Stella Polyhedral Glossary

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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 word coined by George Olshevsky for stellations with no internal faces, such as those described in The Fifty-nine 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.
Antiprisms need not be semi-regular, but usually that is what is meant. A semi-regular 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 semi-regular 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.



A polyhedron is canonical if all of its edges are tangent to a unit sphere (ie it is semi-canonical), and the average of all the points of contact between edge lines and the sphere is the centre of the sphere.
The catalan solids are the duals of the Archimedean solids. See them here.
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 four-dimensional 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.
A polyhedron which is not its own mirror image. These come in left and right pairs, such as the snub cube. Opposite of reflexible.
Radius of the circumsphere, if one exists.
All uniform polyhedra have a circumsphere, which is a sphere passing through each vertex of the model.
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.
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.
A polygon is concave if it has at least one angle of greater than 180 degrees between consecutive sides, eg a dart or arrow-head 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.
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.
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.
You should read about what a cupola is first. Cuploids are similar, having a top face which is an n/d-gon, but now d is even, which would make the bottom 2n/d-gon 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 (non-degenerate). 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.
A polyhedron constructed as follows. Start with an n/d-gon (the top, which may or may not be retrograde), and place a 2n/d-gon 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.
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.


A polyhedron is degenerate when some features align exactly. It may be vertex-degenerate, edge-degenerate, or face-degenerate. 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.
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.
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.


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).
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.
An edge-stellation 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 edge-stellation of the icosahedron.
Elementary region
A 2D region bounded by some lines in the stellation diagram, and not intersected by any others.
The mirror image of some chiral polyhedron.
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 vertex-figure circuit, meaning that more than one topology could be used to represent the model.


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


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 self-intersecting 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.


A face of a polyhedron which passes through the exact centre of the polyhedron.
A polyhedron which has some hemi-faces, ie faces which pass through the exact centre of the polyhedron. For example, the tetrahemihexahedron.
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.
A hyperplane is like a plane but in higher dimensions. For example in 4D a hyperplane could be three-dimensional. Each cell of a polytope lies in a hyperplane. A 3D hyperplane is called a realm.


Radius of the insphere, if one exists.
All duals of uniform polyhedra have an insphere, which is a sphere touching the plane of each face exactly once.
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 vertex-transitive.
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 face-transitive.
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.
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.
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 quasi-regular polyhedra are isotoxal, for example, and the dual of any isotoxal polyhedron will also be isotoxal. Also known as edge-transitive.


Johnson solids
The non-uniform 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).


Kepler-Poinsot solids
The 4 nonconvex regular polyhedra. See them here.


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 hemi-polyhedra, 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).


Main-line stellation
A main-line stellation consists of all the cells in some layer and all the layers below. Thus the number of main-line stellations is relatively small, being the same as the number of layers.
Radius of the midsphere, if one exists.
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:
  1. The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.
  2. All parts composing the faces must be the same in each plane, although they may be quite disconnected.
  3. The parts included in any one plane must have trigonal symmetry, with or without reflection. This secures icosahedral symmetry for the whole solid.
  4. 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).
  5. 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".
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.


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.
Flat patterns which can be cut out and folded up to make a polyhedron.
A noble polyhedron is both isohedral (all faces the same), and isogonal (all vertices the same). Max Brückner studied these in 1906.
A polygon or polyhedron which is not convex.
Opposite of orientable.
A normal to a face or plane is a vector perpendicular to that face or plane.


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 counter-clockwise), 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.


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.
Plural of polychoron.
The name coined by George Olshevsky for a four-dimensional 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.
Two-dimensional closed shape, bounded by line segments, typically with exactly two line segments, or sides, meeting at each vertex. The faces of polyhedra are polygons.
The plural of polyhedron.
Three-dimensional 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.
The equivalent of a polyhedron, but in any number of dimensions. A polyhedron is always three-dimensional. A polychoron is always four-dimensional.
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 well-defined for isohedral polyhedra, and not for chiral polyhedra, where there are no reflection planes.
A prism is a polyhedron made by connecting two identical polygons with rectangles between their corresponding edges. Usually I am referring to semi-regular 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.
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.


A polyhedron is quasi-regular if it is uniform and isotoxal (ie all edges are the same). Example: icosidodecahedron.


Radially convex
Same as fully supported.
A 3D hyperplane.
Opposite of fully supported. Some rays from the centre of a reentrant stellation outwards will cross the surface of the model more than once.
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 Kepler-Poinsot solids).
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/d-gon is a n/(n-d)-gon.


A scaliform polytope is uniform in that it is vertex-transitive 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.
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.
A polygon whose sides cross over each other, or a polyhedron whose faces pass through each other.
A polyhedron is semi-canonical if all of its edges are tangent to a unit sphere. See also canonical.
Semi-regular 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, non-regular 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 (single-peaked), primary, and main-line. See some examples of stellations here.

Stellation diagram
This is a two-dimensional 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 quasi-convex, 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. self-intersection is not allowed within or between faces.
Sub-symmetry group
One symmetry group may be a sub-symmetry group of another if all of its rotations and reflections are included in that group. For example the tetrahedral symmetry group is a sub-symmetry group of the octahedral symmetry group.
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 4-fold axes of symmetry, four 3-fold axes of symmetry, and six 2-fold axes of symmetry.


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.
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.
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. Vertex-transitive is the same as isogonal. Edge-transitive is the same as isotoxal. Face-transitive 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.


A uniform polyhedron consists of regular polygonal faces, and is vertex-transitive. 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 semi-regular 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.

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.


The number of faces meeting at a 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 4-sided 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/(n-p), 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.
NO SUCH WORD! You probably mean vertex.
Plural of vertex.
Vertex figure
The polygon you get as a cross-section 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.


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.




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.

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