The cut has to go somewhere

A polyhedron is flat everywhere except at its corners, so it can be laid out in the plane without stretching anything. It cannot be laid out whole: a corner with an angle deficit cannot be surrounded by flat paper, so something has to be opened.

That is the same statement the globe on a solid makes about curvature, turned into a practical requirement. The curvature is at the corners, and it has to be released somewhere.

How much has to be opened is settled by counting. A net keeps some of the solid’s edges joined and cuts the rest; keeping them all would leave the corners in the way, and keeping too few would break the map into pieces. The right number is exactly F − 1 joins, which by Euler’s formula leaves E − F + 1 = V − 1 cuts. One fewer cut than there are corners, on every convex polyhedron there is.

Every corner is cut, and that is forced #

The duality above has a consequence that is worth extracting before the counting starts, because it explains what the cuts are for.

A spanning tree of the corner graph touches every corner — that is what spanning means. So in every net of every convex polyhedron, each corner has at least one cut running into it. There is no unfolding that leaves a corner intact, and there could not be: an intact corner is a point with an angle deficit surrounded by flat paper, which is exactly what cannot exist.

So the cuts are not a nuisance to be minimised, they are the mechanism. Each one releases the curvature at the corners it joins, and the tree structure is the statement that the release has to reach every corner exactly once over — connected so that nothing is left out, acyclic so that nothing is released twice and the map falls apart.

Compare that with what no map is faithful says about a smooth sphere: there the curvature is everywhere, so the failure is everywhere, and the only choice is how to distribute it. Here the curvature is at twelve points, so the failure is eleven cuts, and the choice is where to put them.

The count is a determinant #

Kirchhoff’s matrix-tree theorem, from 1847, says the number of spanning trees of a graph is any cofactor of its Laplacian — build the matrix with degrees on the diagonal and −1 for each link, delete a row and the matching column, take the determinant.

Applied to the five solids’ face graphs:

Those are exact integers, not estimates, and the first three are checked twice: enumerating every subset of the right size and keeping the ones that are spanning trees gives 16, 384 and 384, which is a brute-force route sharing no code with the determinant.

The count is the number of labelled nets — nets counted as distinct even when a rotation of the solid takes one to another. Counting shapes rather than cuttings gives the familiar smaller numbers: eleven distinct hexomino nets of a cube, and 43,380 of the dodecahedron.

The two trees, and why the counts repeat #

Two of those numbers appear twice, and it is not a coincidence.

The edges a net keeps form a spanning tree of the face graph. The edges it cuts are the complement, and they form a spanning tree of the solid’s own skeleton — the graph of corners and edges. That is checked here rather than asserted: the cut set on every solid has V − 1 edges, is acyclic, and connects every corner to every other.

For a planar graph and its dual, the complement of a spanning tree is a spanning tree of the dual, so the two counts are the same number. The cube’s skeleton has 384 spanning trees and its face graph has 384; the dodecahedron’s skeleton has 5,184,000 and so does its face graph, which is the icosahedron’s skeleton.

The consequence is worth stating in plain terms, because it makes the whole business easier to think about: cutting a solid open is the same operation as choosing a connected path system through its corners. The corners are where the curvature is, and the cuts are a tree that visits all of them.

Whether a net lies flat without overlapping #

Counting nets is one question and drawing them is another. A net can fail in a way a graph cannot see: laid out in the plane, two faces that are nowhere near each other on the solid may end up on the same piece of paper.

That is not hypothetical. It is the reason Dürer’s conjecture — that every convex polyhedron has at least one non-overlapping edge unfolding — has been open since 1975 and is still open.

For the regular solids the question can simply be answered:

The control bar is the part that makes the rest mean anything. A row of zeroes from a test that cannot fire is worth nothing, and the site’s rule is that an assertion must reject something before it is allowed to pass. So the same code is given a tetrahedron with one vertex at the pole and three at a colatitude of 150° — a legitimate convex polyhedron inscribed in the same sphere — and it reports overlaps in a quarter of the unfoldings.

The regular solids are simply well behaved. Any of their nets can be used, and the choice of which is free.

Which makes the cut placement the only free lunch on this site #

That is a stronger statement than it looks, and it is the practical content of this rung.

