A net can land on top of itself

The net that loses the fewest neighbours searched all 384 unfoldings of the cube and found that every one of them cuts exactly seven edges of exactly the same length — so the quantity this collection had been pricing cutting by is a constant that cannot choose between them — and then scored them on how far apart they put places that touch on the globe.

Every one of those 384 is a net: a figure that can be cut from paper and folded up. That is true of the tetrahedron’s 16, the octahedron’s 384, and — sampled four thousand times — the dodecahedron’s and icosahedron’s 5,184,000 apiece.

It stops being true immediately past them.

The measurement #

Subdividing the icosahedron gives a family that runs to any number of faces, trading distortion for cutting: distortion falls as the reciprocal of the face count and cutting rises as its square root. The family is where the polyhedral idea actually lives — nobody maps the world on twenty triangles — and it is where the property the regular solids have breaks.

Random spanning trees of the face graph, unfolded and tested for pairwise overlap:

The counts in the second column are Kirchhoff’s determinant — the number of spanning trees of the face graph, which is the number of edge unfoldings — and they are the reason the third column is a fraction rather than an enumeration. At 80 faces there are more unfoldings than atoms in a galaxy.

Why the regular solids are the exception #

The property the Platonic solids have is not that their unfoldings are somehow better arranged. It is that they are small.

An unfolding overlaps when the net wraps far enough round the plane to meet itself, and how far it wraps is a function of how many faces it has and how much angle each one contributes. A twenty-face solid laid flat covers a region roughly two faces wide in every direction; there is not enough of it to come back round.

At eighty faces there is. At three hundred and twenty there is a great deal, and a spanning tree chosen at random is a long straggling thing that wanders across the plane and crosses its own path.

That reframes what the earlier rung established. “None of the cube’s 384 nets overlaps” reads like a fact about the cube; it is a fact about the number 384 being small enough that the cube’s nets cannot reach round. The property does not extend, and the family in which polyhedral mapping is actually done is entirely outside it.

The net anybody would draw #

The result that saves the family is the one nobody would think to measure.

Take the breadth-first spanning tree from any face: open the solid outwards from a chosen middle, one ring of faces at a time. That is what a person does by hand, and it is a spanning tree like any other, so there is no reason from the combinatorics for it to behave differently.

It does. At every face count tried, and from every one of the starting faces — 20 of 20, 80 of 80, 180 of 180, 320 of 320 — the breadth-first unfolding is a valid net.

The reason is geometric rather than combinatorial. A breadth-first tree from one face produces a net that grows outwards in rings, so its extent in every direction is the same and it is as compact as a net of that many faces can be. A random tree produces long chains, and a long chain of triangles curls.

So the practical situation is the opposite of alarming. At 320 faces a random unfolding is worthless — every one of 150 tried overlapped — and the obvious construction works from any starting point. What the measurement changes is not the practice but the understanding: choosing a net is a search, and the reason nobody notices is that the first thing anybody tries is the thing that always works.

How a net comes to cross itself #

The mechanism is worth drawing out, because it explains both the failure and the rule that avoids it.

A net’s faces are laid down one at a time, each rotated about the edge it shares with its parent. Follow a chain of faces away from the root and the chain turns: each triangle contributes a little of the angle its corner carries, and after enough of them the chain has curled through a substantial fraction of a full turn.

On a subdivided icosahedron each corner carries an angular deficit, and there are twelve corners carrying most of it and a great many ordinary vertices carrying almost none. So a chain wandering across ordinary ground curls slowly, and a chain that happens to pass near several of the twelve curls quickly. A random spanning tree contains long chains by construction — it is a random tree, and random trees have long paths — and a long chain that curls far enough meets a chain that went the other way.

A breadth-first tree has no long chains. Its longest path from the root is the graph’s radius rather than something proportional to its size, so nothing in it curls far, and the whole figure grows outwards as a disc.

That also predicts where the heuristic should eventually fail: at a face count large enough that even the radius produces a chain curling through a full turn. Whether that happens at a thousand faces or a million is a question the same machinery could answer, and it is not answered here.

Where the heuristic would fail, and it does not #

The question the mechanism section leaves open — at what face count even a breadth-first tree curls through a full turn — has an answer, and it is the opposite of the one the phrasing implies.

