               Introduction 

This file tries to explain what the following sentence means.

!Stellate has stored instructions for drawing all the uniform polyhedra and
their stellations using stellation diagrams (except the prisms and antiprisms) 

 Look in your paper manual page 3, or at the Draw file in
Tutorials.Pictures.Defns for explanations of what !Stellate means by a
face, and in Reference.Glossary for any other terms not explained below
which you do not understand.

What is a regular polygon?

A plane figure bounded by straight edges of equal length and having
identical vertices. The regular polygons used by stellate are shown in
Reference.Faces. These are formally all convex polygons, having vertices
with angles less than 180. The star polygons have vertex angles 36, 45
and 72 with 5, 8 and 10 edges; their edges intersect but this does not make
them irregular.

What are Uniform Polyhedra?

A uniform polyhedron is one which has all its vertices surrounded by the
same number and type of faces which are made of regular polygons. They are
all closed polyhedra, this means that the edges of the faces all join one
another, leaving no holes through which you can see inside the polyhedron. 

The simplest polyhedra all have the same kind of faces, made from regular
polygons, they are called the PLATONIC solids:

 tetrahedron       4 triangular faces  (3 sided polygons)
 hexahedron (cube) 6 square faces      (4 sided polygons)
 octahedron        8 triangular faces 
 dodecahedron     12 pentagonal faces  (5 sided polygons)
 icosahedron      20 triangular faces

The list is arranged in increasing order of the number of faces.
Look at the Draw file Tutorials.Pictures.Platonic for drawings of
these. They are called Platonic after the Ancient Greek philosopher
Plato, who was one of the first people to write about polyhedra.

Since all the faces of a Platonic solid are the same, all you need to know
to draw it is how many sides are there on each face, how many faces there are,
the size of the angle between any two faces and how big you want it to be.

You may want to try using !Stellate to look at these polyhedra using the
Name option from the icon bar menu. For example click over dodecahedron 
to see the Parameters window which shows the name and radius of the
polyhedron. Then for each type of face (component) it gives the number of
them, their type, radius (measured from the centre of the face to a vertex)
and the way they are arranged (symmetry). The in-radius is the distance from
the centre of the polyhedron to the centre of a face while the ex-radius is
to a vertex. Each uniform polyhedron has the same radial distance to all its
vertices. All the radii shown are relative to half the length of an edge.

The next simplest type of uniform polyhedra are made from two or more
regular polygons, they are called the ARCHIMEDEAN solids, named after
another Ancient Greek, Archimedes, who studied them.

Again all their vertices are similar; they have the same number of faces 
meeting at each vertex; for example, at each vertex of the truncated 
octahedron there are 2 hexagons and a square. 
( Look at the Draw files TryThese.Answer6a, b, c for drawings of them in the
left hand column, with their stellations diagrams in the centre and 1st
Stellation on the right.)

Their names, face types and number of each face are:

  number of sides of the polygons = 3   4   5   6   8  10
  name                   
 truncated tetrahedron              4           4
 truncated octahedron                   6       8
 truncated cube                     8               6
 truncated icosahedron                     12  20
 truncated dodecahedron            20                  12
 cuboctahedron                      8   6
 icosidodecahedron                 20      12
 rhombicuboctahedron                8  18
 rhombicosidodecahedron            20  30  12
 rhombitruncated cuboctahedron         12   8   6
 rhombitruncated icosidodecahedron     30      20      12
 snub cube                         32   6
 snub dodecahedron                 80      12
 
Notice that there are always an even number of faces, and that only a few
numbers are included.  For example, look at the column for faces made from
triangles, (polygons with 3 sides). The only numbers which appear are
4,8,20,32 and 80. It is not possible to make a uniform polyhedron using any
other number of triangles, if you try other numbers you find that either you
have a polyhedron with a hole in it or you have some pieces over.

So there are only a few ways you can arrange the faces to form these
polyhedra. Platonic solids have one face type, Archimedean have 2 or 3 face
types. !Stellate has instructions for drawing each polygon and each possible
arrangement of the faces, called components. A polyhedron such as
the rhombicosidodecahedron in the list above has 3 face types and so more
instructions are needed to describe them. 

(These instructions are stored in Resources.Components, their definition
is in FormComps and there are 2 Draw files Faces and Components which
summarise all the face types and components known to !Stellate)

Now try looking at these Archimedean solids using the Name option again
checking the numbers of each type of face in the Parameters window and
plotting the polyhedron with the Plot Solid option.

