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

The globe on a solid

Cylinder, cone and plane are not the only surfaces a sphere can be laid on. Project it onto a polyhedron and the curvature goes entirely to the corners — π at each of the tetrahedron's four, π/5 at each of the dodecahedron's twenty, and always 4π in total, which is exactly the curvature of the sphere it replaced.
15 min read 8 figures A theorem, not a limitationMeasured, not namedEvery figure here is a projection

The family taxonomy this collection has used from the start is cylinder, cone and plane, and cylinders, cones and planes argues that the taxonomy says less than it appears to: the surface a projection is notionally rolled from does not settle any property anybody cares about.

It also leaves something out. Those three are the developable surfaces — the ones that unroll flat without stretching — and there is a fourth kind of surface with the same property that no essay here has drawn. A polyhedron is flat inside every face and flat along every edge, so it unrolls too.

The icosahedron's faces, drawn on the sphere. The edges of the icosahedron projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 25.8° from its centre to its own boundary, and the gnomonic map onto it reaches 6.6° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion.
Fig. 1 The icosahedron’s edges projected radially onto the sphere. That is the partition a polyhedral map works with: everything inside one spherical triangle is drawn on one flat face, each face reaching 25.8° from its own centre to its boundary. The construction has been used for real maps — Fuller’s Dymaxion is this solid, Cahill’s butterfly is the octahedron — and this site had never computed one.

Where the curvature goes

The interesting thing about a polyhedron is not that it is flat in the large; it is where it fails to be.

Walk about on a face and the surface is a plane. Walk across an edge and it is still a plane, in the only sense that matters — the two faces can be opened out into one without stretching anything, which is what makes a net possible at all. The surface is flat everywhere except at a finite set of points.

At a corner, the face angles meeting there do not add to a full turn. Six equilateral triangles round a point would give exactly 2π and lie flat; the icosahedron puts five, which gives 5π/3, and the shortfall of π/3 is the amount by which the surface is curved at that corner.

Every corner's share of one sphere. The angle deficit at one corner of each solid — how far the face angles meeting there fall short of a full turn — with the number of corners beside it. The products are all the same: 4 × 3.142, 6 × 2.094, 8 × 1.571, 20 × 0.628, 12 × 1.047, each equal to 4π. That is Descartes' theorem, and 4π is also the total Gaussian curvature of the sphere the solid stands in for. A polyhedron is flat everywhere except at its corners, and its corners carry exactly the curvature of a sphere.
Fig. 2 The angle deficit at one corner of each solid, with the number of corners beside it. Every product is the same: four corners at π, six at 2π/3, eight at π/2, twelve at π/3, twenty at π/5 — and 4 × π = 6 × 2π/3 = 8 × π/2 = 12 × π/3 = 20 × π/5 = 4π. The deficits are computed from the vertex coordinates rather than quoted.

Descartes’ theorem is Gauss–Bonnet with the curvature at points

That every solid’s deficits total 4π is Descartes’ theorem, from about 1630, and it is not a coincidence about regular solids. It holds for every convex polyhedron, however irregular.

It is also exactly the statement the curvature ladder spent eight essays on, in a different notation. Total curvature and the scale rule computes that the Gaussian curvature of a sphere, integrated over the whole sphere, is 4π regardless of its radius — a topological invariant rather than a geometric one. A polyhedron has zero curvature almost everywhere and a spike of it at each corner, and the integral comes out to the same 4π.

So the polyhedron is not an escape from the impossibility. It is the impossibility rearranged: the curvature that a smooth sphere spreads evenly over its whole surface is swept into a finite number of points, where it becomes a set of corners the map cannot lie flat around.

That reframing is what makes the family worth having. Every other projection on this site distributes the unavoidable failure continuously and argues about how; a polyhedral one concentrates it and then cuts the surface open at the concentrations.

The face’s boundary is a great-circle arc, and that is why the trigonometry is short

One detail of the construction is worth stating because it makes the machinery cheap and because it is easy to assume the opposite.

A face of the solid is a flat polygon whose edges are straight lines in space. A straight line and the centre of the sphere define a plane, and the intersection of a plane through the centre with the sphere is a great circle. So the radial projection of a face’s edge onto the sphere is an arc of a great circle, and the spherical face is a spherical polygon with geodesic sides — not some curve needing its own approximation.

The consequence is that the boundary radius at a given azimuth from a face’s centre is the solution of one equation per edge: with ĉ the centre direction, d̂ the direction of travel and m̂ the unit normal of the edge’s plane, a point at arc ρ satisfies cos ρ (ĉ·m̂) + sin ρ (d̂·m̂) = 0, so tan ρ = −(ĉ·m̂)/(d̂·m̂). Three of those for a triangle, five for a pentagon, and the smallest positive answer is the boundary. No iteration, no tolerance, and the whole face-projection machinery rests on it.

