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Concept

Volume

The amount of space a solid occupies, measured against a chosen unit cube. It is additive over pieces that do not overlap, and the factor by which a linear map multiplies it is that map's determinant.

Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

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.

computation · Scissors congruence A parallelepiped of volume 2.94. The image of the unit cube under a three-by-three matrix, beside the six signed products whose sum is its volume.

The only function that behaves like a volume

Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.

algebra · Determinant Four solids with the same counts and every volume. The Reeve tetrahedra at heights 1, 2, 3, 5, drawn in wireframe with a table of their lattice-point counts and volumes. All have four boundary points and none inside; their volumes run from 0.17 to 0.83.

The theorem that has no version in space

A lattice polygon's area is decided completely by two counts of dots. The obvious guess is that a lattice solid's volume is decided by the same two counts in three dimensions, and there is a family of tetrahedra with identical counts and every volume that says otherwise.

discrete · Pick theorem The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones.

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

dynamics · Strange attractor The Reuleaux tetrahedron's width runs from 2.83 to 2.9. The largest and smallest width of the intersection of four balls, plotted against the polar angle of the measuring direction, with the constant width it would need to have drawn flat.

The same question in space

Intersect four balls at the corners of a tetrahedron and the result is not of constant width — it misses by two and a half per cent, computed exactly. Repairing it gives a body that is, and whether that body is the smallest of its kind has been open for a century.

geometry · Constant width

Named alongside it

The objects these essays reach for when they reach for this one.

AreaAxiomBasisCantor setChaosCongruenceConstant widthCounterexampleCounting two waysDeterminantDihedral angleDimension

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