No map of the whole sphere is one to one

This collection has one impossibility in it and has had since its first page. No map is faithful: a sphere has Gaussian curvature 1/R², a plane has none, curvature can be computed from inside a surface without reference to anything outside it, and so no map preserves every distance. Eleven essays develop that, and it is the reason the rest of the site is a series of measurements rather than a series of opinions.

Every step of that argument needs a distance. Curvature is built from the first fundamental form, the first fundamental form is a rule for measuring lengths, and the whole apparatus of scale factors, areal ratios and angular deformations that follows from it is a way of saying how far off a map is. That is why it produces numbers, and it is why a better projection can make those numbers smaller.

There is a second impossibility sitting underneath it, and this collection has never stated it.

A sphere is compact and has no boundary. The plane is not. So there is no continuous one-to-one map of the whole sphere into a page at all — not an inaccurate one, not a badly distorted one, none. The statement uses no distance, no derivative and no tolerance. It holds for any surface a sphere can be deformed into, for any body, for any projection anyone will ever invent, and there is no quantity in it that a cleverer construction could reduce.

The theorem, in three lines #

A continuous injection of a compact space into a Hausdorff space is a homeomorphism onto its image. That is a standard fact and its proof is short: a closed subset of a compact space is compact, a compact subset of a Hausdorff space is closed, so the map takes closed sets to closed sets, so its inverse is continuous.

The sphere is compact. So if a projection were continuous and one-to-one on the whole of it, the sphere would be homeomorphic to a subset of the plane.

It is not. Invariance of domain says the image of an open set under a continuous injection between spaces of the same dimension is open, so the image would have to be an open subset of the plane — and an open subset of the plane is not compact, while the continuous image of a compact space is. The two cannot both hold, and there is nothing to negotiate.

Nothing above mentions area, angle, scale or a projection formula. That is the whole point of it. The impossibility the rest of this field measures is quantitative and its size can be traded against; this one is not a quantity.

What that forces a real projection to do #

The theorem says a projection cannot be defined, continuous and one-to-one everywhere. It does not say which of the three a given projection gives up, and the interesting content is that the library gives up different ones.

Leaving ground out. The map is simply not defined on part of the Earth. The orthographic is the honest case: it draws a hemisphere and declines the rest.

Drawing one place as a curve. The map is defined everywhere, but at some place its answer depends on a number the place does not possess. A plate carrée’s north pole is the standard example — the formula returns the longitude as its x coordinate, and the pole has no longitude, so the one place is drawn as the whole top edge.

Cutting. The map is defined everywhere and single-valued everywhere, and it is discontinuous across a curve, which it then draws twice. Every uncut world map with a rectangular or elliptical outline is of this kind.

The table above is that classification run over the library, with each column measured rather than asserted. What it comes to is that of eighteen projections, five leave ground out with area, two leave out a single place, nine draw at least one place as a curve, thirteen are cut, and eleven do more than one of those. None escapes by none of them.

The pole, on three maps #

The clearest picture of the second escape is the north pole drawn on maps that treat it differently.

Three maps, one place. On the first it is 40,000 kilometres of paper, on the second it is 17,000, on the third it is a dot. And the reason this belongs in a different essay from the distortion work is that no measurement of distortion distinguishes them. Tissot’s indicatrix at 89.9° north is a perfectly well-behaved ellipse on all three. The indicatrix at the pole itself does not exist on any of them, which the distortion ladder has already established — but it establishes it as a failure of the derivative, and what is happening here is one step earlier than that. The map is not a function of the place. It is a function of the coordinates, and the coordinates have a place they do not describe.

The measurement that separates a place from a curve #

Calling the plate carrée’s pole “a line” is a claim, and this site does not print claims it has not tested. The test is the one that decides whether a map has a derivative at a point at all: draw a small circle of ground radius ρ round the place, look at its image, and shrink ρ.