Which way up the solid goes

The solid’s orientation relative to the Earth is free — three degrees of freedom, and none of them changes any number in the table above.

That is the same statement the aspect is a free choice makes about an ordinary projection, and it has a sharper consequence here. On a conic or an azimuthal projection the aspect moves the good region about; on a polyhedral one there is no good region to move — every face is equally good and equally bad — so what the orientation actually moves is which parts of the world land near a corner, where the distortion is worst, and where the tears fall.

Fuller spent his three degrees of freedom on the second. The Dymaxion map’s orientation is chosen so that no cut crosses a continent, which is a decision with no metric consequence whatever and a large presentational one. Fitting the aspect to the region is the essay about spending the same freedom on distortion instead, which is the choice a polyhedral map does not have to make.

The map onto a face

Inside a face, the natural projection is the gnomonic: the point where the ray from the centre of the sphere through the place meets the face’s plane. It is the projection the gnomonic companion is about, used here on twenty small pieces instead of one large one.

Its defining property carries over exactly. A great circle within a face is drawn as a straight line, and the flexion machinery says its flexion is zero to the noise floor of the arithmetic — the only zero in the library.

What it costs is that the face’s corners are further from the centre of the sphere than its middle is, so they are stretched:

solid faces face radius worst ω areal spread
tetrahedron 4 60.0° 43.9° 9.63
cube 6 48.2° 28.3° 4.18
octahedron 8 41.4° 20.7° 2.85
dodecahedron 12 33.6° 11.6° 1.78
icosahedron 20 25.8° 8.0° 1.50

Every number is measured with the same distortion() the rest of the site uses, on a projection object built for the face, so they are comparable with every other number here. For scale: the Mercator projection reaches an areal factor of 1.7 at 40° north, and a gnomonic icosahedron face never exceeds a spread of 1.5 anywhere.

What those numbers are worth, beside an ordinary world map

A table of angular deformations means nothing without something to compare it against, and the comparison is the argument for the whole family.

Measured over the whole sphere with the same code, at the same sampling:

projection worst ω worst areal error scale spread
gnomonic icosahedron face 8.0° 0.52 1.32
Winkel tripel 76.8° 1.98 4.75
Robinson 99.5° 1.06 7.44
Mollweide 125.0° 0.00 16.68
sinusoidal 113.2° 0.00 11.11

The polyhedral map is better than every compromise world map in the library by an order of magnitude on the shape measure and by a factor of four on the scale spread, and it is not close. A reader looking only at that table would conclude the argument about world projections had been settled in 1943 and everyone had failed to notice.

They have not failed to notice. What the table leaves out is the cutting, and the cutting is what the rest of this essay is about — the numbers above are the distortion inside a face, and a map that is excellent inside each of twenty pieces and torn between them is not the same object as a map that is poor and whole.

The trade-off nothing else in the library has

Twenty faces are better than four, and the reason is simply that each face covers less of the sphere. Nothing stops the count rising further — a Goldberg solid with 500 faces would drive the distortion lower still.

That is a knob no other projection has. Every other construction on this site trades one distortion against another: the trade-off is two lines shows that conformal and equal-area cannot hold at once, and every compromise projection is a choice about how to divide a fixed failure. A polyhedral map instead buys distortion down with something else entirely.

More faces, less distortion, more cutting. The trade-off the polyhedral family exists to make. Across the five regular solids the worst angular deformation inside a face falls from 43.9° on the tetrahedron to 8.0° on the icosahedron, while the total length that has to be cut rises from 4.9 to 11.6 sphere radii. No projection in the ordinary library has a knob that does this: distortion is bought down with cut length rather than traded against another distortion.
Fig. 3 The worst angular deformation inside a face and the total length that must be cut, against the number of faces. Distortion falls from 43.9° on the tetrahedron to 8.0° on the icosahedron; the cut length rises from 4.9 to 11.6 sphere radii. Every published polyhedral map sits somewhere on this curve, and where it sits is a decision about how much cutting a reader will accept.

What is being spent is cut length. A net is the solid opened along enough edges to lie flat, and the number of cut edges is E − F + 1, which by Euler’s formula is one fewer than the number of corners. So the tetrahedron needs three cuts and the dodecahedron nineteen, and the cuts are where the map tears the world apart.

That is the same currency giving up continuity spends on an interrupted pseudocylindrical projection, and the polyhedral family is the case where the interruption is not a design decision made to save the oceans but a structural requirement of the surface.

