A polyhedron is said to be *uniform* if its faces are regular polygons and its vertices are all alike, that is, for any two vertices there is a symmetry of the whole polyhedron that moves one vertex into the other.
It is *regular* if it is uniform and all the faces are alike, that is, for any two faces there is a symmetry of the whole polyhedron that moves one face into the other.

There are five regular convex polyhedra. They were known to Plato, and are often called "the Platonic solids". They are familiar to most people interested in geometry: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Other convex uniform polyhedra are often called "Archimedean solids" or "Archimedean polyhedra." There are two infinite families of these, namely the prisms and antiprisms, and thirteen others.

A *prism* has two n-gonal faces; one of these faces
may be obtained from the other by translation in the direction perpendicular to the
plane of the face. Between the n-gonal faces is a
belt of n squares.

An *antiprism* also has two n-gonal faces; one may
be obtained from the other by translation as before, followed by a rotation through
π/n radians. Between these is a belt of
2n equilateral triangles, having edges in common with the
first and second n-gons alternately.

If a convex uniform polyhedron is enclosed in a sphere concentric with the polyhedron, then the faces of the polyhedron may be projected onto the sphere. The result is a tessellation of the sphere; it is "uniform" in that each face is a regular spherical polygon and the vertices are all alike in the same sense as for polyhedra.

The symmetry group of a uniform polyhedron or tessellation can be a kaleidoscope group of rotations
and reflections, or the *even* subgroup, which consists of the rotations and
translations. (These are products of even numbers of generating reflections.)
For a polyhedron or spherical tessellation, the kaleidoscope group must be one of the
spherical groups.
The other kaleidoscope groups are also symmetry groups of uniform tessellations,
of the Euclidean plane or the hyperbolic plane.

The reflections in two edges of the fundamental triangle generate a dihedral group of rotations and reflections; the rotations are centered at the intersection of the two sides. Any other point V is moved by the elements of this group into the vertices of a polygon. If the angle of the intersection is π/p, then the polygon has p sides if V is on the edge defining one of the two reflections; in this case, the polygon is regular. If V is elsewhere, then the polygon has 2p sides; it is regular if (and only if) V is equidistant from the two edges that define the reflections.

We can assemble such polygons into uniform tessellations.
A systematic way of doing this is called *Wythoff's
construction*.
In this construction, we select a point V in the fundamental
triangle, and generate regular polygons from V as
described in the preceding paragraph.
The images of these polygons under all the symmetries of the kaleidoscope group
(or the rotational subgroup) make up our tessellation or polyhedron.
There are several ways to select V, and for each of them
a concise notation.

Let the angles of the fundamental triangle be π/p, π/q, π/r; then (p q r) is a notation for the Kaleidoscope group. The vertices of the fundamental triangle are labeled P, Q, R.

The regular and quasi-regular cases

Let V be a vertex of the fundamental triangle.
If the vertex is P, then the symbol is
p | q r. Typical faces are a
q-sided polygon centered at
Q and an
r-sided polygon centered at
R; around each vertex, this pair is repeated
p times.

Special cases arise if one or more of the numbers p,q, r is equal to 2.

Regular tessellations p | q 2

A 2-sided polygon, or "digon," looks like a single line and may be omitted from
the list of faces. (But the digons show up in the kaleidoscope applet, as lines or arcs.)
The polyhedron or tessellation has q-sided faces, of which
p meet at a vertex. Thus it is a *regular* figure.

The dihedron 2 | q 2

If p = 2 also, then the polyhedron collapses to a pair
of q-gons back to back, or a "dihedron." The spherical
tessellation consists of two hemispheres, with the "equator" between them divided
into q arcs.

A bundle of sticks: p | 2 2

If q = r = 2 then the faces of the regular polyhedron
collapse into digons, and the polyhedron is nothing but 2p
of these. Strictly speaking, there are two families of digons in the
bundle: one from the q-gons and one from
the r-gons.
In the kaleidoscope applet, these digons show up as broken lines if
the polyhedron option is chosen, or as semicircular arcs in a tessellation.
The two families can have different colors.