There are also 4 interesting Quasi-regular solids often associated with
the names Kepler and Poinsot. These have just one type of regular polygon at
a vertex and all the vertices are the same. They are:

 small stellated dodecahedron    12 5-pt stars     12 vertices
 great stellated dodecahedron    12 5-pt stars     20 vertices
 great dodecahedron              12 pentagons      12 vertices
 great icosahedron               20 triangles      12 vertices

The difference from Platonic solids is that their planes intersect.
Try looking at these, in particular at their vertices.
When you have understood about stellation diagrams (below), see how the
regular polygon in the stellation diagram fits onto the 3-D solid. Note that
parts of the regular face in the stellation diagram are missing because
these pieces are inside the 3-D solid and are not visible. They are caused
by the planes intersecting inside the solid.

There are two kinds of uniform polyhedra which !Stellate does not make:
the prisms and the antiprisms; there are an infinite number of both of
these. Prisms are made from two parallel regular polygons with their
vertices aligned so that they can be joined by squares (the cube is a
special case, the square prism); antiprisms are similar except that the
parallel polygon faces are twisted and joined by twice the number of
equilateral triangles. The Octahedron is also the triangular antiprism,
having two parallel triangles joined by six more triangles.  


Stellation diagrams

!Stellate decides on the shape of face to draw using a stellation diagram
for that face.  A stellation diagram is a set of lines made on the plane on 
which one face of the polyhedron stands by projecting other faces down
until they meet that plane. Look at page 6 of the paper manual or at the
Draw file in Tutorials.Pictures.Page6 for a diagram explaining this. The
stored stellation diagrams have pre-defined polygons which you see as filled
cyan or blue shapes. You need only define one of a set of faces
symmetrically arranged in a plane, the others are filled for you by the
program using the stored rotational symmetry.

Try using !Stellate to look at a dodecahedron and compare its stellation 
diagram with those in Page6.   

The polygon in the centre is the pentagon which forms the faces of the 
dodecahedron itself. You can select any closed polygon from a stellation 
diagram such as the set of 5 triangles which are labelled 1st stellation. 
You only have to define one of the triangles, because !Stellate knows the
rotational symmetry of the component (see the table in Draw file
Component) and will calculate the positions of the others after you have
told it whether you are looking at the front or the back of that particular
face. The number between ( ) in the window title is the rotational symmetry
for the face of that component. (see Tutorial2 for detailed instructions). 

When a polyhedron is made from 2 or more components, these all have the same 
centre. They also should have axes with the same rotational symmetries which
!Stellate then aligns. The two rotational symmetries for the components are
listed in the second column in the table in Draw file Component. The first
number is the symmetry of a face and the second that of a vertex; for
example, the dodecahedron has faces with 5 sides and vertices where 3 edges
meet. The last five components in the table have the two symmetries shown in
brackets but these refer to their two kinds of symmetric vertices.
The Snub- faces are not shown extended to show their symmetric vertices;
in the diagram they appear as symmetric white faces.

The faces of one component intersect another, if their radii are suitably
chosen, to make a new polyhedron; (see Draw File Tutorials.Pictures.Ex2Comp
for explanation of a 2 component diagram).

The lines created by projecting the planes of the 1st component are drawn
solid on all the stellation diagrams; those of the 2nd component are dotted
lines, those of the third are drawn dashed and those of the fourth are
dot-dashed. (See any Draw File with more than one component,
Tutorials.Pictures.Ex2Comp or the Draw files in TryThese.) 

Stellation numbering

Our convention is to number the stellations of the polyhedra in order going
outwards from the centre in shells of the 3D space. This does not always
agree with that used in Wenningers book Polyhedral Models

To make the faces for the 1st stellation of the dodecahedron, you choose the
next set of polygons going outwards from the central pentagon. These are the
triangles which fit on the edges of the pentagonal faces.

The next set of closed triangles are the ones with points just touching the
central pentagon. The 3rd and last are the much larger triangles with faces
touching the bases of the triangles of the 2nd stellation.  This is not drawn
in the Draw file Page6 because it is much bigger than the others.

(Now try drawing all 3 stellations of the dodecahedron for yourself)

Some polyhedra have many stellations; the icosahedron has 59 as explained in
the booklet The Fifty Nine Icosahedra by H.S.N Coxeter et al. (reprinted
by Tarquin in 1999)

The uniform snub polyhedra (Wenninger numbers 110 to 119) have three or four
components and are often very complicated. You can find some notes on these
in Reference.Snubs.