Every other choice in a projection costs something. Choosing a standard parallel moves the distortion, choosing an aspect moves it again, choosing a compromise trades one failure for another. Choosing which edges of a polyhedron to cut, on a regular solid, changes nothing measurable at all: the distortion inside each face is decided by the solid, and the cuts do not touch it.

So the cut placement can be spent entirely on the cartographer’s other concerns. Fuller’s Dymaxion map spends it on the oceans: the icosahedron is rotated so that no cut crosses a continent, and the result is a map where the land is unbroken and the sea is in pieces.

What does change between nets: the shape of the paper #

Metrically identical is not the same as interchangeable, and there is one quantity the choice of net genuinely moves.

Across 400 unfoldings of the icosahedron, the area of the smallest rectangle the net fits into runs from 25.0 to 50.0 square radii — a factor of exactly two between the tightest and the loosest — against a map area of 15.2 in the same units. So the best net wastes 65 per cent of its sheet and the worst wastes 230 per cent, and a printer paying for paper has a real reason to prefer one spanning tree to another.

That is the sort of consideration cartography is full of and this site rarely gets to measure, because it usually cannot be separated from the geometry. Here it can: the distortion is fixed by the solid, the tear positions are fixed by the tree, and the sheet efficiency is a third quantity that depends on the tree and on nothing else.

The cut length, which is what is being spent #

The cuts are free to place and not free to have. Their total length is a real quantity and it grows with the face count:

Against the sphere’s own great-circle circumference of 2π ≈ 6.28 radii, the icosahedron’s cutting is nearly two full circuits of the world.

A cut is also a place where the map’s coordinates stop being a function of position in the ordinary way: a point on a cut edge is drawn twice, once on each side, exactly as the antimeridian is on a whole-world map. The antimeridian is a cut in the numbers is about the one cut every ordinary world map has; a polyhedral map has eleven or nineteen of them, and every one behaves the same way.

That is the honest comparison with an interrupted projection. Giving up continuity measures what an interrupted sinusoidal buys — three lobes, a tear down each of two meridians, and a large reduction in the shape error — and the polyhedral construction is the same bargain taken much further: twenty lobes, eleven tears, and an error an order of magnitude smaller than any whole world map’s.

The octahedron, and a map somebody actually printed #

The octahedron also sits at the crossover the previous rung’s ladder shows: its 20.7° is already better than the best whole-world compromise projection’s 76.8°, and going to twenty faces buys another factor of two and a half in shape at the cost of six more cuts. Where a pseudocylindrical puts its error is the essay about making that decision without cutting anything, and the comparison between the two is what the family is for.

Bernard Cahill’s map was patented, printed, and proposed as an international standard, and its arrangement is one of the octahedron’s 384 nets chosen so the cuts fall in the Pacific and the Southern Ocean. The choice is exactly the free one described above, made by a cartographer who did not need to know how many alternatives there were.

What the cuts do to a route #

A reader tracing a journey across a polyhedral map meets the cuts in a way no ordinary projection makes them meet one.

On the icosahedron with gnomonic faces, a great circle is a straight line within a face and a straight line within the next face, joined at the shared edge with a kink. The kink is real and it is not distortion: the two faces meet at a dihedral angle on the solid, and laying them flat opens that angle out, so the route’s direction changes by exactly the amount the flattening added.

Where the route meets a cut rather than a shared edge, it does something worse: it leaves the map and reappears somewhere else, on the far side of a tear that may be half the sheet away. A route from London to Sydney on an eleven-cut net can cross three of them.

That is the practical objection to polyhedral maps and it is not a small one. The site’s usual verdicts are about how wrong a drawn quantity is; this is a map where the quantities are unusually right and the continuity is unusually broken, and no distortion measure on this site reports continuity at all. Giving up continuity is the essay that raises the same point for an interrupted projection with two tears, and the polyhedral case is that argument multiplied by five.

Where the model stops #

The nets here are geometric, not aesthetic. Which of the 5,184,000 a cartographer should choose is a question about coastlines, and this site holds no coastline data on purpose — the reasoning is in computing an area needs a surface, and it applies here too: a net chosen to avoid cutting land is chosen against a particular generalisation of the coastline, and a different simplification would recommend a different net.

