duality :

The dual of a polyhedron is the polyhedron obtained
by switching the rôles of vertices and faces:
Edges of the dual connects nodes associated with adjacent primal faces
(dual polyhedra have the same number of edges).

Geometric Duality : [ skip on first reading ]

Motivation :
The above topological relationship
holds between any convex polyhedron and its
polar (relative
to any center O inside it).
Therefore, the polar of a convex polyhedron is a proper geometric
embodiement of its topological dual, called the
geometric dual.

Elegant and fundamental as it may be, the polar transformation
doesn't generalize immediately to nonconvex
polyhedra. However, the following elementary construction
does (which rely on the lemma proven
below). It's equivalent to the polar transform in the
convex case and always defines the precise geometric characteristics
of a polyhedron whose topology is the dual of that of the original
(i.e., primal) polyhedron, in the above sense.
So, we may as well take this as the general definition
of the geometric dual of a given polyhedron
(convex or not) with respect to some (arbitrary)
sphere of center O and radius R
(the value of R merely provides a scaling factor which is
often considered irrelevant):

Definition / Construction :
Consider the orthogonal projection H of O
onto the plane of a primal face.
The dual vertex associated to that primal face is, by definition, the point
M of the ray OH such that:

OH . OM = R^{2}

To every primal edge between two primal faces correspond
a dual edge connecting the two dual vertices so associated with those primal faces.

Modern Viewpoint :
Using the geometric concept of inversion
introduced by Steiner in 1826,
we may state that the vertices
of the dual polyhedron and the projections of the center onto [the planes of]
the faces of the primal polyhedron are inverses of each other.

What makes the above construction "work" is the following geometric
fact: Lemma : In a plane, Let I be a point such that
IH is perpendicular to OH and IH' is perpendicular to OH'.
If M and M' are points respectively located on OH and OH',
then I, M and M' are aligned if the following numerical relations hold:

OH . OM =
OH' . OM' = OI^{2}

Proof : Consider the configuration
of O,I,H and M if IH is perpendicular to HM (and/or to OH, since
O, H and M are aligned). IM is then a diameter of the circle containing
IH and M.

The power of point O with
respect to that circle is OH.OM. It's also equal to OI^{2}
if and only if OI is tangent to the circle, which is to say, if and only if
OI is perpendicular to the diameter IM.

The same argument holds for O,I,H' and M'.
So, if the advertised numerical equalities hold, then IM and IM'
are perpendicular to the same line (OI) which means that I, M and M' are aligned.

For a given edge of a polyhedron, we may apply the lemma to the orthogonal projection I
onto that edge of the (arbitrary) center of inversion O
in the plane orthogonal to the edge (that plane contains the orthogonal projections
H and H' of O on both of the faces adjacent to the featured edge).

The dual edge MM' thus belongs to the plane perpendicular to
the primal edge
(that conclusion holds for every edge separately, whether or not
the inversion radius is equal to OI, which we chose for convenience).

A so-called canonical polyhedron is endowed with
a midsphere tangent to all its edges.
In that special case, the midsphere is the preferred inversion sphere
(or invariant sphere) for the above construction
of the dual.
The dual of a canonical polyhedron is canonical and both
share the same midsphere. Every primal edge intersect its dual at right
angle at the point where both are tangent to the midsphere.

The canonical case is important because any polyhedron is topologically
equivalent to a canonical polyhedral shape (uniquely defined, modulo
a rotation and a scaling and/or translation).

(2000-11-28) Enumeration of Polyhedra
How many polyhedra have a given number of faces and/or edges?

Polyhedra which are mirror images of each other are not counted as distinct.
In the above, we counted 7 types of hexahedra.
That would be 8 if both chiralities of the tetragonal antiwedge
were tallied.

There's only one tetrahedron:

There are two types of pentahedra:
the triangular prism and
the square pyramid.

There are 7 hexahedra (see previous article),
34 heptahedra, 257 octahedra, 2606 enneahedra, 32300 decahedra,
440564 hendecahedra, 6384634 dodecahedra, 96262938 tridecahedra, 1496225352 tetradecahedra,
etc.

BenO (Ben Ocean, 2001-07-28; e-mail)
Platonic Solids
What are the x,y,z coordinates of the vertices of the 5 Platonic Solids?

Up to rotation and/or scaling, there are only
5 convexregular polyhedra.
These very special polyhedra are known as
Platonic solids.
(See below for a generalization to n dimensions.)

Regular Tetrahedron

A

B

C

D

x =

2

2

-4

0

y =

2Ö3

-2Ö3

0

0

z =

-Ö2

-Ö2

-Ö2

3Ö2

Shown at right are the Cartesian coordinates of the vertices
of a regular tetrahedron ABCD centered at the origin.
These may be scaled and/or rotated.
As given, this tetrahedron has:

A side equal to 4Ö3.

A height equal to 4Ö2.

A circumscribed sphere of radius
3Ö2.

An alternate set of Cartesian coordinates for a smaller regular tetrahedron
(of side 2Ö2, inscribed in a sphere of radius
Ö3)
would consist of every other vertex
in a cube of side 2 (see below) centered at the origin.
This highlights some geometrical relationships and dims others:

A = [+1,+1,+1], B = [+1,-1,-1], C = [-1,+1, -1], D = [-1,-1,+1]

In such a regular tetrahedron, two vertices are seen from the center at an angle
known as the tetrahedral angle (which is very familiar to chemists)
whose cosineis -1/3 and whose value is 109.47122°...
The dihedral angle between a pair of faces is supplementary to
that angle; its cosine is 1/3 and its value is about 70.52878°
(this may be called a cubic angle, for a reason which follows).
Expressed in radians, three times this angle minus a flat angle
(p) gives the value [in steradians, sr]
of the solid angle at each corner of the tetrahedron, namely 0.55128559843...
This is about 4.387 % of the solid angle of a whole sphere
(4p).
Astronomers may use the square degree as a unit of solid angle
(a square degree equals
p^{2}/180^{2} sr);
the solid angle at the corner of a regular tetrahedron is
1809.7638632... square degrees
(or 6515150 square minutes).

We may choose 3 coordinates from the set
{-1,+1} in 8 different ways.
Those correspond
to the coordinates of the 8 vertices of a cube of side 2,
centered at the origin
(and inscribed in a sphere of radius Ö3).
Seen from the center of a cube, the angular separation between corners is
either a flat angle (180° between diametrically opposed vertices),
a tetrahedral angle of cosine -1/3
(about 109.47° between the opposite corners in a face),
or a cubic angle whose cosine is 1/3 and which is
supplementary to a tetrahedral angle (about 70.53° between adjacent corners).
The solid angle at each corner of a cube is clearly p/2, namely
1/8 of a whole sphere (4p).