Quasi-regular tessellations 2 | q r

The polyhedra 2 | 3 4 and 2 | 3 5
are the cuboctahedron and icosidodecahedron respectively.
These are the only polyhedra or spherical tessellations p | q r
for which q > 2, r > 2, and q ≠ r.
However, there are plenty of hyperbolic tessellations, such as
3 | 3 4, which seem to me to qualify as "quasi-regular."

The truncated and rhombic cases

Let V be on one side of the fundamental triangle, for instance
PQ.
Then V generates a regular p-gon
centered at P, and a regular q-gon
centered at Q.
It also generates a 2r-gon centered at
R, which will be regular if V
lies on the bisector of the angle at R. With this choice, we
generate a uniform tessellation for which the symbol is p q | r.
In general, the polygons surrounding a vertex are of orders
p, 2r, q, 2r.

Truncated tessellations 2 q | r

If p=2 then the figure has, at each vertex,
polygons of order 2r, 2r, q.
In the polyhedral case, this can be constructed from a regular
q | 2 r by "truncating," that is, removing a
little pyramid at each vertex.

Rhombic tessellations p q | 2

When r=2, the polygons around a vertex are of orders
p, 4, q, 4.
The Archimedean polyhedra of this kind are the rhombic cuboctahedron
3 4 | 2 and the rhombic icosahedron
3 5 | 2.

The prisms 2 q | 2

When p = r = 2 this form of the Wythoff
construction is a prism.
It could be regarded as a rhombic form of the dihedron, or the result
of truncating a bundle of digons. Note that only one family of digons
get truncated into squares; the others remain as edges between the squares.

The great-rhombic case

Let V be interior to the fundamental triangle.
Then it will generate regular polygonal faces around each vertex, if (and only if)
it is equidistant from the three edges, that is, if it is the intersection
of the three angle bisectors.
The symbol for this construction is p q r |.
The polygons surrounding a vertex are of orders 2p, 2q, 2r.

The Archimedean polyhedra 2 3 4 | and 2 3 5 | have traditionally been called the "truncated cuboctahedron" and "truncated icosidodecahedron." Actual truncation of a quasi-regular polyhedron would produce rectangular faces, not squares. For this reason, the names "great rhombicuboctahedron" and "great rhombicosidodecahedron" are becoming more current.

The snub case

There is one more way to generate a uniform tessellation from a kaleidoscope.
Take a vertex interior to a triangle, and use its images
under the even subgroup.
These will define regular p-gons, q-gons,
and r-gons; and there will be triangular gaps between them.
In general, the gaps are not equilateral.
There is one point in the fundamental triangle for which these triangles are equilateral;
if it is chosen, the resulting figure is called a "snub" polyhedron or tessellation.
The symbol for this case of the Wythoff construction is | p q r.
The faces surrounding a vertex are of orders p, 3, q, 3, r, 3.

The choice of V for this construction is more difficult than for the other cases. The condition that it generate equilateral triangles can be expressed as requiring it to be at an intersection of two conics. Thus its coordinates are the roots of quartic equations, and in general are not constructible with ruler and compass.

The Archimedean polyhedra | 2 3 4 and | 2 3 5 have traditionally been called the "snub cube" and "snub dodecahedron;" recently people have started calling them "snub cuboctahedron" and "snub icosidodecahedron." When they are displayed by the kaleidoscope applet, the snub triangles can have a different color from the triangles that are centered on kaleidoscope vertices.

The antiprisms | 2 2 r

When two families of faces collapse into digons, the remaining faces surrounding a vertex
have orders 3, 3, 3, r.
The result is an anti-prism.

The snub tetrahedron | 2 3 3

It comes as something of a surprise that when the snub construction is applied to
the tetrahedron, the result is a regular icosahedron.

