2-dimensional polyhedra and incidence matrices
No. 174, 1975
(Combinatorial topology)
India ink and pencil on paper, 44x61.5 cm.
Consider, for a moment, a simple smooth surface suspended in space. Imagine then that it is broken up into distinct units, as if it were a piece of dry parchment that cracks from being bent out of shape. Imagine too that each parchment chip were a brick in a wall—a collection of individual units that together comprise that space. Indeed, the notion of a multitude of individual pieces comprising a whole is a central theme in this image. The picture shows how to turn a smooth surface into a polyhedron by partitioning it into small building blocks, things like rectangles, triangles, lunes, and other simple shapes. The stronger the bend in the surface, the smaller the individual building blocks. In fact, it is even possible to number the resulting building blocks in terms of their dimension and create a matrix. We can then compute the homology groups of the surface, the polyhedron. The main topological idea here is that representing complex objects by gluing together simple ones can reveal important properties of the original complicated object itself. A whole is, more often than not, the sum of its parts.