There are 6 ways to choose 3 coordinates from the set {-1,0,+1}
so that only one of them is nonzero.
These correspond to the coordinates of the 6 vertices
of a regular octahedron of side
Ö2 centered at the origin
(and inscribed in a sphere of radius 1).
Seen from the solid's center, two vertices are separated either by
a right angle (90°) or by a flat angle (180°).

The volume
of a regular dodecahedron is (15+7Ö5)/4
times the cube of its side.
The dihedral angle has a cosine of
-1/Ö5 and a value of about 116.565°

(2013-01-30) Symmetrical Polyhedra
Polyhedra having more than one isometric automorphism.

A polyhedron is symmetric if it's stable under at least one nontrivial
isometric transformation (i.e., a mirror reflection or a nonzero rotation).

As previously remarked, a tetragonal antiwedge
can only have one such symmetry.
More complicated polyhedra are usually studied only if they have
so many symmetries that there are only
very few different types of nodes, edges or faces to consider...

Two commensurate components of a polyhedron (e.g., two vertices, two edges or two faces) are
said to be equivalent if there's an isometry (rotation or reflection)
which transforms one into the other.
When this results in only one equivalence class the following adjectives qualify
the polyhedron:

Isogonal (or vertex-transitive) when all vertices are equivalent.

Isotoxal (or edge-transitive) when all edges are equivalent.

Isohedral (or face-transitive) when all faces are equivalent.

Isochoric (or cell-transitive) when all 3-cells are equivalent.

Since all polyhedra possess only one 3-cell,
the term isochoric
applies vacuously to all of them (it's only interesting for polychora
or other polytopes in more than three dimensions).
An isohedral polyhedron is called an isohedron.
Convex isohedra are intrinsically fair dice.

A polyhedron is said to be regular when it possesses
all of the above symmetries (in more than three dimensions, a polytope is
said to be regular when all its
flags are equivalent).
Other special terms denote polyhedra whose symmetries go beyond
at least one of the above:

Quasiregular polyhedra are isotoxal and isogonal.

Parahedra polyhedra are isotoxal and isohedral (neologism).

Noble polyhedra are isogonal and isohedral.
(Hess & Bruckner, 1900)

"Semi-regular" is an ambiguous term, usually meaning uniform.

The only regular convex polyhedra are the
five Platonic solids.
There are also four nonconvex regular polyhedra,
dubbed Kepler-Poinsot
polyhedra (two pairs, obtained by
stellation of the regular
convex dodecahedron and the regular convex icosahedron).

Quasiregular polyhedra are uniform.
The converse isn't true:
Besides the cube and octahedron, uniform prisms and antiprisms aren't isotoxal.
Besides platonic solids, only two convex
quasi-regular polyhedra exist: The cuboctahedron and
icosidodecahedron
(other Archimedean solids have several types of edges).

The duals of these are the two rhombic polyhedra
which are isotoxal
and isohedral (we call them isorhombic)
but not isogonal, namely the
rhombic dodecahedron
(at left) and the
rhombic triacontahedron
(at right).

The only non-regular convex noble polyhedra are
the disphenoids.

Inertial Symmetry :

For completeness, let's mention the type of symmetry
that makes moments of the polyhedron with respect to something
invariant under transformations of that thing.
The only example with any recognized practical importance
pertains to moments of inertia: A solid is
said to be inertially symmetric when it has the same moment of inertia
with respect to any axis (and/or any plane) containing its center
of gravity. This happens when the three eigenvalues of
its tensor of inertia are identical
(physicists call such a tensor scalar ).
This characterization would make it possible to classify inertial symmetry
among the equimetric properties discussed next.

(2013-02-13) Equimetric Polyhedra
Polyhedra in which some commensurate measures are constant.

In this context, the qualifier commensurate is simply
used for things that can be compared to each other (it doesn't
make any sense to compare an edge to a face, for example).
A measure is a numerical function;
it's said to be constant when it's the same for all elements.

The prefix equi- indicates equality of some specific measures
of such things, whereas the prefix iso- (which dominates
the previous section)
indicates their complete equivalence, with respect to any
possible measure or criterion, on account of a global symmetry.

Arguably, the term "equifacial" belongs to neither category.
It denotes a polyhedron whose faces are all congruent
(they need not be equivalent).
The simplest example of an equifacial polyhedron which isn't isohedral
is the pseudo strombic icositetrahedron.
The lesser requirement of faces of equal area seems of little or no interest
(if needed, the qualifier "equiareal", listed below, could be used to indicate this).

Simple examples of the measures that can be used to qualify polyhedra as
equimetric include the length of all edges or the surface areas of all faces.

Other measures involve a prescribed point,
which will naturally be called a center in case of
equimetry.
Examples of such central measurements, include radial distances (to vertices, edges or faces)
angles subtended by edges,
solid angles subtended by faces, etc.

In symmetrical cases
the centers with respect to different measures will often coincide with some
center of symmetry, but this needn't be so in general...
There may even be several centers with respect to
which a given measure is the same for all relevant elements.
One example is the distance to [the planes of] faces in any tetrahedron:
There are 5 different centers that are equidistant to all 4 faces.

Among the lesser-investigated measures are various moments
related to a central point. That category
includes the volume of so-called radial pyramids
(apex at the center and one face as a base) or the area of
radial triangles (apex at the center and one edge as a base).

A traditional nomenclature exists for some equimetric concepts.
Others are best denoted by neologisms of recent origin:

In equiradial polyhedra, all vertices are equidistant from the center.

In canonical polyhedra, all edges are equidistant from the center.

In orthohedral polyhedra, all faces are equidistant from the center.

In equilateral polyhedra, all edges have the same length.

In equiareal polyhedra, all faces have the same surface area.

In equicircular polyhedra, all edges subtend the same angle.

In equispherical polyhedra, all faces subtend the same solid angle.

In equitrigonal polyhedra, all radial triangles have the same area.

In equipyramidal polyhedra, all radial pyramids have the same volume.

