
 Stella Polyhedral Glossary 
 A  B  C 
D  E  F 
G  H  I 
J  K  L 
M  N  O 
P  Q  R 
S  T  U 
V  W  X 
Y  Z 
This glossary contains terms relating predominantly to
polyhedra and
stellation theory in three dimensions. I'm afraid
you'll have to look elsewhere for anything relating to higher dimensions!
A
 Aggregate

A word coined by George Olshevsky for stellations with
no internal faces, such as those described in The Fiftynine Icosahedra.
This is also the kind of stellation made by
Great Stella.
They look identical from the outside to other stellations with internal faces,
but may be topologically different.
 Antiprism

Antiprisms need not be semiregular, but usually
that is what is meant. A semiregular antiprism is a
polyhedron made by starting with two
identical regular polygons. These are placed
parallel to each other, and with a twist so that the vertices of one lie
between consecutive vertices of the other. These are then connected using
equilateral triangles which attach to the edge of one polygon, and extend to
the vertex of the other polygon which lies above/below that edge. Note that
sometimes the polygons align exactly, eg when using a
pentagram, because the polygon's own vertices already
lie between two other consecutive vertices.
See some examples here.
 Archimedean solids

The 13 convex semiregular
polyhedra, excluding
prisms and antiprisms. Note,
they are named after Archimedes, and it is not spelt any of the
following ways: archimedian, archimidean, archimidian, archemedean,
archemedian, archemidean, or archemidian! See some examples
here.
 Augment, Augmentation

To add a pyramid (or sometimes a
cupola) to a face of a
polyhedron. This creates one new
vertex, and a new face for each side of the face being
augmented (and of course you lose that original face).
This is the dual process to truncation.
B
C
 Canonical

A polyhedron is canonical if all of its
edges are tangent to a unit sphere (ie it is
semicanonical), and the average of all the points of
contact between edge lines and the sphere is the centre of the sphere.
 Catalan

The catalan solids are the duals of the
Archimedean solids. See them
here.
 Cell

A stellation cell is a convex region of space
bounded by some of the stellation planes, and not intersected by any others.
Usually the cells of interest are finite, but there are also infinite cells,
where the region is not bounded on all sides. See
stellate for more information.
A cell may also refer to a polyhedron that
forms part of the surface of a fourdimensional
polytope. It is the 4D equivalent of a
face in 3D.
 Cell diagram

A cell diagram is a graph indicating how
stellation cells relate to each
other. The layers of cells are shown as layers of nodes,
each node representing one cell type. Edges of the
graph connect nodes from one layer to nodes of the next layer if the two cell
types share a common face, so the cell type at the bottom of the edge
supports the cell type at the top of the edge.
 Cell type

A minimal set of stellation cells which together follow the
symmetry group of the model being stellated.
See stellate for more information.
 Chiral

A polyhedron which is not its own mirror image.
These come in left and right pairs, such as the snub cube.
Opposite of reflexible.
 Circumradius

Radius of the circumsphere, if one exists.
 Circumsphere

All uniform polyhedra have a circumsphere, which
is a sphere passing through each vertex of the model.
 Coincidic

A polychoron is coincidic if any two of its
cells are corealmic and share
elements that span their realm. That is, they share
vertices that do not all lie in the same 2D plane. Since other elements like
edges and faces are delimited by vertices, it suffices to consider vertices
alone. A typical example of two such cells would be an
icosahedron and a
great dodecahedron sharing the same vertices.
They also share the same edges. Two examples of coincidic
scaliforms are available in
Stella4D:
idfix and irgfix.
 Compound

A model consisting of two or more polyhedra.
Compounds of interest usually have an interesting
symmetry group overall, and the polyhedra usually
have their symmetry groups at least partially aligned, share a common centre,
and are often all the same. It is also common to have a compound of a
polyhedron with its dual.
 Concave

A polygon is concave if it has at least one angle
of greater than 180 degrees between consecutive sides, eg a dart or
arrowhead shaped polygon is concave, but a
pentagram is better referred to as
nonconvex, because although it is not
convex, all of its five angles are less than 180 degrees.
Similarly, a polyhedron is concave if it has
at least one dihedral angle greater than 180 degrees.
Stewart toroids are a good example (like
this one), but the
great dodecahedron would be better referred to as
nonconvex.
 Convex