Duality is an important relation
between different tessellations or different polyhedra.
In it, the vertices of one figure correspond to faces of the other, and *
vice versa*.

In the case of polyhedra related to a spherical kaleidoscope, duality can be defined in
terms of polarity.
Let S be a sphere.
For a point P on S, the *polar*
of P is the plane through P
tangent to S; and P is the
*pole* of that plane.
If P is outside of S, then the
polar of P passes through the points of contact of tangents to
S passing through P.
If a plane intersects S in a circle, then the tangents
to S at the points of intersection are concurrent at
the pole of the plane.
For a point inside the sphere, or a plane which does not intersect it, polarity can be
determined by the rule that if P is on the polar
plane of Q, then Q is on the
polar plane of P.
The *polar line* of a line L is another line,
which is the intersection of all the polars of points on
L, and also the join of all the poles of planes passing through
L.

When a polyhedron has a center of symmetry, it is natural to construct its dual by means of polarity with respect to a sphere which shares the same center. The vertices of the dual polyhedron are the poles of the face planes of the original polyhedron. Two vertices of the dual are joined by a dual edge if the corresponding faces of the original meet in a edge. Then the former edge is on the dual line of the latter. The face planes of the dual polyhedron are the polars of the vertices of the original one.

Let us take a regular polyhedron for an example. The midpoints of its edges lie on a sphere, which we can use to dualize the polyhedron. The dual edges are perpendicular bisectors of the edges of the original polyhedron. Over each face of the original, dual edges meet at the dual vertex. The edges coming from an original vertex have their midpoints arranged in a circle, and in fact they form a regular polygon; the dual edges also form a regular polygon which is a face of the dual polyhedron. If the original polyhedron is p | q 2, which has q-sided faces with p at each vertex, then the dual has p-sided faces with p at each vertex; therefore it is q | p 2.

The other uniform polyhedra can be dualized in a similar way.
Because the vertices of the original, uniform polyhedron are all alike, they lie on a common
sphere, which is the *circumsphere*.
The edges are of equal length; therefore, for any vertex V,
the vertices connected to it by edges are equidistant from it; so, they lie on a small
circle on the circumsphere, and also on the plane containing that circle.
The midpoints of the edges that connect to V also lie on a
circle; they are the corners of a polygon called the *vertex figure*.
Also, we see that the midpoints of all the edges are equidistant from the center of
the circumsphere, so they lie on a common sphere, the *midsphere*.

We construct the dual polyhedron with respect to the midsphere. The dual edges are tangent to the midsphere, and those which correspond to the edges of an original face meet in a point over the center of that face; of course, this is the pole of that facial plane. For the edges that connect to the original vertex V, the dual edges lie in the plane of the vertex figure, and in fact are tangent to the circle in which the vertex figure is inscribed. They are the edges of the dual face.

The dual uniform polyhedra are symmetrical, just as are the uniform polyhedra, but in a different way. The faces are not, in general, regular polygons. But they are all congruent, and for any two there is a symmetry of the whole polyhedron which takes one into the other. The vertices of the dual polyhedron are not all alike in general. But at each vertex, the facial angles are equal, and the dihedral angles between faces are all equal.

Wythoff's construction of uniform polyhedra can be extended to the dual uniform polyhedra.
This may be the best way of seeing that dualization can be extended to uniform tessellations.
Given a uniform tessellation, the dual vertices are at the centers of the original faces.
The dual edges are perpendicular bisectors of the original edges.
Each original vertex is the *in-center* of the corresponding dual face, that is,
it is the center of a circle tangent to the edges of the dual face.
The dual Wythoff construction starts from the fundamental triangle
PQR of the Kaleidoscope group, and a particular point
V in that triangle; but this time, V
is to be the in-center of a face of the tessellation.
Points like the corners of the fundamental triangle are the vertices.
For a polyhedron, V is the pole of a facial plane.

