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 2 essays across 2 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

Named alongside it

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

Area Axiom Basis Congruence Determinant Dihedral angle Dissection Impossibility Invariant Matrix Multilinearity Operation set

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