A disc sewn to a Möbius band

The Möbius band has exactly one boundary curve. A disc has exactly one boundary curve. Two surfaces with one boundary curve each can be sewn together along them, and the result has no boundary at all.

What comes out is not the Klein bottle. It is smaller — the smallest closed surface with one side there is — and it is the fourth of the four ways to glue a square.

Three descriptions, and they are the same surface #

The projective plane arrives under three descriptions that look unrelated, and a good deal of the difficulty people have with it comes from being shown one and then another without being told they coincide.

A disc with opposite boundary points glued. Take a disc and identify each point of its rim with the point diametrically opposite. The rim becomes a single closed curve traversed twice, and the surface has no edge.

A Möbius band with a disc sewn on. Cut a small disc out of the description above and what is left is a Möbius band; putting it back is the sewing.

The lines through the origin in three-dimensional space. Each line meets the unit sphere in two antipodal points, so the set of lines is the sphere with opposite points identified. This is the description the name comes from, and it is the one the projective plane as plane geometry works with — a plane where any two lines meet, because parallel lines meet at the point their common direction names.

That these are one surface is not obvious and is worth the check that the third gives the first: cutting the sphere along its equator leaves two caps, the identification pairs them, so one cap suffices — and the identification that survives on the cap’s rim is exactly the antipodal one. A cap is a disc.

The odd number #

The Euler characteristic of the projective plane is 1, and that single fact does a great deal of work.

A closed two-sided surface is a sphere with $g$ handles and has characteristic $2 - 2g$, which is always even. So a surface with an odd characteristic cannot be two-sided, and no argument about sides is needed to know it — the counting decides.

The counting is done from the gluing word rather than from a picture. Four corners of the square fall into two classes under the identification, there is one edge for each of the two letters, and there is one face; $2 - 2 + 1 = 1$. The figure colours the corner classes so the two can be counted rather than taken on trust.

Compare the Klein bottle, whose four corners all become one vertex: $1 - 2 + 1 = 0$. One extra identification, one less vertex, and a characteristic that collides with the torus’s. The projective plane’s does not collide with anything.

The cross-cap, and why a handle costs twice as much #

Surfaces are built by connected sum: cut a disc out of each of two surfaces and sew them along the resulting boundary circles. Characteristics add, less two, because two discs were removed and one circle was added.

Adding a handle to a surface — connected sum with a torus — costs $2$ of the characteristic, since the torus contributes $0$ and the surgery costs $2$. Adding a cross-cap — connected sum with a projective plane — costs $1$, since the projective plane contributes $1$ and the surgery costs $2$.

A handle costs two and a cross-cap costs one. That is the arithmetic behind the whole classification, and it is why the two lists interleave rather than running parallel.

The word cross-cap names the picture rather than the surface: a disc’s boundary glued to itself antipodally cannot be drawn in three dimensions without a crossing, and the standard drawing is a cap with a seam through it. It is an immersion, for exactly the reason the Klein bottle’s is — a closed surface embedded in three-space separates it and is therefore two-sided.

The classification, both halves #

With the arithmetic in place the classification can be stated in full, and it is one of the cleanest theorems in the subject.

Every closed surface is a sphere with some number of handles, or a sphere with some number of cross-caps, and no surface is both. The first list has characteristics $2, 0, -2, -4, \dots$ and is two-sided; the second has $1, 0, -1, -2, \dots$ and is one-sided.

Two numbers name a surface: the characteristic, and the side count. Neither alone is enough, and the torus and Klein bottle are the standing proof that the characteristic alone is not.

One consequence surprises people and is worth stating. A sphere with a handle and a cross-cap is the same surface as a sphere with three cross-caps. Once a surface is one-sided, a handle stops being a distinguishable thing to add — the cross-cap can be slid around the handle in a way that turns it inside out. So the two lists really are lists rather than a table with two axes, and the two-sided half is the other one.

The lines through the origin, developed #

The third description deserves more than a sentence, because it is the one that gives the surface its name and its uses, and because it turns a topological object into a geometry.

A point of the projective plane is a line through the origin of three-dimensional space. Two such lines span a plane through the origin, and every plane through the origin contains many lines — so a line of the new geometry is a plane of the old one. With those definitions, two points determine a line and two lines determine a point, with no exceptions at all: two planes through the origin always meet in a line.

That absence of exceptions is what the construction buys. In the ordinary plane, two lines meet unless they are parallel, and every theorem about intersections has to carry the parallel case. Here there is no parallel case, because the directions that were missing have been supplied: the lines through the origin that lie in the horizontal plane are the extra points, one for each direction, and together they form one extra line — the line at infinity.

The symmetry between points and lines is exact, and it has a name: duality. Any theorem stated about points and their joins becomes a theorem about lines and their meets by swapping the two words, and the swapped statement is automatically true because the swap is an operation on planes and lines through the origin. That is a strong statement for something obtained by capping a Möbius band with a disc.

The same construction over a field with finitely many elements gives the finite projective planes, where the point count is small enough to draw — seven points and seven lines is the smallest, and it has the same duality for the same reason.

The topology and the geometry are the same object seen at two distances. A topologist asks which surface this is and answers the smallest one-sided one; a geometer asks what its incidence rules are and answers any two lines meet. Both answers come from the identification of opposite points and neither can be derived from the other without going back to it.

Who found it, and how late #

The two halves of this object were found by people who were not looking for the same thing, nearly a century apart, which is part of why the descriptions took so long to be recognised as one.

The geometry came first. Desargues was working projectively in the 1630s, Poncelet systematically in 1822, and the points at infinity were in place long before anybody asked what shape the resulting set of points was. The question did not arise, because a geometry was a system of rules rather than a space.