Generally, the distance
of a point to a set is the smallest distance
from that point to a point of the set (more correctly,
the greatest lower bound of such distances). However,
in the context of polyhedra, the distance to a face or an edge is
understood as the distance to the relevant linear support
(i.e., the plane of a face or the line containing an edge).
That's always obtained as the distance to the center's orthogonal projection.
Here are a few remarks:

An equiradial polyhedron is inscribed in a sphere,
its circumscribed sphere
(whose radius is dubbed circumradius ).
Its dual is orthohedral.
Some authors call
equiradial polyhedra spherical,
although spherical polyhedra are normally understood to be figures drawn
on the surface of a sphere, with edges represented by arcs of great circles
instead of their chords (the latter representation is degenerate or ambiguous
for the monogonal or digonal faces of a spherical polyhedron).

A canonical polyhedron is one that possesses a midsphere
(i.e., a sphere to which all edges are tangent).
The distance from any edge to the solid's center is called midradius.
Isotoxal polyhedra are canonical by symmetry.
So are quasiregular polyhedra, for which
the tangency points are the middles of all the edges.

An orthohedral polyhedron
(or orthohedron) is circumscribed to a sphere,
called its inscribed sphere
(whose radius is dubbed inradius ).
Its dual is equiradial.
In the special case of a tetrahedron,
a distinction is made between the inscribed sphere
inside the tetrahedron and four other escribed spheres
located outside the tetrahedron and tangent to the four planes of its faces.

The term equicircular refers to the circular measure
of planar angles (note that
equiangular
denotes a different noncentral concept).

(2001-07-28) Archimedean solids Catalan solids are the duals of Archimedean polyhedra.

Two vertices of a polyhedron are called equivalent
if one is the image of the other in an isometric transformation (rotation or reflection)
of the polyhedron unto itself. If all its vertices are equivalent,
a polyhedron is said to be isogonal.
If an isogonal polyhedron is also equilateral,
it's said to be uniform.

The other convex uniform polyhedra are
known as Archimedean solids.

All the vertices of an Archimedean solid have the same
arrangement of faces around them, but this condition isn't sufficient.
For example, the elongated square gyrobicupola
(J37,
pictured at left) is not Archimedean
(it's known as the pseudo-rhombicuboctahedron ).

The pseudo-rhombicuboctahedron has been
rediscovered many times, by energetic amateurs or seasoned professionals,
as a supposedly "forgotten" or "overlooked" 14-th Archimedean solid...
As late as 2012,
Thomas C. Hales (1958-)
saw fit to lament about how "the pseudo rhombic cuboctahedron has been overlooked or
illogically [sic] excluded from [convex Archimedean polyhedra]".
He calls that "one of the most persistent blunders in the history of
mathematics" (no less).

To preserve the distinction between actual symmetries and superficial clues,
I beg to differ from the isolated opinion of Pr. Hales.
The duals of Archimedean solids are strict isohedra;
the dual of J37 isn't.
As such, it's not guaranteed to be a fair die, in spite of the fact that all its faces are congruent:
Thoses faces are partitioned into two separate equivalence classes
(8 "polar" faces and 16 "equatorial" ones)
which don't play the same rôle as the die rolls on an horizontal surface.
This may result in a slight statistical bias about the nature of the face (polar or equatorial)
that the die will eventually land on.

There are 13 Archimedean solids (two of which are chiral, the snub
cube and snub dodecahedron).
Archimedes of Syracuse (c.287-212 BC) may have discovered them all,
but only 12 of them were known during the Renaissance.
Kepler (1571-1630) added the snub dodecahedron
when he reconstructed the whole set in 1619.

The truncated icosahedron (the shape of a traditional soccer ball)
is now more commonly known as a buckyball ever since it was found to be
the structure of a wonderful new molecule,
now called fullerene (C60)
in honor of the famous American architect R. Buckminster ("Bucky") Fuller (1895-1983),
who created and advocated
geodesic domes
in the late 1940s.

The buckyball is one of 4
Archimedean
solids without triangular faces.
The other three are the truncated octahedron (at left),
the great rhombicosidodecahedron (at right)
and the great rhombicuboctahedron.

The 4 Archimedean polyhedra illustrated
so far are simplicial
(i.e., only 3 edges meet at each vertex).
There are 3 others such simplicial polyhedra, illustrated next,
which happen to be obtained
(like the buckyball and truncated octahedron above)
by truncating a Platonic solid.

This leaves 4 nonchiral Archimedean solids with vertices of degree 4 :

Finally, 5 edges meet at every vertex of the two chiral
Archimedean polyhedra:

Pappus attributes
the list to Archimedes.
The snub dodecahedron (above right) was lost until
Johannes Kepler (1571-1630)
reconstructed the entire set, in 1619.

(2013-04-22) Isogonal polyhedra generalize uniform ones.
Every uniform polyhedron typifies a d-dimensional isogonal family.

By definition, in an isogonal polyhedron all vertices
are equivalent.
The edges around every vertex can be given d labels.
Let's show that the coordinates of each vertex are porportional to an affibe function
of the length associated to each label.

(2000-11-19) Types of polyhedra named after a polygon :

Take a regular polygon (an hexagon, say)
and construct a polyhedron by considering an identical copy of that hexagon
in a parallel plane.
Join each vertex of the hexagon to the corresponding
vertex in its copy and you obtain what's called an hexagonal prism.
Instead, you may twist the copy slightly and join each vertex to the two
nearest vertices of the copy. What you obtain is an hexagonal antiprism.
In such families, the polyhedron is named using the adjective corresponding to
the name of the polygon
it's built on (e.g., "hexagonal").

There are several other families besides
prisms and antiprisms
for which this pattern applies.
For example, if you cut a prism with a plane containing some edge of either
base polygon (but not intersecting the other), this "half" prism is called
a wedge (it includes the base polygon and its featured edge).

Alternately, if the cutting plane contains only a single vertex,
instead of a whole edge,
the polyhedron we obtain by cutting a prism is an hemiprism.

A deltohedron is what a regular
n-gonal dipyramid becomes if we
twist its upper pyramidal cone 1/2n of a turn with respect to the lower one:
The intersection of the two cones becomes a solid whose faces are quadrilaterals
[see figure at left].
Do not confuse this with the deltahedra defined
below !

Two
n-gons, 2n-4 triangles Only 4 vertices with 3 edges.

Topologically, we obtain what's called an n-gonal antiwedge
by starting with an n-gonal wedge
(as described above)
and splitting each of its n-3 lateral tetragonal
faces into two triangles (by introducing just one diagonal of each
such quadrilateral as a new edge). An equivalent
geometrical construction
starts with two (non-coplanar)
n-gons sharing an edge
(the so-called hinge) and the two
triangular faces formed by an extremity of that hinge together with the adjacent
vertices found on each n-gon. We're then left with n-3 lateral
tetragons (which are not, in general, planar quadrilaterals) from
which we build 2n-6 additional triangular faces (for a grand total
of 2n-2 faces, including the two n-gons).
Note that there's only one way to split a nonplanar tetragon into
two triangular faces to form a convex polyhedron.

