# Regular Maps in the Projective Plane

This page shows all the regular maps that can be drawn in the projective plane. For the purpose of these pages, a "regular map" is defined here.

These include analogues of four of the five platonic solids, each with half as many vertices, faces and edges. The tetrahedron has no analogue in the projective plane.

nameSchläfli
symbol
pictureV
F
E
Eu
dual

Petrie dual

rotational
symmetry
group
hemi-cube{4,3} 4
3
6
1
hemi-octahedron

tetrahedron

S4
hemi-octahedron{3,4} 3
4
6
1
hemi-cube

self-Petrie dual

S4
hemi-dodecahedron{5,3}  10
6
15
1
hemi-icosahedron

self-Petrie dual

S5 The Petersen graph.

If you take hemi-dodecahedra and glue them together five to an edge, you will find that 57 of them form a regular polytope, the 57-cell, Schläfli symbol {5,3,5}. Its symmetry group is PSL(2,19).

3
hemi-icosahedron{3,5} 6
10
15
1
hemi-dodecahedron

C5{5,5}

S5 If you take a hemi-icosahedron and glue another one to each face, and bend them round so that three meet at each edge, you will find that the 11 of them form a regular polytope, the 11-cell, Schläfli symbol {3,5,3}. Its symmetry group is PSL(2,11). 3
hemi-hosohedron{2,2n} 1
n
n
1
hemi-dihedron

n=2: S1:{4,4}(1,0)
n=3: S1:{3,6}(1,1)
n=4: S2:{8,8}
n=5: S2:{5,10}
n=6: S3:{12,12}
n=7: S3:{7,14}

D4n   The image uses 7 as an example value for n.
½
hemi-dihedron{2n,2}  n
1
n
1
hemi-hosohedron

n odd: S0:{n,2}
n even: self-Petrie dual

D4n   The images use 7 as an example value for n. The second image only works for odd n.
½
hemi-digonal hosohedron{2,2} 1
1
1
1
self-dual

dimonogon

C2×C2     Cantellation of the hemi-digonal hosohedron yields the hemi-4-hosohedron.

½

Index to other pages on regular maps;
indexes to those on S0 C1 S1 S2 S3 S4.
Some pages on groups 