Polyhedron
Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.
Why the list of perfect solids stops at five
There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.
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.
Every surface is a sphere with handles
Take a square and say which edges are to be glued to which, and which way round. Four such rules give four different surfaces — and two numbers computed from the rule, without ever building the surface, say which one.
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.
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.
Equal area is enough, and equal volume is not
Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.
Thirteen more when one word is dropped
The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.
The four that are allowed to cross themselves
Drop convexity from the definition of a regular solid and four more appear. Their faces are pentagrams, they pass through one another, and the alternating sum that gives two for every ordinary solid gives minus six for two of them.
Six in four dimensions, and three forever after
The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.
The integers among the quaternions
The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.
The obstruction that was the only one
Dehn showed in 1901 that a cube cannot be cut into a regular tetrahedron of the same volume, because a number built from edges and angles disagrees. For sixty-four years nobody knew whether that number was the whole story. Sydler proved in 1965 that it is: volume and Dehn's number together decide every case.
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.
Every count a solid can have
Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.
Every point at the average of its neighbours
A planar graph can be drawn without crossings, but finding such a drawing looks like a search. Tutte found in 1963 that it is not: pin one face to a convex polygon, put every other point at the average of its neighbours, and the drawing that results has straight edges, convex faces and no crossings at all — provided no two points can cut the graph apart. It is the position a network of equal springs settles into, and it is also where a random walk expects to leave.
Eight points that refuse the cube
Place eight points on a sphere so that the closest pair is as far apart as possible. A cube manages 70.53°. Rotating one face by 45° and pushing the faces together gives 74.86°. A botanist counting pores on pollen grains asked this in 1930, and proofs still stop at fourteen points: one search in five hundred finds the best thirteen, whose failure to reach 60° is why kissing spheres stop at twelve.
Peaks, pits and passes
Take any lumpy closed surface and any direction of height. Peaks plus pits minus passes is 2 on a sphere and 0 on a torus, whatever the lumps, in every direction. Average one vertex's score over all directions and it becomes the corner's angle defect, computed from nothing but comparisons. So Gauss–Bonnet follows from a count of which neighbours lie higher.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicPlatonic solidsDualityGenusAngle defectConvexityCurvatureDihedral anglePlanar graphTorusClassificationConnectedness