A priori, the above constructions yield 2^{n-3} types of
n-gonal antiwedges which may differ either
by their topology or their chirality.
However, some pairs of such configurations may be obtained from each other by
a 180° rotation about an axis perpendicular to the hinge.

Besides the trivial case of the "trigonal antiwedge" (which is just
a fancy name for an ordinary tetrahedron) the simplest such polyhedron
is the tetragonal antiwedge,
a remarkable hexahedron which stands out as the simplest
example of a chiral polyhedron (the two possible
tetragonal antiwedges are mirror images of each other).

(2002-06-14)
Is there a systematic way to name polyhedra?

Only up to a point.
The most "generic" way is to use for polyhedra the same naming scheme as
for polygons,
by counting the number or their faces:
Thus, a tetrahedron has 4 faces, a pentahedron has 5,
a dotriacontahedron (also called triacontakaidihedron)
has 32 faces.

The case of the Icosidodecahedron :

Counting faces is not nearly enough to describe a polyhedron,
even from a topological standpoint.
In some cases, a nonstandard counting prefix is traditionally used for certain
very specific polyhedra.
For example, the dotriacontahedron shown above is
an Archimedean solid
unambiguously known as an icosidodecahedron
(literally, a polyhedron with 20+12 faces)
because it includes 20 triangular faces and 12 pentagonal ones.
Because it's composed of two pentagonal rotundas,
the icosidodecahedron could also be called a
pentagonal gyrobirotunda but that name would mask its much greater symmetry
compared to the pentagonal orthobirotunda (J34)
which is the other way to glue two such halves.
For the same reason, a special name has been given to the cuboctahedron (at right)
which might otherwise be called a triangular gyrobicupola.

If the icosidodecahedron had not claimed the title, for the above reason,
the name could have been given to another Archimedean solid with 32 faces,
the so-called truncated dodecahedron (which has 20 triangular faces
and 12 decagonal ones). It wasn't...

The notoriety of the icosidodecahedron has made it tempting for some (knowledgeable) people
to use the nonstandard icosidodeca prefix
(instead of dotriaconta or triacontakaidi )
to name other unrelated things
(like a 32-sided
polygon). Resist this temptation...

The general situation is similar to the
naming of chemical compounds.
Certain families can be identified and a systematic naming can be introduced
among such families.
The next article gives the most common such examples.

(2000-11-19) Deltahedra
Don't confuse deltahedra with
the aforementioneddeltohedra.

Deltahedra are simply polyhedra whose faces are all equilateral triangles
(a polyhedron whose faces are triangles which are not all equilateral is
best called an irregular deltahedron).
A deltahedron [or an irregular deltahedron] has necessarily
an even number of faces (2n faces, 3n edges, and n+2 vertices).

A noteworthy fact is that there are only 8 convex deltahedra
(disallowing coplanar adjacent faces). Namely:

4 faces

6 faces

8 faces

10 faces

Regular tetrahedron (4 faces).

Triangular Dipyramid (6 faces).

Regular octahedron or "Square Dipyramid" (8 faces).

Pentagonal Dipyramid (10 faces).

Snub Disphenoid (12 faces), J84.
[The Snub Disphenoid was originally called
"Siamese dodecahedron" by Freudenthal and van der Waerden,
who first described it in 1947.]

Triaugmented Triangular Prism (14 faces), J51.

Gyroelongated Square Dipyramid (16 faces), J17.

Icosahedron or "Gyroelongated Pentagonal Dipyramid" (20 faces).

12 faces

14 faces

16 faces

20 faces

All told, the convex deltahedra include

3 Platonic solids (tetrahedron, octahedron and icosahedron).

3 dipyramids (triangular, square and pentagonal).

3 Johnson solids (J17, J51, and J84).

This adds up to 8, instead of 9, because the regular octahedron
happens to be counted twice
(as a Platonic solid and a square dipyramid)...

Johnson Solids and Polyhedral Nomenclature

It's a challenge to enumerate all convex polyhedra whose faces
are regular polygons.
Besides infinitely many prisms and antiprism,
this includes only:

Norman W. Johnson
(1930-) gave the full classification in 1966,
by adding the 92 polyhedra now collectively named after him.

Only 5 of the 92 Johnson solids are chiral, namely:

J44 : gyroelongated triangular bicupola.

J45 : gyroelongated square bicupola.

J46 : gyroelongated pentagonal bicupola.

J47 : gyroelongated pentagonal cupolarotunda.

J48 : gyroelongated pentagonal birotunda.

Nomenclature :

To describe these and other common polyhedra, some systematic nomenclature is useful.
In particular, any polyhedron gives rise to many other types whose names include one
or more of the following adjectives:

Elongated: By replacing (one of) the largest m-sided polygon,
with an m-gonal prism (that polygon may not be a face of the
polyhedron, but an "internal" polygon with apparent edges).
This adds m vertices, 2m edges, and m faces.
The simplest example, shown at right, is the elongated tetrahedron (J7),
which is an heptahedron._{ }

Gyroelongated: By replacing (one of) the largest m-sided polygon,
with an m-gonal antiprism (that polygon is usually not a face of the
polyhedron, but an "internal" polygon with apparent edges).
This adds m vertices, 3m edges, and 2m faces.
Gyroelongation can be performed in
two different ways (often leading to different chiral versions of the same
polyhedron)._{ }

Snub: Snubbing is an interesting chiral process which,
roughly speaking, amounts to loosening all faces of a polyhedron and rotating them
all slightly in the same direction (clockwise or counterclockwise),
creating 2 triangles for each edge and one m-sided polygon for each vertex of degree m.
A polyhedron and its dual have the same snub(s)!
If a polyhedron has k edges, its snub has 5k edges,
2k vertices and 3k+2 faces._{ }

Truncated: By cutting off an m-gonal pyramid at one or more
(usually all) of the vertices. This add (m-1) vertices, m edges and 1 face
for each truncated vertex._{ }

Augmented: By replacing one or more of the m-sided faces
with an m-gonal pyramid, cupola, or rotunda._{ }

etc.

Other terms are available to describe certain interesting special cases:

Cingulum (Latin: girdle; cingere to gird).

Fastigium (Latin: apex, height).

Sphenoid (Greek: wedge).

etc.