A convex
polygon or
polyhedron is one where any line segment drawn from a
point inside the shape to another point inside the shape, will lie entirely
within the shape.
 Convex hull

The convex hull of a
polygon or
polyhedron is the smallest
convex polygon or polyhedron which encloses the given
shape.
 Corealmic

In 4D, two cells are said to be corealmic if they
lie in the same realm (a 3D
hyperplane).
 Crossed antiprism

Similar to an antiprism, but the triangles used to
connect the two polygons reach across the polyhedron, connecting an edge
of one polygon with the opposite vertex of the other polygon. The
vertex figure is thus a crossed quadrilateral.
 Cuploid

You should read about what a cupola is first.
Cuploids are similar, having a top face which is an
n/dgon, but now d is even, which would make the bottom
2n/dgon degenerate, being like two
n/(d/2)gons exactly aligned. This means that we have a square and a
triangle meeting at each degenerate edge, so we can remove the bottom face
completely and the result is a true polyhedron (nondegenerate). This
polyhedron is a cuploid. For example, the top face could be a pentagram, which is 5/2, making the bottom a 10/2, which
appears to be simply a pentagon (5/1). The bottom face is not present, but the
edges form a pentagon.
 Cupola

A polyhedron constructed as follows. Start with an
n/dgon (the top, which may or may not be
retrograde), and place a 2n/dgon in a
parallel plane (the bottom). Here d must be odd. Squares attach
from the edges of the top to alternate edges of the
bottom. The other bottom edges connect to triangles which fill the gaps
between the squares and touch a top vertex. The plural form is cupolae.
 Cupolaicblend

A polyhedron constructed as follows. Take two
cupolae exactly aligned, and rotate one of them until
their bottom faces align again. Now again, as with the
cuploid, the bottom faces can be removed leaving a true
polyhedron, with squares and triangles meeting at the bottom edges. Note that
this model has two top faces, in the same plane, but twisted with respect to
each other.
D
 Degenerate

A polyhedron is degenerate when some features
align exactly. It may be vertexdegenerate,
edgedegenerate, or facedegenerate.
A true polyhedron is normally defined as having exactly two faces meeting at
each edge. There would be more uniform polyhedra if
this restriction was lifted, and these can often also be thought of as
compounds of other uniform polyhedra. Usually in these
cases there are four faces meeting at some edges, so there's more than one way
to think of which face is attached to which face.
 Deltahedron

A polyhedron whose faces are all
equilateral triangles.
 Dihedral angle

The angle between adjacent faces of a
polyhedron, measured on the inside of the model. For
example, 90 degrees for all edges of a cube. It will
be less than 180 degrees for convex edges, or greater than 180 degrees for
concave edges. Note also that some people measure the
dihedral angle as the angle between the normals of the
faces, rather than between the face planes themselves. Subtract the value from
180 degrees to convert between these two representations.
 Dorman Luke construction

A method for finding the shape of a uniform polyhedron's
dual from its vertex figure. The
vertices of the vertex figure lie on a circle. For each of these vertices in
turn, draw a tangent to the circle at that point. An edge of the dual's face
will lie along this tangent, extending from where it hits the previous vertex's
tangent to where it hits the next vertex's tangent.
 Dual

Every polyhedron has a dual, and the dual of
that dual brings you back to the original polyhedron again. The two models
share the same number of edges, but have the number of
vertices and faces exchanged.
Roughly speaking, you create the dual by replacing faces with vertices, and
vertices with faces, the edge generally turning by 90 degrees.
For example, the cube and
octahedron are duals, as are the
dodecahedron and
icosahedron. The
tetrahedron however is its own dual.
For simple models like these, the dual can be create by connecting the
midpoints of faces around each vertex to each other, forming new faces.
More generally though, the exact operaton used to create the dual is called
spherical reciprocation, which is done with respect to a
sphere. The midsphere is usually used if one exists.
Using different spheres will distort the resultant dual, although repeating the
operation with the same sphere will always bring you back to the original
model. Note that any faces in planes passing through the centre of the sphere
will lead to infinite vertices in the dual, and exactly how to draw such a
model becomes hard to define.
E
 Edge

Where the faces of a polyhedron
meet. The edge starts and ends at a vertex. Note that
each edge connects exactly two vertices, and is the boundary between exactly
two faces (except in degenerate cases).
 Edgelet

