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Euler characteristic — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. V − E + F = 2, five times. Vertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row.

    Every corner pays for itself

    Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.

    part 1 · topology
  2. Two trees, sharing every edge between them. The cube flattened into a planar graph, with a spanning tree of its corners drawn solid and the leftover edges drawn dashed; the leftover edges join the faces into a second tree, and the two counts add to the number of edges.

    Two trees, and every edge in exactly one of them

    Euler's formula is usually proved by deleting things until nothing is left. There is a better argument that deletes nothing — a tree through the corners and a tree through the faces, which between them use every edge once and can therefore be counted.

    part 2 · topology
  3. The gap at a corner of the cube. The 3 faces meeting at one corner of the cube, unfolded onto the page. They leave a gap of 90.0 degrees, and the 8 gaps come to 720 degrees in total.

    Seven hundred and twenty degrees of gap

    Unfold the faces around any corner of a solid and they do not close up. The gap left over is different at every corner and on every solid, and the gaps always add to two full turns.

    part 3 · topology
  4. A solid where V − E + F is 0. a slab with one hole through it, drawn as a wireframe. Its 32 vertices, 64 edges and 32 faces give an alternating sum of 0 rather than 2.

    The solid where the answer is not two

    A slab with a hole through it has flat faces, straight edges and sixteen corners, and its alternating sum is zero. It is not a trick and not a degenerate case — it is the object that shows the theorem had a hypothesis nobody had written down.

    part 4 · topology
  5. Twelve units of shortfall, on every solid with three faces at a corner. A bar for each of 8 polyhedra with three faces at every vertex, divided into each face's shortfall from six sides; every bar has total length twelve, and the hexagons contribute nothing.

    Twelve pentagons, whatever the hexagons

    A football has twelve pentagons and twenty hexagons. A molecule of sixty carbon atoms has the same pattern, a molecule of seventy has twelve pentagons and twenty-five hexagons, and a geodesic dome of any size has twelve places where the pattern of six breaks. None of this is a coincidence of design: Euler's formula, rearranged, says that faces meeting three at a corner must fall short of hexagons by exactly twelve in total, and the hexagons are free.

    part 5 · topology
  6. The six regular 4-polytopes, and an alternating sum of 0. A table of the six regular polytopes in four dimensions with their numbers of vertices, edges, faces and cells and the alternating sum, which is zero for each.

    Zero in four dimensions

    Corners minus edges plus faces is two for every solid. One dimension up, corners minus edges plus faces minus cells is zero for every one of the six regular four-dimensional solids, from the five-cell to the six-hundred-cell, and for every other convex solid in four dimensions. The alternating sum does not break when the dimension rises: it alternates, two in odd dimensions and zero in even ones, because it is measuring a sphere and not a solid.

    part 6 · topology

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