The rest of the nomenclature used in the context of Johnson solids,
is best described in the words of Norman W. Johnson himself :

" If we define a lune as a complex of two triangles attached to opposite sides of a
square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lunes.
The prefix dispheno- denotes two such complexes,
while hebespheno- indicates a blunter complex of two lunes separated by a third lune.
The suffix -corona refers to a crownlike complex of eight triangles,
and -megacorona, to a larger such complex of 12 triangles.
The suffix -cingulum indicates a belt of 12 triangles. "

How do polyhedra generalize to 4 dimensions or more?

The equivalent of a polyhedron in dimension 4 is called a polychoron
(plural polychora).
Polychora are discussed extensively on beautifully illustrated pages proposed by
George Olshevsky and
Jonathan Bowers.

Although the introduction of the term polychoron is fairly recent,
it seems now generally accepted, as there's no serious competition
(the etymology of "polyhedroid" is poor and misleading).
The term was coined in the 1990's by
George Olshevsky, whose
earlier proposal of "polychorema" (plural: "polychoremata") was unsuccessful.
Olshevsky's new proposal had the early support of Norman W. Johnson,
after whom the 92 "convex regular-faced solids" are named (Johnson solids).

A polychoron is bounded by 3-dimensional faces, called cells.
The four-dimensional equivalent of the Euler-Descartes formula is a topological relation
which relates the number of vertices (V), edges (E), faces (F), and cells (C)
in any polychoron enclosing a portion of hyperspace homeomorphic to
a 4D open ball (provided edges, faces and cells
are homeomorphic, respectively, to 1D, 2D and 3D open balls):

V - E + F - C = 0

Such higher-dimensional generalizations of the Euler formula were first established
in 1852 by Ludwig Schläfli (1814-1895)
but his results were only published in 1901 (by then, many others had rediscovered them).

In an unspecified number of dimensions, the counterpart of a 2D polygon,
a 3D polyhedron, or a 4D polychoron is called a polytope,
a term coined by
Alicia Boole Scott
(1860-1940) daughter of
George Boole (1815-1865).

The boundary of an n-dimensional polytope consists of hyperfaces
which are (n-1)-dimensional polytopes,
joining at hyperedges, which are (n-2)-dimensional polytopes.
(All the hyperfaces of an hyperface are thus hyperedges.)
This vocabulary is consistent with the well-established term
hyperplane to designate a vector space of codimension 1
(in a hyperspace with a finite number n of dimensions, a hyperplane is
therefore a linear space of dimension n-1).
We also suggest the term hyperline for a linear space of codimension 2
(and, lastly, hyperpoint to designate a space of codimension 3).

To denote the p-dimensional polytopes within a polytope of dimension n, the following
terms may be used: vertex (p=0; plural "vertices"), edge (p=1), face (p=2),
cell or triface (p=3), tetraface (p=4), pentaface (p=5), hexaface (p=6),
... hypervertex (p=n-3),
hyperedge (p=n-2), hyperface (p=n-1), hypercell (p=n).

To establish and/or memorize the n-dimensional equivalent of the Euler-Descartes
formula for "ordinary" polytopes in n dimensions, it's probably best to
characterize each such polytope by the open region it encloses (boundary excluded),
except in dimension zero (the 0-polytope is a single point).
For the formula to apply, each such region should be homeomorphic [i.e., topologically
equivalent] to the entire Euclidean space of the same dimension, or equivalently to
an open ball of that dimension.
An edge is an open segment, a face is an open disk, a cell is an open ball, etc.
[for example, the ring between two concentric circles is not allowed as a face,
and the inside of a torus is disallowed as a cell].
Then we notice that a number can be assigned to any polytope (and a number of other
things) called its Euler characteristic (c),
which is additive for disjoint sets,
equal to 1 for a point and invariant in a topological homeomorphism
(so that topologically equivalent things have the same c).
For our purposes, this may be considered an axiomatic definition of
c.
It may be used to establish (by induction)
that the c of n-dimensional
Euclidean space is (-1)^{n},
which is therefore equal to the c of our
"ordinary" open polytopes
(HINT: A hyperplane separates hyperspace into
three parts; itself and 2 parts homeomorphic to the whole hyperspace).
The c of all "ordinary" closed polytopes in n dimensions
is the c of a closed n-dimensional ball and it may
be obtained by inspecting the components of the boundary of any easy n-dimensional
polytope like the hypercube or the simplex polytope discussed below.
It turns out to be equal to 1 in any dimension n.
If the hypercell itself (the polytope's interior) is excluded from the count,
as it is in the traditional 3-dimensional Euler-Descartes formula, the RHS of the formula will
therefore be 2 in an odd number of dimensions and zero in an even number of dimensions.
For example, in 7 dimensions, if we denote by T the number of tetrafaces, by P the number
of pentafaces (hyperedges), and by H the number of hexafaces (hyperfaces), we have:

V - E + F - C + T - P + H = 2

We may focus on the n-dimensional equivalent of the Platonic solids, namely
the regular convex polytopes,
whose hyperfaces are regular convex polytopes
of a lower dimension, given the fact that the concept reduces to
that of a regular polygon [equiangular and equilateral] in dimension 2.
In dimension 3, this gives the 5 regular polyhedra.
In dimension 4, we have just 6 convex regular polychora, first described
by Schläfli:

The hyper-diamond
(24 octahedral cells, 6 per vertex, 3 per edge).

The dodecaplex
(120 dodecahedral cells, 4 per vertex, 3 per edge).

The tetraplex
(600 tetrahahedral cells, 20 per vertex, 5 per edge).

In dimension 5 or more, only 3 regular polytopes exists which belong to one of the following
three universal families (also existing in lower dimensions):

Family

n-Polytope

dim.

V 0

E 1

F 2

C 3

T 4

p-Faces p

Simplex

n

C(n+1,p+1)

Point

0

1

Segment

1

2

1

Triangle

2

3

3

1

Tetrahedron

3

4

6

4

1

Pentachoron

4

5

10

10

5

1

Cross- Polytope (orthoplex or cocube)

n

C(n,p+1) 2^{p+1}

Point

0

1

Segment

1

2

1

Square

2

4

4

1

Octahedron

3

6

12

8

1

Hexadecachoron

4

8

24

32

16

1

Hypercube

n

C(n,p) 2^{n-p}

Point

0

1

Segment

1

2

1

Square

2

4

4

1

Cube

3

8

12

6

1

Tesseract

4

16

32

24

8

1

The regular simplex polytope is obtained by considering n+1 vertices in dimension n,
so that each one is at the same distance from any other
(its hyperfaces are simplex polytopes of a lower dimension).
Choosing as vertices all points whose Cartesian coordinates are from the
set {-1,+1}, we obtain an n-dimensional hypercube (of side 2).
The hyperfaces of an hypercube are hypercubes of a lower dimension.
The dual of the above hypercube is the regular cross polytope whose
vertices have a single nonzero coordinate, taken from the set {-1,+1}.