This term is similar to facelet, but with reference to
edges instead of faces. An edgelet is what
appears to be an edge from outside a
polyhedron, so these are the edges that you will need
to score/fold/cut to build a physical model. True edges may intersect through
other faces and be partly hidden inside the model.
 Edgestellation

An edgestellation of a polyhedron is a
polyhedron whose edges lie in the same lines as the edges
of the original polyhedron. For example, the
great stellated dodecahedron may be considered
to be an edgestellation of the
icosahedron.
 Elementary region

A 2D region bounded by some lines in the
stellation diagram, and not intersected by any others.
 Enantiomorph

The mirror image of some chiral polyhedron.
 Exopolyhedron

A model which obeys all the defining criteria for a
polyhedron
except that some or all of its edges coincide with other
edges, that is, more than two faces meet at some edges.
However none of the faces may coincide with other faces. This leads to
ambiguity in the vertexfigure circuit, meaning that more
than one topology could be used to represent the model.
F
 Face

What we call the flat sides of a polyhedron.
Each face is a polygon.
For example, a cube has square faces.
 Facelet

The external parts that would need to be cut out and stuck together in order to
build a polyhedron, literally meaning "small face".
The true faces of a polyhedron may intersect each other,
leaving some parts that are hidden from view inside the model. To build a
physical model only the parts that are visible from outside need to be put
together. These facelets could include whole faces, or just smaller
parts of faces. For example, the
great dodecahedron has 12 pentagons for faces, but
there are 60 facelets, 5 on each face. See also
edgelet.
 Facet

A facet of a polyhedron is a
polygon whose vertices are all vertices of that
polyhedron, although it is generally not a face of that polyhedron.
See faceted.
 Faceted, faceting

A polyhedron may have many faceted forms. A
faceted model, also know as a faceting, has the same
vertices as the original model, but different
faces connecting them. These new faces are
facets of the original polyhedron. For example the
tetrahemihexahedron is a
faceted form of the octahedron.
Faceting is the dual operation of stellation.
Whereas stellation keeps the same facial planes, but changes the vertices,
faceting keeps the same vertices, but changes the faces.
 Fissary

A polytope is fissary if it has compound
vertices, edges, faces, or cells, etc. Being fissary rules out a
polychoron from being included in the official list
of uniform polychora, although some people think they
should still be included. They are omitted because such cases dominate in
higher dimensions. Fissaries are still interesting though, and a number of
uniform fissary polychora are included in
Stella4D.
 Fully connected

I call a stellation fully connected if it satisfies
Miller's rules, and the model does not consist of parts
that either don't touch, or only touch at vertices or along edges. This means
that some small enough creature inside could get from any part to any other
part. John Gingrich proposed these criteria while studying the
rhombic triacontahedron, and referred to them as the
Gingrich rules. In a way, they are the most "sensible" rules, and
always produce models which can be physically built (in theory!). However,
they also turn out to be the hardest rules to use!
 Fully supported

A stellation is fully supported if all of its
included cells are supported.
Another way of saying this is that any ray from the centre of the original
polyhedron outward in any direction will only cross the model's surface once.
Another term for this is radially convex. This is the default criteria
for a valid stellation in
Great Stella.
G
 Genus

Roughly speaking, this is the number of holes through a
polyhedron (or other 3D object). A cube has genus 0,
a doughnut has genus 1, and so on. With many selfintersecting polyhedra,
including many uniform polyhedra, it can be hard to say
what the genus is just by looking at them, since the "holes" may not be
apparent due to intersecting faces getting in the way.
 Golden ratio

The value (1 + sqrt(5)) / 2, which is approximately 1.6180339887. This value
turns up all over the place for models with icosahedral symmetry, just as
sqrt(2) turns up a lot for models with octahedral symmetry. The value has many
strange properties. For example, to square it just add 1, or to reciprocate it
subtract 1.
H
 Hemiface

A face of a polyhedron which
passes through the exact centre of the polyhedron.
 Hemipolyhedron

A polyhedron which has some
hemifaces, ie faces which pass through the exact
centre of the polyhedron.
For example, the tetrahemihexahedron.
 Homohedral

A polyhedron is homohedral if all its
faces are congruent, although they might each have a
different relationship to the solid as a whole. If the relationship is the
same for each face, then the polyhedron is isohedral.
Deltahedra are examples of homohedra.
 Hyperplane