At an ordinary point the image shrinks in proportion. That is what a derivative is: the map is locally a linear one plus something smaller, so quartering the circle quarters the picture.

The slope of that log-log line is the whole classification, and it needs no vocabulary beyond the picture got smaller. One means an ordinary point. Zero means the place is drawn as a curve of fixed size. And a slope that goes the other way — the image growing as the circle shrinks — means the place has been sent to infinity, which is the stereographic’s answer at the antipode of its centre.

And the pole is not the special case #

The third curve on that chart is the interesting one. It is the azimuthal equidistant projection, and the place being examined is the antipode of its centre: 180° east, on the equator. Nothing marks that point out. It is not a pole. It is not on the antimeridian in any sense the projection’s formula cares about. The formula is a smooth function of the angular distance from the centre and behaves perfectly well right up to the last step.

One place on the Earth, drawn as a closed curve three times the length of the equator. The Lambert azimuthal does the same thing with a rim of 12.50 units, and both of them are in daily use. What decides it is not the coordinate system and not the formula; it is that an azimuthal projection is a map of the sphere-minus-a-point onto a disc, and the point it removed has to go somewhere. The rim is where.

What “leaves ground out” is worth, and why the fraction does not say #

The first escape looks like the one that admits a number. The orthographic omits 49.86 per cent of the Earth; the polyconic, as this library bounds it, omits 1.52 per cent; the conformal conic omits 0.061 per cent; the stereographic omits 0.0048 per cent. A ranking suggests itself immediately and it is worthless, because those four numbers are not all measurements of the same kind of thing.

The exponent is the dimension of the missing set, obtained by refining rather than by knowing the answer in advance. A region keeps its share, a curve halves it, a point quarters it. So the conformal conic and the stereographic, which both report a fraction well under a tenth of a per cent, are doing entirely different things: one has thrown away a strip of the world and the other has thrown away one place. The gap between their fractions is a factor of thirteen; the gap between what they mean is a whole dimension.

This matters beyond the tidiness of it. The measurement code that produced the table at the top of this essay would happily have reported the stereographic as omitting five thousandths of a per cent of the Earth, and a reader would have taken that as a small version of the orthographic’s fifty per cent. It is not a small version of it. It is a different column.

The cut, and the curve that is drawn twice #

The third escape is the most common and the least remarked on, because every reader has seen it and it looks like the edge of the paper.

The cut is not a property of the picture’s border. It is exactly the set on which injectivity fails, and its length on the page is what the projection paid to be continuous everywhere else. Across the library the two edges are 4.44 units apart on Gall–Peters, 5.14 on the Winkel tripel, 5.66 on Mollweide, 6.28 on the plate carrée and the sinusoidal, and 45.50 on the conformal conic, where they run away from each other towards the pole the cone does not reach. The duplicated ground curve is the same 20,004-kilometre half-meridian in all thirteen cases; what differs is what the page spends on it, which runs from 4.00 units to 51.31.

There is an essay on this site that looks like this one and is not. The antimeridian is a cut in the numbers shows that no choice of where to put the seam removes it, because a circle cannot be numbered by an interval. That is a statement about longitude — about a coordinate, one dimension, and a stored geometry that crosses the wrap. This one is about the map: two dimensions, and a projection that is asked to be continuous and injective at once. They rhyme because they are the same kind of obstruction one dimension apart, and the first is the reason the second is so often mistaken for a convention that could be tidied up.

What the three escapes cannot be traded against #

Suppose someone wanted to rank the library by how much it pays. The orthographic pays 254 million square kilometres of ground. The plate carrée pays 6.28 units of page for a place with no size. Mollweide pays 5.66 units of page for a ground curve 20,004 kilometres long, which it duplicates rather than loses.

Those are a solid angle, a length, and a length attached to a different length. To put them on one axis is to invent an exchange rate between a missing hemisphere and a duplicated meridian, and no such rate exists — not because nobody has worked it out, but because the two quantities have different dimensions and are not measurements of a common thing.