The interactive hypercube at right is from
Kurt Brauchli (details
here).
Click and drag with the mouse to turn the cube aound the chosen axes (H and V)
indicated in the menu bar.
This 5D cube projects like a 3D cube if you rotate
only around axes 0, 1 or 2.
The fourth and fifth dimensions appear with axes 3 and 4.

With the bottom cursor, you may choose a distant (left) or close-up perspective (right).

The word tesseract was coined by
Charles Howard Hinton (1853-1907)
the son-in-law of the logician
George Boole (1815-1864).
C. Howard Hinton was obsessed with visualizations of the fourth dimension involving the tesseract,
which he presented in his 1904 book entitled The Fourth Dimension,
whose third edition (1912)
has been made available online by rkumar, thanks to
Banubula.

EnolaStraight (2002-05-07)
What's the radius of the circle touching 3 touching unit circles?
What's the radius of the sphere touching 4 touching unit spheres?
[In this (edited) question, "touching" means "externally tangent (to)".]

The generalization of this question to any number of dimensions is a classic demonstration
that whatever geometrical intuition we may have developed in two or three
dimensions may not be trusted in a space of more dimensions.
The two-dimensional case [pictured at right] shows 3 congruent circles, centered on the
vertices of an equilateral triangle, touching each other and a much smaller circle
[pictured as a red disk] whose radius has to be determined.
Based on this 2-D case [and, to a lesser extent, on the 3-D case] it would seem that
such an inner ball [disk, sphere, or n-dimensional hypersphere] would always be
small enough to fit inside the simplex [equilateral triangle, regular tetrahedron,
n-dimensional regular simplex] formed by the centers of the congruent balls.
This happens to be true for a dimension equal to 4 or less, but fails for a dimension of
5 or more. In a very large number of dimension,
the (linear) size of the inner ball is about 41% the size of the outer ones.
More precisely,
Ö2-1 = 0.41421356...
is the limit of that ratio when the number of dimensions tends to infinity.
Read on...

Consider the center O of the n-dimensional simplex
formed by the n+1 centers of the congruent balls [each of radius 1].
The critical quantity is the distance D(n) from the center O to any vertex;
the radius of the inner ball is simply D(n)-1.
Well, because O is the center of gravity of n+1 vertices, it is on the line that goes from
a vertex to the center of gravity of the n others. It divides that line in a 1 to n ratio.
The length of that line is therefore (1+1/n)D(n) and it is also one side of a
right triangle whose other side is of length D(n-1) and whose hypotenuse is of
length 2 (it's the side of the simplex, the distance between the centers of two balls).
In other words, D(n) is given recursively by the relations:

D(1) = 1 and
D(n) = Ö[4-D(n-1)^{2}] / (1+1/n)

This recursion can be used to prove, by induction, the following formula:

D(n) = Ö[ 2n / (n+1) ]

The simplicity of this result is a hint that there might be a more direct way to obtain it.
D(2) = 2/Ö3 says that the radius of the
inner circle in the above figure is
2/Ö3-1
[about 15.47 %] of the radius of any outer circle.
Similarly, the corresponding ratio for spheres is
½Ö6-1
[about 22.4745 %].
The limit of D(n) is Ö2:
In a space with a very large number of dimensions,
the ratio of the radius of the inner hypersphere to the radius of any outer hyperspheres is
thus slightly less than
Ö2-1
[about 41.42 %].

In dimension n, the distance from the center O to any of the hyperplanes
(of dimension n-1)
of the "faces" is D(n)/n.
Therefore, if the radius D(n)-1 is greater than is D(n)/n.,
the inner hypersphere bulges outside of the n-dimensional regular simplex formed
by the centers of the outer hyperspheres.
This happens as soon as n^{2}-5n+2 > 0,
which is the case when n is at least equal to 5.
This higher-dimensional configuration is contrary to the intuition we would form
by looking only at the two-dimensional and/or three-dimensional cases...

Dr. Murali V.R. (2004-02-25; e-mail)
What is the volume of a regular antiprism?

A regular antiprism is a polyhedron whose faces are
two parallel n-gonal bases [regular polygons with n sides]
and 2n equilateral triangles called lateral faces.

Look at the outline of such a solid from above, and what you see
is a regular polygon with 2n sides (every other vertex is on the top base,
and every other one is on the bottom).
The angle at each vertex of this outline is thus
q = p-p/n.

Now, each lateral face is seen as an isosceles triangle having an angle
q at the top
and featuring a base observed at its real size a
(as the direction of observation is perpendicular to it).
The height of such an isosceles triangle is thus:

½ a cotan (q/2)
=
½ a tan (p/2n)

This quantity is also equal to the length of a side of a right triangle
whose hypotenuse is the true height
of a lateral face (namely ½ aÖ3)
and whose other side is the height h of the antiprism
(namely, the distance between its bases).
This gives the height h of the antiprism in terms of
the length a of its edges:

h = ½ a

Ö

3 - tan^{2}(p/2n)

Consider the circumscribed prism of height h
whose base is the 2n-gonal outline.
Each side of this outline is equal to
½ a / cos(p/2n).
Its surface area is therefore:
(n a^{2}/4) /
sin(p/n) and the volume
of the prism is h times that.

The antiprism is obtained from this prism by removing
2n triangular pyramids of height h
whose bases are all congruent to the above isosceles triangle,
for a combined base area of
(n a^{2}/2) tan(p/2n)
and a total volume h/3 times that.

The volume V of the antiprism is the difference between these two volumes:

V = (n a^{3} / 24)
[ 3 / sin(p/n)
- 2 tan (p/2n) ]

Ö

3 - tan^{2}(p/2n)

This can be rewritten in a much more palatable form, using
t = tan (p/2n) :

V = n a^{3}
( 3 - t^{ 2 })^{ 3/2}/ 48t

h = ½ a

Ö

3 - tan^{2}(p/2n)

V = n h^{ 3}/ 6 tan(p/2n)

Two noteworthy special cases (for a = 1):

V = 1 / Ö72 when n=2. A regular tetrahedron!
[A degenerate but valid case.]_{ }

V = ½ Ö3 when n=3.
A regular octahedron...

(2006-01-18) Szilassi Polyhedron & Császár Polyhedron
Two polyhedra of genus 1 [topology of a torus] dual of each other.