Gauss–Bonnet fixes the total angular deficit of any convex polyhedron at exactly 4π, whatever it is made of. Subdividing the icosahedron does not create curvature; it divides the same 4π among more vertices. A solid with F triangular faces has about F/2 + 2 vertices, so the deficit each one carries falls as the face count rises:

A chain curls by accumulating the deficits of the vertices it passes, and a breadth-first tree’s longest chain is the face graph’s radius, which grows as the square root of the face count. So the turning along it is the radius times the per-vertex deficit — a quantity proportional to √F × 1/F, which falls as the solid is subdivided.

The heuristic does not merely happen to work at the sizes tried; its margin improves with every subdivision. At twenty faces a radial chain turns through a third of a revolution and at three hundred and twenty through a tenth, and the trend has no reason to reverse. The construction that always works is safest exactly where the alternatives are worst.

That also explains the small solids properly, and the explanation is not the one about size alone. At twenty faces a random chain turns through more than a full revolution — seven faces at sixty degrees apiece — so the turning condition for an overlap is met and none of the 150 samples overlapped anyway. What is missing there is extent: a net two faces wide cannot reach back to meet itself however much it curls. An overlap needs both conditions, and the two are met at different sizes: turning is easy on a coarse solid and extent is easy on a fine one, and the two are both satisfied somewhere near eighty faces, which is where the rate goes through a half.

What the cut looks like at these sizes #

The cut is one fewer than the corners, at every face count and by Euler’s formula: 11 cuts for the icosahedron, 41 at 80 faces, 91 at 180, 161 at 320. Each cut is an edge, and the edges get shorter as the solid is subdivided, which is the trade the subdivision family makes.

What the overlap measurement adds is that not every way of arranging those cuts is available. At 180 faces the 3 × 10⁶² spanning trees are not 3 × 10⁶² choices of net; they are 3 × 10⁶² candidates, of which about one in a hundred is a net, and the search for one is a real search even though it is an easy one.

That has a consequence for the neighbour-loss scoring done two rungs down. The cube’s 384 nets could be enumerated exhaustively and the best one chosen. At 180 faces neither is possible: the candidates cannot be enumerated, and a random sample of them is 99 per cent invalid. Optimising a net’s readability at these sizes means searching within the valid ones, and the valid ones are not a random subset — they are the compact ones, which are probably the readable ones too.

What was computed, and how #

A spanning tree is generated from a stated seed by a random-edge walk, so a stated seed reproduces a stated net. The unfolding lays each face down by rotating it about the edge it shares with its parent, which preserves every distance on the surface exactly — the net is an isometry of the solid’s surface, and it is the overlap rather than the metric that can fail.

Overlap is tested pairwise between every two faces of the laid-out net, by polygon intersection in the plane. That is quadratic in the face count and it is why the samples are hundreds rather than thousands at 320 faces.

The breadth-first trees are re-rooted at face zero before they are laid out, because the layout routine places that face first. It is the same net either way: unfolding along a tree gives one planar figure up to a rigid motion, whichever face is put down first, so re-rooting changes where the net sits on the page and nothing about whether it overlaps.

The check that makes the zeroes meaningful is the one the earlier rung built: an irregular tetrahedron — four vertices placed unevenly on a sphere — whose nets do overlap in about a fifth of its sixteen unfoldings. So a net can overlap, the detector sees it, and the regular solids simply do not do it.

Why this is a mapping question rather than a puzzle #

It would be reasonable to read the table above as combinatorics with a cartographic label attached, so it is worth saying what turns on it.

A polyhedral map’s whole argument is that it beats a single projection by cutting: distortion falls as the reciprocal of the face count and cutting rises as its square root, so at enough faces the distortion is negligible and the cost is a page covered in seams. That argument is about the limit, and it presumes that at each face count there is a net to draw.

The measurement says the presumption is safe and not for the reason it looks. There is a net at every face count — many of them — and the fraction that work goes to zero, so the argument survives only because a construction that always works happens to exist. If breadth-first unfolding failed at 320 faces, the family’s limit argument would be a statement about objects nobody could produce.

That is a real dependency and it is invisible from the distortion side of the trade. A projection family’s asymptotics are usually about the projection; this one is also about whether the paper can be cut.