A map, drawn

The sphere on an icosahedron, unfolded. A polyhedral map: the sphere projected face by face onto an icosahedron and the solid cut open along 11 of its 30 edges. The face projection here is the gnomonic, which draws every great circle as a straight line within a face, and the worst angular deformation inside a face is 6.6°. Each cut is a place where two pieces of the world that touch are drawn apart; each corner of the solid is a place where the surface has curvature and the map has an angle deficit.
Fig. 4 The sphere on an icosahedron, cut along eleven of its thirty edges and laid flat, with the graticule drawn on each face by the same face projection the table above measures. Twenty pieces of world, each of them a nearly undistorted small map, and eleven tears. Fuller’s Dymaxion map is this construction with the solid rotated so that the cuts fall in the oceans.

Two things about that picture are worth pointing at rather than leaving to be noticed.

The first is that the graticule breaks at every face boundary — the meridians change direction as they cross an edge, because two faces meet at an angle on the solid and are laid flat at a different angle in the net. The map is continuous as a surface and its coordinate lines have kinks.

The second is that the world is nowhere badly distorted and is everywhere cut up. That is the polyhedral bargain in one image, and whether it is a good bargain is a question about the reader rather than about the geometry.

The same construction on a smaller solid

The sphere on a cube, unfolded. A polyhedral map: the sphere projected face by face onto a cube and the solid cut open along 7 of its 12 edges. The face projection here is the gnomonic, which draws every great circle as a straight line within a face, and the worst angular deformation inside a face is 21.2°. Each cut is a place where two pieces of the world that touch are drawn apart; each corner of the solid is a place where the surface has curvature and the map has an angle deficit.
Fig. 5 The sphere on a cube, seven cuts, with a worst angular deformation inside a face of 28.3° against the icosahedron’s 8.0°. Fewer, larger pieces and less cutting: this is the other end of the same trade-off, and it is the construction every screen-map indexing scheme reaches for, because six square faces index far more conveniently than twenty triangles.

The cube is the interesting middle case because its faces are the shape a computer wants. That is a separate argument and it belongs to the cells ladder, which is about addressing the sphere rather than drawing it.

The tetrahedron, which shows the mechanism by failing

Four faces is the fewest a solid can have, and it makes the mechanism visible by exaggerating it.

The tetrahedron's faces, drawn on the sphere. The edges of the tetrahedron projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 60.0° from its centre to its own boundary, and the gnomonic map onto it reaches 38.7° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion.
Fig. 6 The tetrahedron’s four faces on the sphere. Each face reaches 60° from its centre — a spherical triangle covering a quarter of the world — and the gnomonic map onto it stretches the corners by a factor of 9.6 in area and shears them by 43.9°. The angle deficit at each of the four corners is a full π, which is half a turn: the surface is so sharply pointed there that a flat neighbourhood of a corner is a half-plane.

The deficit of π is worth dwelling on because it is the largest a convex polyhedron can have at a vertex, and it is where the discrete and smooth pictures separate most clearly. A smooth sphere has a small amount of curvature everywhere; the tetrahedron has none at all over 99 per cent of its surface and a half-turn’s worth at each of four points. Both integrate to 4π, and no continuous deformation between them changes that number.

The cube's faces, drawn on the sphere. The edges of the cube projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 48.2° from its centre to its own boundary, and the gnomonic map onto it reaches 21.2° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion.
Fig. 7 The cube’s six faces on the sphere, between the tetrahedron’s four and the icosahedron’s twenty. Each face reaches 48.2° from its centre and the deficit at each of its eight corners is a right angle — eight times π/2 is 4π again, which is the theorem holding on a solid whose faces are not triangles.

Where the model stops

The faces here are regular polygons and the solids are the five regular ones. Nothing in the machinery requires that: the angle deficit, the face projection and the net all work for any convex polyhedron, and the geodesic solids that real polyhedral maps use — subdivided icosahedra with hundreds of faces — are the obvious extension. What they lose is regularity, so the faces are no longer identical and the table above becomes a distribution rather than a single number for each solid.

The corners are not drawn. A polyhedral map has a genuine singularity at each corner of the solid: the projection is fine everywhere except exactly there, where the map has an angle deficit and no amount of care removes it. The figures sample the interior of each face and stop short of the vertices, which is honest about the construction rather than a limitation of the sampling.

The face projection is a choice. The gnomonic is the obvious one and it is not the only one, and the third rung of this ladder measures what the alternatives buy: an area-preserving face map exists, holds its areal factor at one to a part in a million, and pays for it in angle.