In a polyhedron, a line between two nonadjacent vertices
is called a diagonal.
When every pair of vertices is connected by an edge, the polyhedron
has thus no diagonals.
The tetrahedron is an example of a polyhedron without diagonals,
so is Császár's polyhedron.
By duality, this means that every face
of the Szilassi heptahedron has an edge in common
with each of the other 6 faces...

Geometrically, the Szilassi polyhedron has an axis of 180° symmetry:
3 pairs of congruent faces and a symmetrical face (darkest in the picture).
This symmetry allows one to build a full mental picture of the polyhedron
from the image at right (obtained from David Eppstein's
Geometry Junkyard).

Beyond Császár and Szilassi...

The Descartes-Euler formula for a polyhedral surface of genus G is:

V - E + F = 2 - 2G

In a polyhedron where all pairs of faces share an edge, E = (F-1)F/2.
Also, we have V = 2E/3, since any vertex must belong to only 3 faces.
Eliminating E and V using these two relations makes the above Euler formula boil down to:

G = (F-4)(F-3) / 12

As G is an integer, F must have definite values modulo 12,
namely 0, 3, 4 or 7.
Beyond F=4 (the tetrahedron of genus 0) and F=7 (the Szilassi heptahedron of genus 1)
the next possibility would thus be F=12, a dodecahedron of genus 6...

Curved toroidal maps which can't be straightened :

The number of colors required to color a map (or a graph) so that no two
adjacent patches (or nodes) are of the same color is called
its chromatic number.
For any given surface other than the Klein bottle
(a surface of genus 1 on which any map can be colored with only 6 colors)
the maximal chromatic number of all maps drawn on it depends only on
its geometric genus G, according to the following formula, proposed by
P.J. Heawood in 1890
(A000934).

Maximal Chromatic Number =
ë
½ ( 7 + ( 48G + 1 )^{ ½} )
û

Geometric Genus

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Heawood Number

4

7

8

9

10

11

12

12

13

13

14

15

15

16

16

16

17

For genus 0, this amounts to the celebrated
4-color theorem for planar or spherical maps,
as proved by Appel and Haken in 1977.
Otherwise, the formula was shown to be an upper bound by Heawood in 1890.
It was found to be exact (except for the Klein bottle) in 1968,
when Ringel and Youngs showed how a map with this many countries could always be drawn on
a surface of genus G so that two countries always have a common boundary.

When G is (F-4)(F-3) / 12,
the Heawood formula gives precisely
a maximal chromatic number equal to F [check it].
The challenge met by Szilassi for genus 1 (a topological torus)
was to draw a Ringel-Youngs map with 7 flat countries.
Could the same feat be possible for the next cases,
starting with a genus-6 surface consisting of 12 planar faces?

Miro Dabrowski
(of Bunbury,
Australia. 2006-07-21)
What are the dihedral angles in a buckyball ?
How do I taper 32 ready-to-glue wooden faces ?

First, let's consider how the overall size of the buckyball relates to the length
of its edges (a). One easy way to do so is to consider
the equator of the ball if the polar axis
goes through the centers of two opposite pentagonal faces
(for example,
let the polar axis be vertical in the above picture).
This equator is a regular decagon
whose sides are of length 1.5 a
(namely, the distance between the middles of two nonadjacent edges
in a regular hexagon of side a).
The radius of the circle circumscribed to that decagon
is the so-called midradius
(r) of the buckyball
(i.e., the distance from the ball's center to the middle of any
edge).
Thus, 3a/ 4r
is the sine
of p/10 (or 18°)
which is 1/2f.
This yields the first relation below.
The other equation gives the
circumradius (R) of the buckyball
(i.e., the distance from the center to any vertex )
obtained from the Pythagorean theorem
(r^{2} + a^{2}/4 = R^{2 }).

r / a =

^{3}/_{4 }
(1+Ö5) =
^{3 f}/_{2}

= 2.42705098312484227230688025...

R / a =

^{1}/_{4 } Ö

58 + 18 Ö5

= 2.47801865906761553756640791...

A solid buckyball
(a truncated regular icosahedron) consists of 32
straight regular pyramids (12 pentagonal and 20 hexagonal ones)
with a common apex at the center O.
All the lateral faces of those pyramids are congruent to an isosceles triangle
of base a and sides R.

A thick buckyball "shell" may be assembled from wooden faces (hexagonal or pentagonal)
whose sides are tapered at an angle equal to the angle between the base and any lateral
face of such a pyramid.
This is the only way to have the same taper for all hexagonal sides,
in order to simplify tooling and final assembly.

The interior shape will not be a perfect buckyball itself
if boards of the same thickness are used for both types
of faces. If needed, this flaw (which is normally hidden)
could be eliminated by shaving about 5.1% off
the boards used for the hexagonal blocks
(since the height of an hexagonal pyramid
is about 94.9% of the height of a pentagonal one).

Each n-gonal pyramid consists of n slices (as pictured at left)
sharing an edge from O to the center
M of the n-gon (n is 5 or 6).

a is the length of every edge.
R is the circumradius
(distance from the center O to any vertex).
r is the midradius
(distance from O to [the middle of] any edge). b_{n} = ½ a / tan(p/n)
is the inscribed radius of the n-gonal face.

The aforementioned taper angles (q = 0
would be a straight cut) are obtained from the formula
q_{n} =
arcsin ( b_{n} / r ). Numerically, this boils down to:

q_{5} =
arcsin Ö

(5+Ö5) / 90

=
16.4722106927603966099...°

q_{6} =
arcsin [

(Ö5 - 1)Ö3 / 6 ]

=
20.9051574478892990329...°

Between an hexagon and an adjacent pentagon, the dihedral angle is
q_{5}+q_{6}.
The dihedral angle between two adjacent hexagons is
2q_{6}.

Angular
distances seen from the center of a buckyball :

Angle...

Symbol

Expression

Value in degrees

subtended by an edge

a

2 arcsin

a

2R

23.28144627 °

from face center to vertex

b_{5}

arcsin

a

2R sin p/5

20.07675127 °

b_{6}

arcsin

a

2R sin p/6

23.80018260 °

from face center to [middle of] edge

q_{5}

arcsin

a

2r tan p/5

16.47221069 °

q_{6}

arcsin

a

2r tan p/6

20.90515745 °

between middles of edges, 1/3 of the way around an hexagon

p/5

36 °

A useful check
is obtained by considering any journey along half a
great circle (180°) which includes an edge between hexagons:

180° = p =
a +
2 (b_{5} +
q_{5 }) +
4 q_{6}

A similar "trick" was used to obtain trivially the last entry of the above table
(actually, this was our starting point
to obtain the midradius-to-edge ratio).

