This page is obsolete. See the current version of Regular Maps in the Projective Plane

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

symmetry
group
Commentsqy
hemi-cube{4,3}hemi-cube4
3
 6 
1
hemi-octahedron


tetrahedron

S4replete
hemi-octahedron{3,4}hemi-octahedron3
4
 6 
1
hemi-cube


self-Petrie dual

S4replete
hemi-dodecahedron{5,3}hemi-dodecahedron
hemi-dodecahedron
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).

replete

3
hemi-icosahedron{3,5}hemi-icosahedron6
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).

replete

3
hemi-hosohedron{2,2n}hemi-hosohedron1
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 polygons with too few edges Faces share vertices with themselves Vertices share edges with themselves

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


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

D4n Vertices with too few edges Faces share vertices with themselves Faces share edges with themselves

The images use 7 as an example value for n. The second image only works for odd n.
½
hemi-digonal hosohedron{2,2}hemi-digonal hosohedron1
1
 1 
1
self-dual


dimonogon

C2×C2 Faces with too few edges Vertices with too few edges Faces share vertices with themselves Faces share edges with themselves Vertices share edges with themselves

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 C4 C5 S3 C6 S4.
Some pages on groups

Copyright N.S.Wedd 2009,2010