This is why the world-map argument never settles. Choosing between projections is a question with an answer as soon as a purpose is named, and the purpose supplies the units. There is no purpose that makes an omitted hemisphere commensurable with a doubled meridian, so the choice between an azimuthal world map and an elliptical one is not a smaller version of the choice between Mercator and Gall–Peters. It is a different kind of question, and the reason is above.

Where the model stops #

Everything here is about the sphere. Every statement in this essay holds for any surface homeomorphic to a sphere, which includes the ellipsoid, the triaxial bodies and every irregular body this site has mapped — but it says nothing about a torus, and a torus can be covered by a plane, which is why a doughnut has a perfectly good flat map with no cut and no missing point. The obstruction is the sphere’s topology and not surfaces in general.

“One to one” is being asked of the whole sphere. Every projection in the library is one to one on a large open piece of it, and that is what makes them useful. UTM works because a zone is a disc; a tile is a disc; a national grid is a disc. The theorem bites exactly when a single sheet is asked to carry everything at once, which is a thing atlases avoid and world maps cannot.

The measurements are of this library. Eighteen projections is not all projections, and the table is evidence for the theorem rather than a proof of it. The proof is three lines and is above; the table is what the three lines look like when a computer is made to check every member of a real collection, including the ones whose escape is not the one their family suggests.

Who found it, and when #

The compactness argument is nineteenth-century point-set topology in its final form, but the cartographic version is much older than the language for it. Every maker of globe gores knew that a sphere will not lie flat without cuts, and knew it as a fact about paper rather than as a theorem. The step from “will not” to “cannot” happened twice: once for distance, in Gauss’s Theorema Egregium of 1827, which this field opens with; and once for continuity, in the topology that Poincaré, Brouwer and their successors built between 1895 and 1912. Brouwer’s invariance of domain, published in 1912, is the piece that closes the argument above, and it was proved to settle a question about dimension rather than about maps.

The two impossibilities are usually presented as one, and the presentation matters. A reader told that “the Earth cannot be flattened” and shown a curvature argument will reasonably conclude that a sufficiently small piece of the Earth can be. That is true of the curvature obstruction and false of this one, which does not get smaller and does not go away: it lives on the whole sphere and nowhere else, and it is the reason the smallness of the piece is what makes the rest of this site’s measurements possible.

What it settles about a seamless world #

One modern question falls straight out of the theorem, and it is worth answering because the marketing language obscures it.

A seamless global map — a single continuous sheet on which any place can be reached from any other without crossing an edge — is the exact thing the theorem forbids. Not difficult, not expensive: forbidden, on the same three lines.

So every system that appears to offer one has taken one of the escapes, and knowing which is knowing what the system will do at the wrong moment.

A slippy web map takes the cut, at the antimeridian, and hides it by wrapping the tiles horizontally so a reader panning west arrives at a fresh copy of Asia. The map is not continuous there; it is periodic, and a geometry crossing the line is torn exactly as the theorem requires. The same system takes the omission at the poles, cutting the pyramid off near 85°.

A globe viewer looks seamless because it has declined to flatten anything at all. It renders a sphere, and a sphere is not a map — the theorem has nothing to say about it, and neither does any of this site’s arithmetic about distortion, because there is none.

There is no third arrangement. A reader told a product is seamless should ask where the cut went, and there is always an answer.

Where this goes next #

Two more consequences follow from the same source and neither is about position.

The first is about direction. Ground north is a field of arrows on the sphere, and the next rung asks whether a map can carry it consistently: it cannot, the failure is counted rather than measured, and the count is the same 2 that Gauss–Bonnet reaches by integrating curvature — the same invariant from the metric and from counting.

The second is about opposites. A projection may be continuous everywhere or injective everywhere. When it chooses the first, the failure of the second is not arbitrary: it always includes a pair of antipodal places, drawn on the same spot. That is a theorem too, and it is a stronger statement than anything on this page.