The net that loses the fewest neighbours

The polyhedral ladder ends on a trade-off: more faces means less distortion inside a face and more cutting between them, with the distortion falling as the reciprocal of the face count and the cutting rising as its square root. Both halves are measured, both are honest, and one of them is measured on the wrong side of the paper.

Cut length is what a manufacturer pays. It is how much scissor work a net needs, how much edge has to be glued, how much of the sheet is boundary. It is not what a reader pays. A reader pays when two places that are next to each other on the globe are drawn in different parts of the sheet — and how far apart the sheet puts them is a different number, which nothing here has computed.

For the Platonic solids the difference between the two is stark, because the first of them is a constant.

Why the length is a constant #

A net is a spanning tree of the face graph — the same object the cut has to go somewhere counted: keep an edge and the two faces stay joined, cut it and they come apart. A spanning tree of F faces has F − 1 edges, so the number of cut edges is E − F + 1 — twelve minus six plus one, seven for the cube — for every net, without exception, because that is what a spanning tree is.

On a Platonic solid every edge has the same length. So the total cut length is (E − F + 1) times the edge length, a number that depends on the solid and not on the net. All 384 nets of the cube cut 8.083 units of edge; all 384 of the octahedron cut 7.071; all 16 of the tetrahedron cut 4.899.

That is a stronger statement than it looks. The cut has to go somewhere counted the nets and tested them for overlap and found no overlaps at all among the regular solids, which left the choice between them entirely open. Cut length does not close it either. Two of the three quantities this collection has used to describe a net cannot tell any two nets of a solid apart.

What the reader pays #

The measurement on the reader’s side is exact and needs no sampling. Every cut edge appears twice in the laid-out net — once in each of the two faces it used to join — and both positions are known once the net is placed. The distance between the midpoints of those two images is how far the net has thrown one severed neighbourhood.

Summed over the cut edges and divided by the solid’s own edge length, that is the net’s separation. For the cube it runs from 11.30 to 17.97, a factor of 1.59. For the octahedron, five cuts, 6.06 to 10.39, a factor of 1.71. For the tetrahedron, three cuts, exactly 3.00 to exactly 4.00.

The tetrahedron’s numbers are worth a moment: 3.00 means all three severed edges land exactly one edge length apart, which is the best a cut can possibly do — the two copies are as close as two distinct positions in the net can be. The worst tetrahedral net achieves 4.00 by throwing one of the three cuts twice as far as the other two. With only sixteen nets and three cuts the whole space is small enough to read.

The seven distances, read individually #

The best cube net’s seven severances are not seven similar numbers. They are 0.707, 0.707, 0.707, 0.707, 2.236, 2.236 and 4.000 edge lengths.

The 0.707 is the smallest separation a cut on a cube can have, and it happens when the two faces that used to share the cut edge end up meeting at a corner in the net. Put the centre square of a cross at the unit square and a neighbour above it and another to its right: the cube edge they share is severed, its copy in one face is a vertical edge and its copy in the other is a horizontal one, and the midpoints are half a unit apart in each direction — 1/√2. Two places that touch on the globe are drawn a third of a face apart, on opposite sides of a corner a reader’s eye crosses without effort.

The 4.000 is the opposite case: one pair of faces at the two ends of the net, four squares apart, with no relationship visible at all.

The worst net’s seven are 0.707, 0.707, 2.236, 2.236, 3.808, 3.808 and 4.472. It has the same two easy cuts and no third; everything else has been thrown to the far end. The difference between the best and worst net is not that one has better cuts on average — it is that the best net concentrates its damage into a single hopeless severance and keeps the other six local, and the worst spreads it.

And the space of outcomes is far smaller than the space of nets. The cube’s 384 nets take only eight distinct values of total separation, and forty-eight of them are optimal. The octahedron’s 384 take seven values with twenty-four optimal. So the choice is not a delicate optimisation over hundreds of nearly equal options; it is a choice between eight outcomes, one of which is available in an eighth of all nets and is very unlikely to be hit by accident.

Does compactness decide it? #

The available intuition is that a compact net — the cube’s familiar cross — keeps neighbours together and a straggling snake of six squares does not. That is a claim about a correlation between two computable numbers, and all 384 nets are available, so it can be settled rather than asserted.

0.949, and not 1. Compactness predicts most of the damage and does not determine it, which means the arrangement of a net matters as well as its extent. Two nets can occupy the same amount of sheet, with the same radius of gyration, and separate their severed neighbours by measurably different amounts, because which edges were chosen for cutting is not the same question as how far the pieces sprawl.

This is the useful form of the result for anybody actually laying out a polyhedral map. Minimising the bounding box is a good proxy and is not the objective; the objective is computable directly, and for a cube or an octahedron it is computable exhaustively in under a second.

The face-count ladder, re-priced #

Two things come out of that ladder, and neither was predictable from the cut-length version.

The ordering by face count is not the ordering by separation. Per cut, the three solids with triangular faces score 1.00, 1.21 and 1.36 as the face count rises from four to twenty; the cube and the dodecahedron sit above all of them at 1.61 and 2.05. Face shape matters more than face count, because a triangle’s severed edge has less far to travel round the remaining triangles than a pentagon’s does round the remaining pentagons. That is the same reason hexagons cannot tile the sphere matters to a cell system: what a face shape does to its neighbourhood is not a detail of the face.

The spread between the best net and the worst widens with the solid, from 1.33 on the tetrahedron to at least 2.41 on the dodecahedron. So on a solid with many faces, the choice of net matters more — and there are five million more of them to choose badly from. More faces, less distortion, more cutting prices the third column as a fixed cost of the face count. It is not fixed. It is a range whose width grows faster than its floor.