The overlap test is about the drawn net, not the projection. Whether the map on a net overlaps is the same question as whether the net does, because the face maps take each face’s boundary to its own boundary — which is the property what a face can preserve shows both face projections have and which an arbitrary face map would not.

Only edge unfoldings are counted. A net that cuts across a face rather than along an edge is also a valid flattening, and there are infinitely many of those. Allowing them changes Dürer’s question completely — every convex polyhedron is known to have a non-overlapping general unfolding — so the open problem and the counting here both belong specifically to cuts along edges.

Overlap is tested for the drawn faces, in the plane. The test is a separating-axis check on convex polygons, with a tolerance so that two faces meeting along a shared edge count as touching rather than overlapping. Faces meeting at a single point also count as touching, which is the right answer for a printed map and is a decision rather than a fact.

The nets sampled for the two large solids are 2,000 of 5,184,000. Nothing here proves the icosahedron has no overlapping net; it says that none was found in a sample, and that the three solids small enough to check exhaustively have none at all.

Who found it, and when #

Kirchhoff’s theorem is from 1847 and was about electrical networks; the spanning trees were a step in solving a circuit, not an object of interest. Dürer drew polyhedral nets in 1525 without asking whether they always work, and the question that carries his name was raised by Geoffrey Shephard in 1975.

The connection to cartography is more recent than either. Cahill’s and Fuller’s maps were designed by hand, one net apiece, and the observation that a polyhedral map’s cut pattern is a spanning tree — and therefore that there are exactly this many of them — belongs to computational geometry rather than to cartography, which is why the two literatures state it in vocabularies that share almost no words.

The free lunch has an algorithm #

If cut placement is the only choice on a polyhedral map that costs nothing, it is worth knowing that choosing it well is a solved problem rather than a search over an astronomically large set.

The number of nets is a determinant and it is enormous, which makes enumeration hopeless for anything past the coarsest solids. That is the discouraging reading and it is the wrong one, because the objective decomposes.

The damage is a sum over edges. Each edge of the solid is either cut or not; a cut edge tears whatever crosses it; and the cost of tearing a given edge — how much land it crosses, how much of a coastline, how much of a route — is a number computable for that edge alone, without reference to which other edges are cut.

And the set of legal cut patterns is the complement of a spanning tree of the face-adjacency graph. So minimising total cut damage is maximising the damage retained inside the tree, which is a maximum-weight spanning tree problem: Kruskal’s algorithm, sorting the edges by weight and taking them greedily, in time that is linear in the edges up to the sort.

That turns an intractable-looking enumeration into a few milliseconds for any solid anybody would print, and it produces a provable optimum rather than the best of however many nets somebody had patience to draw by hand.

One honest caveat, and it is the same one the flatness question raises. The spanning-tree formulation says nothing about whether the resulting net lies flat without overlapping — that constraint is not a sum over edges and cannot be folded into the weights. So the algorithm returns the best candidate, and its cut cost is a lower bound on what any valid net can achieve; whether that candidate is valid has to be checked afterwards, and if it is not, the next-best trees are enumerable in order.

Which is a much better position than the one the design tradition has been in. Cahill and Fuller each drew one net, by hand, and defended it; the argument here is that the best net under a stated cost is computable, and that the cost is the only part of the problem requiring judgement.

There is one more consequence of stating the cost as a sum over edges, and it is what makes the formulation worth having beyond this subject. Because the cut cost of a net is the total edge length of the solid minus the length kept by the tree, minimising the cut is maximising the tree, and the two are the same problem read in opposite directions. So a designer who wants the shortest cut and a designer who wants the longest run of unbroken coastline are asking for the same net, and neither has to be told about the other’s objective.

Where the ladder goes next #

The solid is chosen, the faces are measured and the cutting is counted. What has been assumed throughout is the map onto a face: the gnomonic, chosen because it is obvious and because it draws great circles straight.

It is not the only choice, and the next rung measures what the alternatives buy. A face can be mapped with its area preserved exactly — to a part in a million, by a construction with an ordinary differential equation in it — and what that costs is angle. A face cannot be given both, for the same two-line reason no whole projection can.