The sphere on an octahedron, unfolded. A polyhedral map: the sphere projected face by face onto an octahedron and the solid cut open along 5 of its 12 edges. The face projection here is the gnomonic, which draws every great circle as a straight line within a face, and the worst angular deformation inside a face is 17.5°. Each cut is a place where two pieces of the world that touch are drawn apart; each corner of the solid is a place where the surface has curvature and the map has an angle deficit.
Fig. 8 The octahedron’s net, which is Cahill’s butterfly arrangement: eight faces, five cuts, and 20.7° of angular deformation at the face corners. It sits between the cube and the icosahedron on the ladder above, and it is the shape a printed polyhedral map has most often taken.

Who found it, and when

Polyhedral maps are old and their inventors are unusually easy to name because each one is associated with a solid. Albrecht Dürer drew nets of polyhedra in 1525, for reasons that were about solid geometry rather than about the Earth. Bernard Cahill patented a butterfly map on the octahedron in 1913. Buckminster Fuller published the Dymaxion map on the cuboctahedron in 1943 and moved it to the icosahedron in 1954. John Snyder gave the family its equal-area member in 1992.

Descartes’ theorem is older than any of them and was not published in his lifetime: it survives in a manuscript copied by Leibniz and was not printed until 1860, two centuries after it was written and thirty years after Gauss had proved the smooth version nobody had connected it to.

What kind of error a cut is

The trade this family makes is unlike anything else in the library, and it is worth being precise about why a reader might prefer it, because the usual comparison — how much distortion, at what cost — misses the property that matters.

Distortion is continuous, invisible and everywhere. A reader of a Mercator sheet cannot see the areal factor; nothing on the page marks where it is 2 and where it is 14; and no part of the map is free of it. The error is smooth, so there is no edge at which a reader could be warned, and it contaminates every measurement taken anywhere on the sheet by an amount the sheet does not state.

A cut is discrete, visible and localised. It is drawn. A reader can see exactly where it is, can see that a feature has been torn, and can see that everywhere else the sheet is intact. The polyhedral family has moved the error out of the interior of every face and concentrated it on a set of curves that are marked on the map.

That is a different kind of honesty, and it is the family’s real argument. Not that the total error is smaller — the measurements in this rung say what it is and it is genuinely good — but that the error is declared. Nothing else in the library declares its worst behaviour by drawing it.

The preference between the two is a question about the reader rather than about the map. Somebody measuring within a region wants the cuts, because their region is intact and the distortion inside it is small and bounded. Somebody tracing a route across the world wants the compromise, because a route crossing three cuts is a route in pieces, and a route crossing a smoothly distorted sheet is at least continuous.

And the two failure modes are not commensurable, which is the same point no map of the whole sphere is one to one makes about the three escapes. There is no exchange rate between this shape is eight per cent too large and this ocean is in four pieces, so the choice between a polyhedral map and a compromise cannot be settled by any distortion score, however carefully constructed.

It also explains why the family’s advocates argue about the solid and its detractors argue about the cuts. Both are talking about the same trade from the two sides of it, and neither has a measurement that could settle the other’s objection, because the two quantities being weighed have no common unit and the family’s whole design consists of choosing between them.

Which explains something about how these maps are received. A polyhedral world map is either found revelatory or found unusable, with very few readers in between, and the split is not about taste. It is about whether the reader’s question survives being cut — and that is decided before they look at the map, by what they wanted from it.

The measurement in this rung takes a side without saying so, and it is worth admitting which. Reporting the distortion inside the faces and counting the cuts separately is the polyhedral family’s own framing — it presents the good number and leaves the bad one uncombined, because there is no way to combine them. A reader should treat the two as a pair rather than as a headline and a footnote.

Where the ladder goes next

The solid has been chosen and the faces measured. What has not been decided is where to cut — and that turns out to be a question with a number attached: which edges to open is a spanning tree of the face graph, so the count of distinct nets is a determinant, and it comes out at 384 for the cube and 5,184,000 for the icosahedron.

Whether any of them overlaps itself when laid flat is a second question, and it is the next rung.

Where this goes next

Construction against property. Every projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has. Three entries are picked out: Mercator, Albers equal-area conic, Orthographic. Cylinders, cones and planes The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on. A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere. What can be unrolled A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not. The least distortion possible over a 30° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison. Total curvature and the scale rule The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 13 that link here.

The objects this essay names

Each one links to every other essay that touches it.

Angle deficitDevelopable surfaceFaceGauss–Bonnet theoremGaussian curvatureGnomonicInterruptionPlatonic solidPolyhedral projectionTissot's indicatrix