The History of Mathematics: A Brief Course

388 12. MODERN GEOMETRIES

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Sphere Torus

FIGURE 14. Left: The sphere, regarded as a square with edges

identified, is disconnected by a diagonal. Right: The torus requires

two cuts to disconnect.

4.3. Mobius. One fact that had been thought well established about polyhedra
was that in any polyhedron it was possible to direct the edges in such a way that one
could trace around the boundary of each face by following the prescribed direction
of its edges. Each face would be always to the left or always to the right as one
followed the edges around it while looking at it from outside the polyhedron. This
fact was referred to as the edge law (Gesetz der Kanten). The first discovery of
a closed polyhedron that violated this condition^44 was due to Mobius, sometime
during the late 1850s. Mobius did not publish this work, although he did submit
some of it to the Paris Academy as his entry to a prize competition in 1858. This
work was edited and introduced by Curt Reinhardt (dates unknown) and published
in Vol. 2 of Mobius' collected works. There in the first section, under the heading
"one-sided polyhedra," is a description of the Mobius band as we now know it
(Fig. 14). After describing it, Mobius went on to say that although a triangulated
polyhedron whose surface was two-sided will apparently contain only two-sided
bands, nevertheless a triangulated polyhedron with a one-sided surface can contain
both one- and two-sided bands.
Mobius explored polyhedra and made a classification of them according to the
number of boundary curves they possessed. He showed how more complicated poly-
hedra could be produced by gluing together a certain set of basic figures. He found
an example of a triangulated polyhedron consisting of 10 triangles, six vertices, and
15 edges, rather than 14, as would be expected from Euler's formula for a closed
polyhedron: V — Ε + F = 2. This figure is the projective plane, and cannot be em-
bedded in three-dimensional space. If one of the triangles is removed, the resulting
figure is the Mobius band, which can be embedded in three-dimensional space.

4.4. Poincare's Analysis situs. Poincare seemed to be dealing constantly with

topological considerations in his work in both complex function theory and dif-

ferential equations. To set everything that he discovered down in good order, he

wrote a treatise on topology called Analysis situs in 1895, published in the Jour-

nal de I'Ecole Polytechnique, that has been regarded as the founding document of

modern algebraic topology.^45 He introduced the notion of homologous curves—

curves that (taken together) form the boundary of a surface. This notion could

be formalized, so that one could consider formal linear combinations (now called

(^44) In fact, a closed nonorientable polyhedron cannot be embedded in three-dimensional space, so
that the edge law is actually true for dosed polyhedra in three-dimensional space.
(^45) Poincare followed this paper with a number of supplements over the next decade.