The dual regular and dual semi-regular cases

Let V be the vertex P.
As a symbol for this construction, we may use D(p | q r).
The face is a 2p-gon, with vertices like
Q and R alternating.
These vertices are of orders q and r
respectively.

Dual regular tessellations D(p | q 2)

A vertex of order 2 is completely unnoticeable, and we may as well omit it.
We are left with p-gonal faces, meeting at
vertices of order q.
Thus D(p | q 2) is essentially the same as
q | p 2.
The dihedron 2 | q 2 may be constructed as
D(q | 2 2), and the "bundle of sticks"
p | 2 2 as D(2 | p 2).

Dual quasi-regular tessellations D(2 | q r)

When p = 2, the dual face is a rhombus.
The polyhedra D(2 | 3 4) and D(2 | 3 5)
are called the "rhombic dodecahedron" and "rhombic triacontahedron;"
the latter is sometimes also called "Kepler's solid."

The dual truncated and dual rhombic cases

Let V be the intersection of PQ
with the angle bisector at R.
The dual construction will be denoted D(p q | r).
The vertices of a face are of types P, R, Q, R; their
orders are p, 2r, q, 2r.

Dual truncated tessellations D(2 q | r)

If p=2 we disregard the "order-2" vertices; we then
have isosceles triangular faces, meeting at vertices or orders
q, 2r, 2r.
A polyhedron of this kind resembles a r | 2 q with
a low q-sided pyramid raised on each face.

Dipyramidal tessellations D(2 q | 2)

The dual polyhedron to a q-sided prism is a "dipyramid,"
with 2q triangular faces. These are arranged in two
q-sided pyramids which meet at an equatorial
q-gon.

The Kaleidoscope as dual uniform tessellation

Let V be the in-center of the fundamental triangle.
The notation for this case is D(p q r |). The dual face is
a triangle, with vertices like P, Q, R of order
2p, 2q, 2r.
In fact, the dual tessellation is nothing else than the fundamental regions of the
Kaleidoscope itself.

The dual snub case

In the dual snub case, to make the construction follow the original as closely as
possible, V should be the reflection of the snub point in
one of the sides of the fundamental triangle.
The symbol for this case is D(| p q r).
The vertices include points like P, Q, R of order
p, q, r; these alternate with vertices of order 3 that are
the centers of the snub triangles.

The trapezohedra D(| 2 2 r)

If p = q = 2, then two of the six vertices of the
dual face may be disregarded; the remainder make a quadrilateral with one
vertex of order r and the others of order
3.
Two families of r faces each meet in a skew
polygon of 2r sides.

The Wythoff construction may be found in the paper by H. S. M. Coxeter and others, listed below.
In this paper, one can find out about a much wider class of uniform polyhedra, most
of which are non-convex. Fr. Wenninger's book *Polygonal Models* gives photographs
of cardboard models of these, and advice on their construction.
His book *Dual Models* treats of the non-convex dual uniform polyhedra and some
of the stellations of uniform polyhedra.
Cromwell's book is an interesting recent survey of the geometry of polyhedra.
Finally, *Mathematical Models* by Cundy and Rollett is a book dear to my heart,
full of instructions and advice on the construction of these models and other interesting topics.

Coxeter, H. S. M; Longuet-Higgins, M. S.; Miller, J. C. P: Uniform Polyhedra, Philos. Trans. Roy. Soc. London Set. A 246(1954),401-450.

Cromwell, Peter R.: Polyhedra, Cambridge, Cambridge University Press, 1997.

Cundy, H. Martyn; Rollett, A. P.: Mathematical Models, Stradbroke (England), Tarquin Publications, 1997.

Wenninger, Fr. Magnus J., OSB: Dual Models, Cambridge, Cambridge University Press, 1983.

Wenninger, Fr. Magnus J., OSB: Polygonal Models, Cambridge, Cambridge University Press, 1971.