Cartographic Projection
every map is wrong, and by exactly how much
Concept

Goldberg polyhedron — where it appears

The dual of a subdivided icosahedron: hexagons everywhere except for exactly twelve pentagons, however fine the subdivision. It is the shape every hexagonal global grid actually is, and the twelve pentagons are where such a grid's addressing and its neighbourhood queries go wrong.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Past the five solids: what more faces buy, and what they cost. The icosahedron subdivided 2, 3, 4, 6, 8 ways, giving 20, 80, 180, 320, 720, 1280 faces. The worst angular deformation inside a face falls from 13.1° to 0.26°, with a fitted exponent of -0.947 against the face count — the reciprocal, as it must be, because a face's angular size goes as the inverse square root of the count and the gnomonic's deformation goes as the square of that. The total cut length rises from 11.6 to 96 sphere radii, fitted at 0.509. Both exponents together say the whole economics of the family in one line: halving the distortion costs √2 times the cutting, for ever.

More faces, less distortion, more cutting

The five regular solids are where polyhedral mapping stops, and they stop because there are only five rather than for any reason about maps. Subdivide instead, and the family runs to any number of faces with two fitted exponents: distortion falls as the reciprocal of the count and cutting rises as its square root.

families · Polyhedral A hexagonal tiling of the sphere, and its pentagons. 362 cells — 350 hexagons and 12 pentagons, the pentagons marked — drawn on Orthographic. The twelve are not a defect of the construction and cannot be removed by subdividing further: Euler's formula requires exactly twelve however many hexagons there are. Each pentagon here has 0.52 times the area of an average hexagon, so a count aggregated over these cells has twelve entries that mean something different from all the others.

Hexagons cannot tile the sphere

Hexagons are the best cell shape a plane offers and the sphere will not take them. Euler's formula forces exactly twelve pentagons into any such tiling — twelve at 42 cells and twelve at 642, while the hexagon count rises twenty-one-fold — and each of the twelve is measurably smaller than the hexagons around it.

applied · Cells Two unfoldings of the same 80-face solid. Both are edge unfoldings of the same solid along different spanning trees of its face graph, so both preserve every distance on the surface exactly. The left one is a net. The right one is not: 3 pairs of its faces occupy the same ground, so it cannot be cut out of paper and folded up. Nothing in the unfolding procedure prevents this, and past the regular solids most trees produce it.

A net can land on top of itself

Every one of the cube's 384 unfoldings is a net, and every one of the icosahedron's five million is too. Past the regular solids that stops being true: at 180 faces, 99 of every 100 randomly chosen unfoldings have faces sitting on top of each other, so choosing a net stops being a choice and becomes a search — except that the net anybody would actually draw works every time.

families · Polyhedral

Named alongside it

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

IcosahedronTopologyAngle deficitEuler characteristicInterruptionPlatonic solidPolyhedral projectionSpanning treeAggregationCellCombinatoricsDiscrete global grid

All concepts