The real projective plane coxeter djvu

Advertisement Hide. This service is more advanced with JavaScript available. The Real Projective Plane. Authors view affiliations H. Coxeter George Beck. Front Matter Pages i-xiv. A Comparison of Various Kinds of Geometry. Pages Order and Continuity. One-Dimensional Projectivities. Two-Dimensional Projectivities. Projectivities on a Conic. Affine Geometry.

The real projective plane exists by mere construction, but we cannot be limited only by what we see. Nevertheless, what we see can help us understand it more. It is the only non-prismatic uniform polyhedron with an odd number of faces. It has 4 equilateral triangles and 3 squares with 2 triangles and 2 squares meeting alternately at each of its 6 vertices. The edges of the tetrahemihexahedron coincide with the 12 edges of a regular octahedron. In fact, one may obtain the tetahemihexahedron by replacing 4 triangles of the octahedron by the 3 squares formed by these 12 edges.

Notice that the square faces intersects along "false" edges, these being diagonal lines of the squares. If one accepts that the tetrahemihexahedron is an immersion of a closed surface, then one must conclude that this surface is none other than the real projective plane. A polyhedron closely related to the tetrahemihexahedron is the cuboctahedron. This uniform polyhedron has 8 equilateral triangles, 6 squares, 24 edges, and 12 vertices. Notice that these data are obtained by doubling the corresponding data for the tetrahemihexahedron.

Moreover, the vertex configurations for both polyhedra are topologically identical, with 2 squares and 2 triangles meeting alternately.

If one takes the cuboctahedron as a polyhedral decomposition of the sphere, then the cuboctahedron serves as a universal cover of the tetrahemihexahedron. In this case the covering index is 2, meaning that each point on the surface of the tetrahemihexahedron corresponds to 2 points on the cuboctahedron. One may use the tetrahemihexahedron to see that the complete graph on six vertices K 6 may be imbedded in the real projective plane.

These edges are diagonals in the 3 squares, although there are two distinct ways to place them. However, the embedding of the 2 remaining edges is uniquely determined once one of these edges is drawn. Due to the above considerations, one may describe the tetrahemihexahedron as a "topological" model of the real projective plane.

Perhaps the biggest deficiency with this model is that one must penetrate the ideas of "immersion" and "false intersections". The 3 line segments where the squares intersect are not to be regarded as edges of the tetrahemihexahedron. One may build this model if one has a large number of rubber bands. It is obtained merely by successively wrapping more and more rubber bands around a central core. Due to the average of forces exerted inward by the rubber bands, the model becomes increasingly spherical as more rubber bands are added.

Ideally, each rubber band is placed along a great circle of the sphere.