A hyperplane is like a plane but in higher dimensions. For example in
4D a hyperplane could be threedimensional. Each cell of a
polytope lies in a hyperplane. A 3D hyperplane is
called a realm.
I
 Inradius

Radius of the insphere, if one exists.
 Insphere

All duals of uniform polyhedra have
an insphere, which is a sphere touching the plane of each
face exactly once.
 Isogonal

A polyhedron is isogonal if any of its
vertices may be rotated and/or reflected to any other
by one of the polyhedron's symmetries. In other words,
all vertices are the same (ie have the same relationship to the whole
polyhedron). All uniform polyhedra are isogonal.
Also known as vertextransitive.
 Isohedral

A polyhedron is isohedral if any of its
faces may be rotated and/or reflected to any other by one
of the polyhedron's symmetries. In other words, all
faces are the same (ie have the same relationship to the whole polyhedron).
All duals of uniform polyhedra are
isohedral. Also known as facetransitive.
 Isomer

One polyhedron is an isomer of another (ie
they are isomeric) if they have the same number of
faces of each kind, the same number of vertices, and the
same number of edges, but they are not
isomorphic. Also, the number of vertices of each
type must match, eg one type of vertex may be surrounded by three
squares and one triangle. For example the
rhombicuboctahedron and its
pseudo version.
 Isomorphic

Two polyhedra are isomorphic if they
share the same topology. For example the
icosahedron and the
great icosahedron both have five triangles meeting
at each vertex, so they are isomers.
 Isotoxal

A polyhedron is isotoxal if any of its
edges may be rotated and/or reflected to any other
by one of the polyhedron's symmetries. In other words,
all the edges are the same (ie have the same relationship to the whole
polyhedron). All quasiregular polyhedra are isotoxal,
for example, and the dual of any isotoxal polyhedron will
also be isotoxal. Also known as edgetransitive.
J
 Johnson solids

The nonuniform convex
polyhedra with regular
faces. That is, all the convex polyhedra with regular
faces other than the Platonic solids,
Archimedean solids, prisms, and
antiprisms. There are 92 such models, labelled J1 to
J92. Two examples are the
snub disphenoid (J84) and the
bilunabirotunda (J91).
K
 KeplerPoinsot solids

The 4 nonconvex regular
polyhedra. See them here.
L
 Layer

When stellating a polyhedron,
cells form layers from the centre outwards.
Generally there is a single central cell, which is the region under all
the face planes, where the volume under a face is whichever side
contains the centre if the polyhedron. In other words, the central cell is the
one which contains the centre of the polyhedron. For
hemipolyhedra, there are several central cells, each
having a vertex at the centre of the polyhedron. The central cell/cells form
the innermost layer (usually referred to as layer 0). Each layer after that is
made up of the minimal set of cells required to completely cover the previous
layer (or cover as much as possible for the outer layers where sometimes the
previous layer can not be completely covered).
M
 Mainline stellation

A mainline stellation consists of all the
cells in some layer and all the layers
below. Thus the number of mainline stellations is relatively small, being the
same as the number of layers.
 Midradius

Radius of the midsphere, if one exists.
 Midsphere

All uniform polyhedra and their
duals have a midsphere, which is a sphere touching
each of their edges exactly once. Actually, it's the line
containing the edge which must touch the sphere once, since sometimes they
touch beyond the end of the edge for some dual models. Uniform models always
touch the sphere exactly half way along each edge.
 Miller's rules

J. C. P. Miller came up with a set of rules for deciding which
stellations should be counted as valid. They are
worded specifically for stellations of the
icosahedron, and are as follows:
 The faces must lie in twenty planes, viz., the bounding planes
of the regular icosahedron.
 All parts composing the faces must be the same in each plane, although
they may be quite disconnected.
 The parts included in any one plane must have trigonal symmetry, with
or without reflection. This secures icosahedral symmetry for the whole
solid.
 The parts included in any plane must all be "accessible" in the
completed solid (i.e., they must be on the "outside". In certain
cases we should require models of enormous size in order to see all
the outside. With a model of ordinary size, some parts of the "outside"
could only be explored by a crawling insect).
 We exclude from consideration cases where the parts can be divided
into two sets, each giving a solid with as much symmetry as the whole
figure. But we allow the combination of an enantiomorphous pair having no
common part (which actually occurs in just one case).
These rules can easily be extended for finding stellations of any polyhedron.
The first rule is really just the definition of a stellation. The next two
rules specify that the stellation should have the same full symmetry, possibly
without reflection, of the original polyhedron. The fourth rule asks that the
nodes in the cell diagram be connected, that is,
that the cell types be connected to each other. This
does not necessarily mean that the final model will be connected though, since
individual cells within a particular cell type may not be
physically connected. And the fifth rule requires that all the nodes of the
cell diagram which aren't used are also connected, with one exception.
There has been some debate over exactly what Miller meant by that exception
though, in the last part of the fifth rule (see
here).
 Monoacral stellation