The surface came with the classification. Möbius and Listing found the band in 1858 while cataloguing surfaces; the projective plane appears in the classification that followed as the odd entry — literally, the one with the odd characteristic. Its impossibility in three dimensions was clear immediately, and for forty years it was assumed no smooth immersion existed either.

Boy’s surface, in 1901, settled that, and the way it was settled is worth the sentence: Hilbert set Boy the problem of proving that no such immersion exists, and Boy came back with one. It is drawn in every topology department and it is genuinely hard to see; the algebraic description that makes it computable was not written down until the 1980s.

Cutting it, and what falls out #

The band’s essay is largely about what happens when scissors are applied, and the question is worth asking here.

Cut the projective plane along the closed curve that cannot be shrunk — the identified rim of the disc, which is the band’s core. Nothing falls off. What is left is a single disc, and that is the reverse of the construction: the surface was a disc all along, with its rim identified, and cutting along the image of the rim undoes the identification.

Cut it instead along a small circle that can be shrunk, and a disc falls out, leaving a Möbius band. That is the other reverse, and the two together say that the surface really is the two pieces named in the title and nothing more.

A third cut is the interesting one. Cut along a curve that goes round the non-shrinkable loop twice: that curve can be shrunk, so cutting along it is like cutting along a small circle, and a disc comes away — but the disc’s boundary is a doubled curve, and reassembling it is where the two-to-one map of the rung above comes from.

Why it is not a sphere with a hole plugged #

A frequent confusion is worth heading off, because it is the natural first guess.

The projective plane is not a disc with its rim glued to a point, and it is not a sphere with anything removed. The identification on the rim is by pairs, not to a single point: gluing the whole rim to one point gives a sphere, which is two-sided and has characteristic 2.

The difference is one word — antipodal — and it is the difference between a surface with two sides and a surface with one. The rim, after the antipodal identification, becomes a closed curve; and the striking property of that curve is that going round it once does not return the way it left. Going round it twice does. That curve is the Möbius band’s core, seen from the other description.

The loop that has to be gone round twice #

The curve above deserves its own paragraph because it is the projective plane’s most useful feature and the one that outlives the surface.

A closed loop on a surface can sometimes be shrunk to a point and sometimes not; the essay on loops is about the distinction. On the projective plane there is a loop that cannot be shrunk, and the surprise is what happens when it is traversed twice: the doubled loop can be shrunk. So the loops form a group with two elements — go round an even number of times, or an odd number.

That two-element group is what the rung above is about. It is the reason the projective plane has a two-sheeted cover and the reason the cover is a sphere, and it is also why the projective plane is the natural home for anything that comes back reversed after one circuit and right after two: a spin, a sign, a frame.

The phenomenon has an everyday demonstration in the rotation group, where a full turn and a double turn are famously not the same thing to a belt or a cup of water held in the hand — and the reason is that the rotations of space form a three-dimensional projective space, whose loops behave exactly as this surface’s do.

Counting the cells a second way #

The Euler characteristic is a count over a cell structure, and a count is only worth trusting when the same number comes out of a different structure. The gluing square gives $2 - 2 + 1 = 1$. Here is another route.

Build the projective plane as a disc with an antipodally identified rim, and give the rim two marked points opposite each other. Those two points are identified with each other, so they become one vertex. The rim is cut by them into two arcs, and each arc is identified with the other, so they become one edge. The disc is one face. That is $1 - 1 + 1 = 1$.

A third route uses no square at all and is the one the rung above turns into a construction: take the icosahedron, whose $12 - 30 + 20 = 2$ is a sphere’s, and identify each cell with the one opposite it. Every cell has a distinct opposite, so the counts halve to $6 - 15 + 10 = 1$.

Three cell structures with different numbers of vertices, edges and faces, and one characteristic. That is what an invariant is, and it is the reason the number can be used to tell surfaces apart at all — a count that depended on how the surface was cut up would be a fact about the cutting. The same argument, on a solid rather than a surface, is what every corner pays for.

What the picture cannot show #

Every drawing of a projective plane is a drawing of something else.

The gluing square is honest and does not look like a surface. The cross-cap picture looks like a surface and crosses itself. Boy’s surface — a third immersion, and the prettiest — is smooth and has no pinch point, and still crosses itself along a curve. There is no fourth option in three dimensions.

The disc-with-identified-rim picture has a subtler failure. It is drawn as a flat disc, which invites the reading that the surface has a boundary and the arrows are decoration. They are not decoration: the arrows are the surface, and a point on the rim is the same point as the one opposite, in the way that two edges of a screen that wraps are the same edge.

And no picture at all shows the odd characteristic. The number is a count over a cell structure, and it is the one property of this surface that is completely reliable and completely invisible.

Where the ladder goes next #

Above this rung: the orientation double cover, which puts a sphere over the projective plane and a torus over the Klein bottle, and turns one-sidedness into a statement about a two-to-one map. And orientation stated as a sign, where the surface disappears entirely and what is left is a determinant.

One debt this essay opens. Boy’s surface is named and not drawn, and it deserves to be: it is the projective plane immersed with no pinch points at all, its self-intersection is a single curve with one triple point, and it was found by Werner Boy in 1901 after Hilbert had asked him to prove no such immersion existed.

What the sewing was for #

The smallest one-sided closed surface is a disc and a Möbius band, sewn along the one edge each of them has.

The construction is worth carrying because it explains the arithmetic. A cross-cap costs one of the Euler characteristic rather than two, and the reason is that it is built from a band and a disc rather than from a handle — one identification with a flip in it, where a handle needs two without. Every odd characteristic in the classification is that single flip, counted.