Stellation of Regular Maps

The concept of stellation is normally applied to polygons, polyhedra, and other polytopes. This page extends the concept of stellation to apply to regular maps.

 Great stellated dodecahedron.

The standard method of stellation

When we think of a stellated polyhedron, we usually think of a conformation in ℝ3. An example is the great stellated dodecahedron, shown to the right.

The standard procedure for constructing a stellated polyhedron from a regular polyhedron is as follows:

• Extend each face until it reaches the extension of some other face. In general there is choice about how to do this; different choices, consistently applied, lead to different regular stellations of the original polyhedron.

We would like to apply some such procedure to apply to other regular maps. We describe a geometric method, then a topological method.

Geometric method of stellation

The process above applies to regular maps in the sphere, with the sphere embedded symmetrically in ℝ3. It may also be applied to regular maps in other surfaces that can be embedded in a higher space. For example, the genus-3 orientable surface can be regarded as the Klein quartic, embedded in ℂ3. Then the same process can be applied, allowing us to stellate regular maps in that surface, such as the Klein map.

However, it is not obvious how to generalise this method to all genera, or even that this is possible. We will therefore prefer the topological method described below.

Topological method of stellation

A regular map consists of a network of edges which meet at vertices, all embedded in a surface. A face of a regular map is defined by the set of edges which one traverses if one sets out from a vertex along an edge, and takes the next edge on one's left at every vertex, until returning to one's starting place.

Topologically, we define a "stellated regular map" as above, except that:

• instead of each edge of a stellation being an edge of the regular map, it may be a "path" of the regular map, defined as, for example, "set off along an edge; take the m1th left at the next vertex, then the m2th left at the next vertex; ...". The path is then regarded as an arc from its start vertex to its final vertex (rather than a zigzag route that visits the intervening vertices).
• instead of a face being bounded by a set of edges or paths such that each is the next on the left after traversing the previous one, it may be bounded by a set of edges or paths such that each is the nth on the left after traversing the previous one.

Notation for "paths"

We specify a path from one vertex to another using angle-brackets < > containing a list of numbers that indicate which way to turn at each successive vertex along the path. Thus 1 means "first edge from the left", 2 means "second edge from the left", -1 means "first edge from the right", etc.

So, for example,
< >   the path is just one edge
< 1 >   the path is two successive edges of a face
< 1,2 >   the path is three edges long. At the first vertex you follow the first edge on the left, at the second you follow the second edge on the left.
< 1,-1 >   the path is three edges long, the first two having a common face and the second two having a different common face

Generally we will want to specify paths that are palindromic, so that if there is a path from vertex a to vertex b there is also a path from vertex b to vertex a. < 1,-1 > is the simplest non-trivial such path. "< 1,2 >" specifies a path that is generally not palindromic; but "< 1,2; 2,1 >"uses a semicolon to add an alternative, and means, "first left then second left or second left then first left", which is palindromic.

Notation for circuits of "paths"

A face of a stellated regular map is defined as a circuit of paths. We specify the circuit by following the path specification with /n. If no /n is given, assume /1, meaning that having traversed a path from a to b, we take the next such path to the left. When a path specification /n is given, having traversed a path from a to b, we then take the nth such path to the left.

A list of stellations

This list is work in progress. It may contain errors. It does not aim to be complete in any respect.

Original regular map pathold and
new faces
genusnameSchläfli formulaC&D no. genusnameSchläfli formulaC&D no.
S0cube{4,3}R0.2' < 1 > S0tetrahedron{3,3}R0.1 2 × 1"stella octangula"

Core is octahedron,
convex hull is cube

icosahedron{3,5}R0.3 <   >/2 S4great dodecahedron{5,5/2}R4.6 3Core is dodecahedron,
convex hull is icosahedron
< 2 > small stellated dodecahedron{5/2,5} Core is dodecahedron,
convex hull is icosahedron
< 2 >/2 S0great icosahedron{3,5/2}R0.3 7Core is icosahedron,
convex hull is icosahedron
dodecahedron{5,3}R0.3' < 1,-1,1 > great stellated dodecahedron{5/2,3}R0.3' Core is dodecahedron,
convex hull is dodecahedron
< 1,-1 > tetrahedron{3,3}R0.2p 5 × 1"five tetrahedra"

Core is dodecahedron,
convex hull is dodecahedron

< 1 >/2 cube{4,3}R0.1 5 × 1"five cubes"

Core is dodecahedron,
convex hull is dodecahedron

S1square tiling{4,4} < 1 > S1square tiling{4,4} 2 × 1even number of squares
2odd number of squares
< 2,1 > 5 × 15 divides number of squares
55 does not divide it
< 2,1; 1,2 > S2,
S3, etc.
various{8,8}R2.6,
R3.10, etc.
4
hexagonal tiling{6,3} < 1 > S1hexagonal tiling{6,3} 6 × 13 divides number of hexagons
2 × 33 does not divide it
< 1,1 > 4 × 14 divides number of hexagons
44 does not divide it
< 1,-1 > 7 × 17 divides number of hexagons
77 does not divide it
< 1,-1; -1,1 > S2,
S4, etc.
various{6,6} R2.5,
R4.8, etc.
2
triangular tiling{3,6} < 1,-1 > S1triangular tiling{3,6} 3 × 13 divides number of triangles
33 does not divide it
< 3 > 4 × 14 divides number of triangles
44 does not divide it
< 3,2 > 7 × 17 divides number of triangles
77 does not divide it
<  >/2 hexagonal tiling{6,3} 3
< 1,-1 >/2 9
< 3,2; 2,3 > S3,
S9, etc.
various{12,12}R3.12,
R9.26/27, etc.
6
S2{3,8}R2.1 < 1,-1 > S2{4,8}R2.3 3 × 1
<  >/2 {8,4}R2.3' 3 × 1
S3dual of Klein map{3,7} R3.1 <  >/2 S19"great Klein map"{7,7/2} R19.23 3analagous to the great dodecahedron
< 1,-1 > "small stellated Klein map"{7/2,7} analagous to the small stellated dodecahedron
< 2,-2 > {7,7} 9
<  >/3"3rd-order hole" S10{4,7/3} R10.9 ?
dual of Dyck map{3,8}R3.2 < 1,-1 > S3{4,8|4}R3.5 3 × 1
S4{3,12}R4.1 < 1,-1 > S4{6,12}R4.9 3 × 1