The term monoacral was suggested by Peter Messer for a
stellation consisting of a single
cell type and all its
supporting cells (and their supporting cells etc).
Another way of saying this is that for some cell type, it is the minimal
fully supported stellation which includes that cell
type. The number of such stellations is thus generally quite small, being the
same as the number of different cell types (unless some cell types are
unsupportable).
These stellations are often very appealing, as they are generally simpler and
not too "messy".
 Monohedral

The word you're looking for is probably homohedral.
It was argued that monohedral would mean a polyhedron with only one
face, rather than only one kind of face.
 Monostratic section

A section of a polyhedron between two parallel
planes, each passing through some of its vertices, and
with no other vertices in between.
N
 NearMiss

Often used to describe polyhedra that are nearly
Johnson solids. This generally means that the faces are
almost regular, so you could build a model using
regular faces without noticing the error.
 Net

Flat patterns which can be cut out and folded up to make a
polyhedron.
 Noble

A noble polyhedron is both
isohedral (all faces the same), and
isogonal (all vertices the same).
Max Brückner studied these in 1906.
 Nonconvex

A polygon or
polyhedron which is not
convex.
 Nonorientable

Opposite of orientable.
 Normal

A normal to a face or plane is a vector
perpendicular to that face or plane.
O
 orientable

A polyhedron is orientable if an orientation
may be given to each face, ie which order to visit the
face's vertices in (either clockwise or counterclockwise),
such that an edge is always traversed in opposite directions according to the
two faces either side of it. This implies that if you start on one side of a
face, and move from face to face, you will never be able to end up on the
opposite side of the original face. Think about moving over the surface of a
cube, you can't end up inside. If a polyhedron is nonorientable, then
the surface behaves like a Mobius strip, and you can end up on the opposite
side of a face by moving over its surface.
P
 Pentagram

A five pointed star, usually regular, in which
case it has the same vertices as a regular pentagon. It has the symbol 5/2.
 Platonic solids

The 5 convex regular
polyhedra. See them here.
 Polar reciprocation

See spherical reciprocation. Actually polar reciprocation
is a more general term because some curved surfaces other than spheres can be
used.
 Polychora

Plural of polychoron.
 Polychoron

The name coined by George Olshevsky for a fourdimensional
polytope. In 3D, the surface of a
polyhedron is made up of 2D
polygons called faces, but in 4D the
surface consists of 3D polyhedra called cells.
 Polygon

Twodimensional closed shape, bounded by line segments, typically with exactly
two line segments, or sides, meeting at each vertex.
The faces of polyhedra are
polygons.
 Polyhedra

The plural of polyhedron.
 Polyhedron

Threedimensional object bound by polygons.
The polygons, or faces, are typically planar and finite,
and meet with exactly two at each edge. If more than two
faces meet at each edge, the model is sometimes called
degenerate.
 Polytope

The equivalent of a polyhedron, but in any number of
dimensions. A polyhedron is always threedimensional. A
polychoron is always fourdimensional.
 Primary line

A line of a stellation diagram which happens to lie in
a reflection plane of the polyhedron.
 Primary region

A region of a stellation diagram which is bounded only
by primary lines.
 Primary stellation

A primary stellation is one which only has
edges which lie on primary lines. So all the faces
are primary regions. If you think about it, this
implies that the stellation must be isohedral.
Thus this kind of stellation is only welldefined for isohedral polyhedra,
and not for chiral polyhedra, where there are no
reflection planes.
 Prism

A prism is a polyhedron made by connecting two
identical polygons with rectangles between their corresponding edges. Usually
I am referring to semiregular prisms, where the
rectangles are squares, and the two polygons being connected are
regular polygons.
See some examples here.
 Pseudo uniform

