The formation of full shurikens, half shurikens, stretched full shurikens, and stretched half shurikens, are diifferent ways of constructing maps (generally not regular maps), in non-orientable surfaces, from known regular maps. They all involve the insertion of crosscaps, so the resulting maps are necessarily in non-orientable surfaces.

S*n*: {p,q}_{ } V F E

(meaning, it is in the orientable manifold of genus C*2n*: {p,q}_{ } V F E

(similar, but in the non-orientable manifold of genus C(V+2*n*): {2p,4}_{ } E F 2E

(meaning, it is in the non-orientable manifold of genus V+2To convert a regular map to its full shuriken,

- replace each vertex by a crosscap
- replace each edge by a 4-valent vertex (as shown to the right)
- each face becomes a new face with twice as many edges

The full shuriken will **probably not** be regular. If p,
the number of edges per face, was even, the full shuriken will have some digonal
faces, and will definitely not be regular.

An example of a full shuriken which **is** a regular map is shown here. We start with the
tetrahedron (left), and get
C4:{6,4}_{3} (right).

S*n*: {p,q}_{ } 2V F E

(meaning, it is in the orientable manifold of genus C*2n*: {p,q}_{ } 2V F E

(similar, but in the non-orientable manifold of genus C(V+2*n*): {p,2q}_{ } V F E

(meaning, it is in the non-orientable manifold of genus V+2To convert a regular map to its half shuriken,

- replace alternate vertices by crosscaps
- double the valency of the remaining vertices, with two edges connecting them to each crosscap (as shown to the right)
- each face continues to be a face with the same number of edges

The half shuriken will **probably not** be regular.

An example of a half shuriken which **is** a regular map is shown here. We start with the
cube (left), and get
C4:{4,6}_{3} (right).

S*n*: {p,q}_{ } V F E

(meaning, it is in the orientable manifold of genus C*2n*: {p,q}_{ } V F E

(similar, but in the non-orientable manifold of genus C(V+2*n*): {2p,4}_{ } E F 2E

(meaning, it is in the non-orientable manifold of genus V+2To convert a regular map to its full shuriken,

- replace each vertex by a crosscap
- replace each edge by two 3-valent vertices (as shown to the right)
- each face becomes a new face with three times as many edges

The stretched full shuriken will **not** be regular. If q,
the number of edges per vertex, was even, the full shuriken will have some digonal
faces.

The stretched full shurikens which I have looked at are Petrie duals of truncated regular maps (which are never themselves regular).

The stretched full shuriken of the hemicube is the Petrie dual of the truncated tetrahedron,

the stretched full shuriken of the tetrahedron is the Petrie dual of the truncated hemicube,

the stretched full shuriken of the cube is the Petrie dual of the truncated {6,3}_{(2,2)}, and

the stretched full shuriken of {6,3}_{(2,2)} is the Petrie dual of the truncated cube.

S*n*: {p,3}_{ } 2V F E

(meaning, it is in the orientable manifold of genus C*2n*: {p,q}_{ } 2V F E

(similar, but in the non-orientable manifold of genus C(V+*n*): {2p,3}_{ } E+V/2 F 6V

(meaning, it is in the non-orientable manifold of genus V+2To convert a regular map to its stretched half shuriken,

- replace alternate vertices by crosscaps
- retain the remaining vertices as vertices
- replace each edge by new vertex, with one edge connecting it to an old vertex and two connecting it to a new crosscap (as shown to the right)
- each face continues to be a face but with twice as many edges

The stretched half shuriken will **not** be regular.

The stretched half shurikens which I have looked at are Petrie duals of alternately truncated regular maps — regular maps which have had half their vertices truncated. These are never themselves regular.

The stretched half shuriken of the cube is the Petrie dual of the alternately truncated cube, and

the stretched half shuriken of {6,3}_{(2,2)} is the
Petrie dual of the alternately truncated {6,3}_{(2,2)}.

A shuriken (strictly, a hira-shuriken) is a Japanese throwing star, as supposedly used by ninjas. I call structures like the one shown to the right shurikens, from their resemblance. Unfortunately real shurikens generally had an even number of blades, while the ones useful in generating regular maps have an odd number.