What this says about the published maps #

Every polyhedral world map anybody has seen is one net out of an enormous number, chosen by its author. Fuller’s Dymaxion map is a particular unfolding of an icosahedron, laid out to keep the continents unbroken; Cahill’s butterfly is a particular unfolding of an octahedron, in four lobes joined at the poles.

Both authors chose their nets by eye, against a criterion about land, in the same spirit as choosing which face projection to use — a decision made once and then inherited by everybody who copies the map. Neither had this number, and the number does not replace their criterion — a net that scores well on separation may still tear a continent in half, because the measurement here knows nothing about what is on the sphere. What it provides is the other axis: given a family of nets that all keep the land intact, this is how to rank them, and the range within such a family is wide enough to be worth ranking.

The same argument applies to an interrupted projection, which is a net of a much simpler kind — one central meridian per lobe, cut along the parallels of a chosen boundary. Its separations are computed by the same construction and its lobe boundaries are chosen by the same eye.

What was computed, and how #

All 384 spanning trees of the cube’s face graph, all 384 of the octahedron’s and all 16 of the tetrahedron’s are enumerated by brute force over subsets of the twelve face-graph edges — 792 subsets tested, the spanning ones kept — which is the same enumeration the overlap test uses and which agrees with the determinant formula for the number of spanning trees.

Each tree is laid flat by walking it from a root face, placing each child by reflecting it about its shared edge with its parent. Positions are keyed by (face, vertex) rather than by vertex, because a cut is exactly the situation where one vertex of the solid has several positions in the plane.

The separation is then the distance between the midpoints of the two images of each cut edge, in units of the net’s own edge length as measured from the placement rather than from the solid — so an error in the layout’s scale would show up as a wrong ratio rather than being divided out.

The dodecahedron and icosahedron are sampled with four hundred random spanning trees each, built by shuffled Kruskal. That sampler is not uniform over spanning trees — it favours trees with many short paths — which is stated in the library rather than corrected, because what the sample is used for is a range rather than a distribution, and a non-uniform sample of a range is still a lower bound on it.

The assertions require the cut count to be identical across every net of a solid — the constant half — and the separation to differ by more than 1.3, which is the refusal: a bug that returned the same separation for every net would pass the first and fail the second.

Where the model stops #

Separation as measured here is a straight-line distance in the plane of the net, between two points that were adjacent on the sphere. A reader following a route does not travel in a straight line across the sheet; they find the other piece, which is a search rather than a distance. The number is a proxy for that search’s difficulty and not a model of it.

Only the severed edges are measured, not the ground either side of them. Two points a degree apart across the middle of a cut edge are separated by about the number computed here; two points a degree apart near a corner where three cuts meet can be separated by a great deal more, and the corner case is not in this measurement. It is the polyhedral analogue of the angle deficit being concentrated at the vertices, and it deserves its own rung. The corner case is also where a conformal map onto a face has its own singularity, which is not a coincidence: both are about what happens where three faces meet.

And nothing here knows what is on the sphere. Ranking nets by separation ranks them for a reader of an unmarked globe. The moment there is land on it, the objective becomes a weighted one — separation weighted by how much anybody cares about the ground at each cut — and the weights are a decision this collection does not make.

Who found it, and when #

Nets have been drawn since Dürer’s Underweysung der Messung of 1525, which is where the word net comes from in this sense and which drew them without ever counting them. The count is modern: the number of edge unfoldings of a polyhedron is the number of spanning trees of its face graph, and Kirchhoff’s matrix–tree theorem of 1847 turns that into a determinant.

Whether every convex polyhedron has at least one net that does not overlap itself is Dürer’s problem and is still open. That question is about whether a net exists at all; this one is about choosing between the ones that do, and as far as this collection can tell, nobody has needed to ask it — because a polyhedral map is normally drawn once, by hand, by somebody who is looking at the land rather than at the seams.

What the weighted version would be, and why the weights are editorial #

The objective declines to know what is on the sphere, and the honest version of that refusal is to say what the weighted problem looks like, since the machinery for it already exists.

The weights sit on the edges and nothing else changes. Each edge of the solid is either cut or kept, the cost of cutting it is a number, and the legal cut patterns are the complements of spanning trees of the face graph. So the weighted problem is a maximum-weight spanning tree — the same construction the cut has to go somewhere arrives at from the other side — and it is solved by sorting the edges and taking them greedily.

What changes is only what goes into the weight, and that is where the judgement lives. How much land the edge crosses, weighted by population; whether it severs a coastline a reader traces; whether it breaks a route, an ocean basin, a country. Each of those is a defensible objective and they do not agree with each other.

Which is why this collection computes the unweighted answer and stops. Separation on a bare sphere is a property of the solid and the net, it is the same number for everybody, and it can be measured rather than argued about. A weighted score is a measurement of somebody’s priorities dressed as a measurement of a map, and the dressing is the part worth refusing.

The unweighted answer is still useful as a floor. It says what the cut structure costs before any of the caring is applied, so a weighted design that does much worse than it is spending real separation to protect a particular region — which is a legitimate trade and one the designer should know they are making.

And it makes the published maps legible. Fuller’s arrangement is not the best net by separation, and the gap between it and the best is the price he paid to keep the cuts in water. That is exactly the kind of statement the unweighted measurement licenses: not that the map is wrong, but how much of its cost is geometry and how much is choice.

Where the ladder goes next #

Six rungs have priced a polyhedral map by the distortion inside its faces and by the cost of the cuts between them, with both measured on a solid whose faces are all the same. The obvious remaining question is the one Fuller answered by eye and nobody has answered by measurement: where to point the solid, which is an aspect search over a group with the solid’s own symmetry in it, and which the site now has the machinery to run — the same three-parameter search the third parameter, run had to build a grid for, with the solid’s own symmetry group folded into it.