Similar to just uniform, but we require each vertex to
be locally the same only, that is, each vertex is surrounded by the same
sequence of face types, but the whole model can not always be made to align
with where it was when rotating and/or reflecting from one vertex to another.
For example the
pseudo rhombicuboctahedron.
 Pyramid

A pyramid is a polyhedron with a
face of any shape as its base, and triangles
attached to each side, meeting at the same vertex, called the apex.
Q
 Quasiregular

A polyhedron is quasiregular if it is
uniform and isotoxal (ie all
edges are the same). Example: icosidodecahedron.
R
 Radially convex

Same as fully supported.
 Realm

A 3D hyperplane.
 Reentrant

Opposite of fully supported. Some rays from the centre
of a reentrant stellation outwards will cross
the surface of the model more than once.
 Reflexible

A polyhedron which is its own mirror image, such as
the cube. Opposite of chiral.
 Regular polygon

A polygon where all sides are the same length, and all
angles between consecutive sides are the same, eg the square or regular
pentagon. The centre may be enclosed more than once, in which case the polygon
is nonconvex, such as the
pentagram.
 Regular polyhedron

A polyhedron where all faces are identical
regular polygons, and all
vertex figures are identical regular polygons. There are
five convex regular polyhedra (the
Platonic solids), and four
nonconvex ones (the
KeplerPoinsot solids).
 Retrograde

Refers to faces meeting at a
vertex which circle back the opposite way around the
vertex with respect to other faces. Polygons
represented as n/d are retrograde when the fraction is greater than a
half, eg the pentagram 5/2 is not retrograde, but
written as 5/3 it is. The retrograde version of a n/dgon is a
n/(nd)gon.
S
 Scaliform

A scaliform polytope is
uniform in that it is
vertextransitive and has regular faces, but not all
of its cells are uniform polyhedra.
See the Scaliform section of the 4D Library in
Stella4D
for examples.
 Schläfli symbol

A symbol used to represent a regular polyhedron. It is
written as {p, q}, where p is the number of sides of each
face, and q is the number of faces meeting at each
vertex.
 Schwartz triangle

A sphere's surface may be broken into spherical triangles by intersections with
the various reflection planes of some symmetry group.
Each of these triangles, which tessellate the sphere,
is called a Schwartz triangle. Their vertices lie at intersections
between the sphere and the rotational symmetry axes of the symmetry group.
 Schlegel diagram

For a (typically convex) polyhedron, this is a 2D diagram representing the
faces of the polyhedron flattened out into a single diagram. This can be
achieved by projecting the vertices onto a particular face, towards a point
just above the face, so that the entire projected model lies inside this single
face. For a convex polyhedron this should produce a diagram where all faces
from the original polyhedron are visible and not overlapping (no edges will
overlap).
For a 4D polytope, the Schlegel diagram is a 3D structure, created along
similar lines.
 Secondary line

A line of a stellation diagram which does not lie in
a reflection plane of the polyhedron. See also
primary line.
 Segmentohedron

A polyhedron whose vertices
all lie on a common circumsphere, and also on two
parallel planes. In addition, all edges must be the same
length. This also implies that all the faces are
regular. Note also that such polyhedra are
monostratic. The convex
segmentohedra are also Johnson solids: two pyramids and
three cupolae. The nonconvex ones include
cuploids and
cupolaic blends.
 Selfintersecting

A polygon whose sides cross over each other, or a
polyhedron whose faces pass
through each other.
 Semicanonical

A polyhedron is semicanonical if all of its
edges are tangent to a unit sphere. See also
canonical.
 Semiregular polyhedron

A uniform polyhedron which is not
regular. This includes the 13 convex
Archimedean solids, an infinite array of
prisms and antiprisms (both
convex and nonconvex), and the remaining nonconvex, nonregular uniform
polyhedra.
 Spherical reciprocation

The method used to construct the dual of some
polyhedron. It is done with respect to some sphere.
For uniform polyhedra we usually choose the
midsphere. We create a vertex
for each face of the original polyhedron. Cast a ray from
the centre of the polyhedron out through the face, and perpendicular to it.
The vertex will lie on this ray. The distance from the centre to the vertex is
the reciprocal of the distance to the face. A face is then created for each
original vertex, connecting the new vertices which represent old faces sharing
that original vertex.
 Stellate, stellation