The deficiency at each of the 60 buckyball vertices is 12°, namely
4p/60 = p/15.
It may also be computed directly as
2p-2(2p/3)-3p/5.

The deficiency at a vertex of a polyhedron is defined
to be what's missing from a full
turn (2p or 360°)
when you add up all the angles of the faces which meet at that vertex.
The name is historically linked to
Descartes' deficiency theorem
which states that, in any polyhedron homeomorphic to a sphere,
the sum of the deficiencies at all vertices is equal to an angle of
4p (or 720°).

(2013-02-22) Zonogons, Zonohedra, Zonotopes and Zonoids
A zone is a belt of polygons joined by parallel edges.
In a zonoid, every 2n-gonal face belongs to n zones.

A zonotope is a convex polytope obtained as the
Minkowski sum of line segments.
In three-dimensional space, a zonotope is called a zonohedron.

A zonotope is said to be nonsingular
when it's the Minkowski-sum of n line segments among which no 3 segments are coplanar
(some authors consider only this special kind of zonotopes).
The edges of such a nonsingular zonotope have n possible
directions (all the edges along one direction have the same length).
Each direction defines a single zone containing 2(n-1)
faces. There are n(n-1) faces; every face is a parallelogram that
belongs to two zones.

In a space of dimension d less than n, a nonsingular zonotope of order n can viewed
as the shadow of an n-dimensional hypercube.
In three dimensions, a nonsingular zonohedron of order n ≥ 3
has n(n-1) faces, 2n(n-1) edges and 2+n(n-1) vertices.

This doesn't apply to singular zonohedra whose faces
(zonogons) can be dissected into parallelograms but need not be parallelograms themselves.

My favorite zonohedron
is the scalene isogonal tetradecahedron
pictured at right
(with void rectangular faces, through which some inner
faces are shown, with pencil marks on them).

It's rarely mentioned elsewhere, except in its
more regular incarnation, commonly dubbed
truncated octahedron, where the rectangular faces are square
(isosceles isogonal tetradecahedron).

The equilateral version (uniform truncated octahedron)
is an Archimedean solid
which is space-filling (see below) and serves as
the node-skeleton of Kelvin's foam.

(2013-02-16) Space-filling polyhedra ( Plesiohedra )
There are many of those:

(2016-01-04) Pyritohedron : Space-filling pentagonal dodecahedron.
The only plesiohedron in a parametric family of isohedra.

The pyrithedron belongs to a one-parameter family of dodecahedral shapes
which interpolate continuously between a split-faced cube and a
rhombic dodecahedron (neither extreme case being included in the family).
They all feature 12 congruent pentagonal faces, 30 edges
and 20 vertices. just like the regular dodecahedron,
which is one of them!

This is illustrated by the morphing animation at right,
made in November 2009 by
Tom Ruen
(a computer programmer born in 1968).
His picture shows 8 fixed vertices
(±1, ±1, ±1)
and 12 that depend on h
in the interval ]0,1[ :
(±(1-h^{2 }), ±(1+h), 0)
(±(1+h), 0, ±(1-h^{2 }) )
and
(0, ±(1-h^{2 }), ±(1+h) )

The 6 edges parallel to a coordinate axis have length
2-2h^{2}. The other 24 edges have a length
of Ö(1+h^{2}+h^{4}).

Both quantities are equal when h is (Ö5-1)/2
(the inverse of the golden ratio). In that special case, all edges
have length Ö5-1 while the horizontal diagonal has length 2,
which makes the ratio of the diagonal to the size equal to the golden ratio, thus establishing
that the faces are inddeed regular pentagons and, consequentely that the solid is a a regular dodecahedron.

For the solid to be space-filling, a whole number of them must fit around each of the six special
edges. That number is 2 or 4 in the extreme cases where h is 0 or 1.
For a proper member of the family to be space-filling, the aforementioned number must be 3,
which implies a dihedral angle of 120° and h = 1/Ö3.
The corresponding shape is called a pyritohedron.

h-parameter, edge lengths and volume of special related solids

Split-faced Cube

Pyritohedron

Regular Dodecahedron

Rhombic Dodecahedron

h

0

^{ }

Ö3

=^{ }0.57725...^{ }

3

Ö5-1

=^{ }0.61803...

2

1

e_{6}

2

4/3 = 1.333333...

Ö5 - 1 = 1.236067977...

0

e_{24}

1

Ö13/3 = 1.20185...

Ö3

V

8

8+32/Ö27 = 14.158402871356+

10+2Ö5 = 14.472135955-

16

Proofs of the above statements :

Let 1+h be the altitude (z) of the horizontal top edge and 2d be its length.
The extremities of that edge are A' = (0,-d,1+h)
in the back and A = (0,d,1+h) in the front.

Let's now consider the pentagonal face whose highest point is the latter vertice .
(It's the pentagon with the lightest color in the animation).

In that pentagon, A is connected to the fixed points B=(1,1,1) and E=(-1,1,1).
Those two are respectively connected to two base points on the equator (connected to each
other by an horizontal edge).
Namely: C = (d,1+h,0) and D = (-d,1+h,0).

First, we must establish the relation between h and d from the fact that
A, B, C, D ane E are coplanar. One way to do so is to say that the
determinant of EA, EB and ED vanish (EC will be
in the same plane by symmetry):

0 =

1

d-1

h

_{ } = 2h^{2} + 2d - 2

2

0

0

1-d

h

-1

Therefore, d = 1 - h^{2} as advertised...

Each pentagonal face has a special side of length 2d = 2-2h^{2}.
The other sides have a length whose square is 1+h^{2}+(1-d)^{2}
= 1+h^{2}+h^{4}.
The pentagon is equilateral when both lengths are equal:

(2-2h^{2 })^{2} = 1+h^{2}+h^{4}

This boils down to 3 ( h^{4} - 3 h^{2} + 1 ) = 0 a quadratic equation
in h^{2} whose only solution below 1 is
h^{2} = (3-Ö5)/2.
Therefore, h = (Ö5-1)/2.

Volume :

The polyhedron discussed so far can be dissected into 19 solids:

A cube of side 2 and volume 8.

Six prisms of height 2d, based on a triangle of height h and base 2.

Twelve pyramids of height h based on a rectangle of sides 2 and 1-d.

Thus, the total volume goes from V = 8 (for h = 0)
to V = 16 (for h = 1):