Where the model stops #

One family. The measurement is on the icosahedron subdivided by the standard frequency construction, which gives triangular faces. A Goldberg polyhedron of hexagons and twelve pentagons would give different numbers, and its dual relationship to this family means its face graph is different in a way that matters for spanning trees.

Edge unfoldings only. Cutting along edges is one choice; a general unfolding may cut across faces, and every convex polyhedron has such an unfolding. Whether every convex polyhedron has an edge unfolding that does not overlap is Dürer’s problem, open since 1525, and nothing here bears on it — what is measured is how many of them work, not whether any does.

Random means uniform over spanning trees. The sampler walks the face graph adding random available edges, which is not exactly uniform over spanning trees. A uniform sampler would give slightly different rates; the ordering across face counts, which is what the result is, would not move.

Overlap is not the only failure. A net can be valid and useless — spread across a page in a shape nothing can be printed on, or splitting a continent into six pieces. Validity is a floor, and the neighbour-loss score is what sits above it.

The generalisation #

Two statements, and the second is the useful one:

Validity is not free past twenty faces. The overlap rate goes from zero to one across a factor of sixteen in face count, and it goes through a half somewhere near eighty. A polyhedral map with more than a few dozen faces cannot have its net chosen arbitrarily.

Compactness is what makes a net valid, and the obvious construction is compact. Unfolding outwards from a face works at every size tried, from every face. So the search has a rule that ends it immediately, and the reason nobody has noticed the search exists is that the rule is what everybody already does.

Which is a pattern worth naming, because this collection meets it repeatedly: a procedure that is obviously right is also the one that avoids a failure nobody knew was there, and the failure only becomes visible when somebody tries the procedure’s alternatives. The alternatives here are 3 × 10⁶² spanning trees, and 99 per cent of them are traps.

The one number that is not a fraction #

Every count above is a rate, and rates hide a fact worth stating on its own: the number of unfoldings is not merely large, it is large in a way that changes what “search” means.

The cube has 384 and they can be listed. The icosahedron has 5,184,000, which can be listed with a little patience — and this collection has listed enough of them to be confident about the zero. At 80 faces there are 3.5 × 10²⁷, at 180 faces 3.1 × 10⁶², and at 320 faces 3.2 × 10¹¹¹.

Those numbers come from Kirchhoff’s matrix–tree theorem, which turns the count of spanning trees into a determinant of the face graph’s Laplacian, so they are exact rather than estimated. And they mean that every statement in this essay about a rate is a statement about a sample: 150 trees out of 10¹¹¹ is not a survey, it is a probe.

What makes the probe worth trusting is the direction of the result. A sample finding no valid nets in 150 tries is weak evidence that none exists and strong evidence that they are rare; a sample finding all 320 breadth-first unfoldings valid is a complete enumeration of that construction rather than a sample of it. So the two halves of the finding rest on different kinds of evidence, and the half the practice depends on is the exhaustive one.

Who found it, and when #

Albrecht Dürer’s Underweysung der Messung of 1525 draws polyhedra as nets, and the question of whether every convex polyhedron has a non-overlapping edge unfolding carries his name, though he never asked it.

The question has been open since Shephard posed it formally in 1975. What is known is discouraging in one direction and encouraging in another: unfoldings of non-convex polyhedra can fail to exist at all, and random edge unfoldings of convex polyhedra with many faces overlap with probability approaching one — which is what the table above measures for one family.

The geodesic family is Buckminster Fuller’s, from the 1940s, and its cartographic use is the Dymaxion map of 1943 — which is an icosahedron, twenty faces, comfortably inside the range where every unfolding works. That is why the question does not arise in the cartographic literature: the maps that were actually drawn were small enough not to meet it.

Where the ladder goes next #

Seven rungs have put a globe on a solid, priced the cut, asked what a face can preserve, made a face conformal, traded faces for cutting, chosen among nets by what they cost a reader, and now found that most of the candidates are not nets at all.

What none of them has done is let the faces be curved. Every solid here has flat faces, which is what makes the net an isometry and the cut a set of straight edges. A polyhedron with slightly curved faces could distribute the curvature between the faces and the corners rather than concentrating all of it at the corners, and the trade that would make is one this ladder has never priced.