A stellated polyhedron is called a
stellation. Stellating a polyhedron is a very powerful way of creating
a large number of new polyhedra, most of which often bear little resemblance to
each other. It is NOT just a matter of attaching a pyramid to each
face! (See augmentation).
A polyhedron is made up of faces, and each face lies in
some plane. If we consider the whole of each plane, rather than just the area
bounded by each face, we get a bunch of planes which all intersect each other
many times. You may think of these planes as carving up space. One plane
divides space into two halves. Two planes divides this further into four
parts. A third plane will generally divide space into eight parts, but so far
all the parts are infinite, that is, no part is bounded yet. When we add a
fourth plane, space is divided up again, and this time we might have a single
bounded region of space. The tetrahedron has four faces, and here the four
planes do indeed enclose a region of space: the tetrahedron itself. As we add
more planes, many more bounded regions of space, called
cells, are created.
A collection of cells which together follow the
symmetry group of the model being stellated (and where
no smaller collection does) is referred to as a
cell type. Generally (and from here on) when I say
cell I really mean cell type, since you don't often want to refer
to just one single cell on its own.
So finally, a stellation is some combination of these cells put together to
form a single polyhedron. Sometimes the stellation may have disconnected parts
floating around separately in space, or it may have parts connected by vertex
or edge only, or it could be one solid piece. Due to the huge number of
possible combinations of cells, various people proposed various criteria for
deciding whether a given combination should be considered valid or not. Other
people came up with different criteria in order to help find "interesting"
stellations, or at least ones that could be physically built! Here are some of
the criteria used, ordered roughly from least restrictive (allowing the most
valid stellations) to most restrictive (allowing the fewest stellations):
Miller's rules,
fully connected,
fully supported,
monoacral (singlepeaked),
primary, and
mainline.
See some examples of stellations here.
 Stellation diagram

This is a twodimensional diagram in the plane of some
face of a polyhedron, showing
lines where other face planes intersect with this one. The lines cause 2D
space to be divided up into regions. Regions not intersected by any further
lines are called elementary regions. Usually infinite
regions are excluded from the diagram, along with any infinite portions of the
lines. Each elementary region represents a top face of one
cell, and a bottom face of another. A collection of these
diagrams, one for each face type, can be used to represent any
stellation of the polyhedron, by shading the regions
which should appear in that stellation.
 Stellation pattern

The set of elementary regions within the
stellation diagram which are required for some
particular stellation.
 Stewart Toroid

Professor Bonnie Madison Stewart wrote a book called Adventures Among the
Toroids, in which he describes a great many toroidal polyhedra (ie they
have holes through them). He presented various criteria to narrow his search.
The term Stewart Toroid is used loosely for models adhering to all or
some of these criteria. The criteria are as follows, for some polyhedron P:
 (R) Each face of P must be
regular.
 (A) Faces of P which share an edge must not be coplanar ("A" is for
"aplanar").
 (Q) It must be quasiconvex, ie. every edge of the
convex hull of P is an edge of P.
 (T) P must be tunnelled, ie. P contains an excavation which
amends the genus of P. Excavations which do not add
to the genus of P are not allowed.
 (D) P must have a disjoint interior, ie. selfintersection is not
allowed within or between faces.
 Subsymmetry group

One symmetry group may be a subsymmetry group
of another if all of its rotations and reflections are included in that group.
For example the tetrahedral symmetry group is a subsymmetry group of
the octahedral symmetry group.
 Support

A stellation cell is said to
be supported if all cells from the layer below
which touch it are also included as part of the stellation (unless the cell is
unsupportable). Cells have top and
bottom faces, where bottom faces are the ones that can be "seen"
from the polyhedron's centre if no other cells are present. So a cell is
supported if all its bottom faces are covered by the adjacent cells.
 Symmetry group

A collection of rotations and reflections which define the overall symmetry of
a polyhedron. For example the octahedral symmetry
group (eg think of a cube) has three 4fold axes of
symmetry, four 3fold axes of symmetry, and six 2fold axes of symmetry.
T
 Tessellate (or tesselate)

To fill space completely without leaving gaps. This may refer to space of any
number of dimensions, and sometimes different surfaces too, for example you can
tessellate the surface of a sphere.
 Topology

How things are connected, as opposed to where they actually lie in space. The
topology of a polyhedron specifies how its
faces are attached to each other, and how its
vertices are connected by edges, but is not concerned
with the coordinates of those vertices. Nor is it concerned with false
edges where faces may intersect, but do not really share an edge.
 Transitive

Means that there is only one type of a certain element throughout a
polytope or compound.
One type means that the element may be rotated and/or reflected to any
other by one of the polytope's symmetries.
Vertextransitive is the same as isogonal.
Edgetransitive is the same as isotoxal.
Facetransitive is the same as isohedral.
 Truncate, Truncation

To slice off part of a polyhedron, losing one of the
original vertices. This will create a new
face with the shape of the vertex figure,
as well as one new vertex for each edge which met at the
original vertex. Often truncation will be applied to all vertices at once.
This is the dual process to augmentation.
U
 Uniform

A uniform polyhedron consists of
regular polygonal faces, and
is vertextransitive.
This also implies that each vertex is surrounded by the same sequence of face
types (see vertex description), that all vertices lie on a
sphere, and that all edges are the same length. The set includes the
regular and semiregular
polyhedra. See some examples here.
The term extends to 4D. Again, all vertices are the same and faces must be
regular, but we only require that cells be uniform, not
necessarily regular. If the cells themselves are not uniform, then the
polytope is called scaliform.
 Unsupportable

A stellation cell is said to
be unsupportable if it can not be supported.
This is an unusual case which only happens when a cell has bottom faces
which are adjacent to infinite cells. See here
for more information.
V
 Valence

The number of faces meeting at a
vertex.
 Vertex

Where edges of a polyhedron or sides of a
polygon meet at a point.
 Vertex cycle

The sequence of
face types meeting around a meeting around a
vertex. Similar to the
vertex description.
 Vertex description

A way of describing a uniform polyhedron, by giving the sequence of
face types
meeting around a vertex. Eg "4, 4, 4" represents a
cube, because there is a sequence of three
regular 4sided
polygons (squares) around each vertex. Most people
prefer to use dots rather than commas to separate the faces, but some reason I
like commas a lot more. Each face is specified as n/p, where n
is the number of sides, and p is the number of times the polygon
encloses its centre. Eg 5/2 means a
pentagram. When p is 1 we leave it
out. To specify a
retrograde face, use n/(np), eg 5/3 for a
pentagram which goes the opposite way around the vertex. Finally, if the
vertex is surrounded more than once by the faces, put it in parenthases and add
/q at the end, where q is the number of times the faces circle
the vertex.
 Vertice

NO SUCH WORD! You probably mean vertex.
 Vertices

Plural of vertex.
 Vertex figure

The polygon you get as a crosssection when you cut
a vertex off a polyhedron.
Each face meeting at the vertex will cause a line segment in the vertex figure,
so the vertex figure will have the same number of sides as the number of faces
meeting at the vertex. Faces not meeting at the vertex are ignored. Each edge
meeting at the vertex should be intersected by the cutting plane at the same
distance from that vertex, often half way along the edge.
For uniform polyhedra there is only one type of
vertex figure, since all vertices are the same. The vertex figure can be used
in this case to find the shape of the face of the
dual polyhedron. This method is called the
Dorman Luke construction. Note also that in this case all
vertices in the vertex figure lie on a circle.
W
 Wythoff symbol

A symbol used to represent a uniform polyhedron, based
on symmetric repeats of a point through some
symmetry group, and where that point was positioned
initially with respect to the Schwartz triangles.
Too involved for a full explanation here.
X
Y
Z
 Zonohedron

A polyhedron whose faces all have
sides occuring only in parallel pairs. The faces will always have an even
number of sides (eg parallelograms, hexagons, etc), and opposite sides are
parallel and have the same length. A zone is a loop of faces connected
by opposite (parallel) sides. So a face with 2N sides will be part of
N zones. A whole zone can be removed to create a new model with one
less zone. After removing all faces in a zone, the two disjoint halves
remaining will fit together perfectly to form the new polyhedron, since all
edges across the zone were the same length. Similarly you can add zones to a
polyhedron by splitting it apart and inserting new faces.
 A  B  C 
D  E  F 
G  H  I 
J  K  L 
M  N  O 
P  Q  R 
S  T  U 
V  W  X 
Y  Z 
