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ひそむ秩序 · No.08 · リアルタイム描画の短編 · 6:42

Topology

A short film about topology

Topology in five short chapters. Bend a shape, stretch it, twist it: as long as nothing is torn or glued, some things never change. A strip joined into a loop with a half twist has one side. Draw any grid on a sphere, and vertices minus edges plus faces make two; however the sphere's hair is combed, it has bald spots, and the turns of the hair round them add up to two as well. Two loops linked once stay linked once however they are shaken, as long as neither passes through the other. These counts capture something no deformation can erase. Yet some tangles escape counting in pairs: three rings, no two of them linked, that still cannot be pulled apart. Last comes a sphere in four dimensions made entirely of circles, every pair of them linked once. Every frame is computed by code; on the website, each one is drawn live in your browser as the film plays.

上映時間: 6:42
構成: 五つの章と、プロローグ、エピローグ
解像度: 再生エリアに合わせて自動調整（最大 440 万画素）
フレームレート: 最大でディスプレイのリフレッシュレート（実際は端末の性能による）
事前に録画したコマ: なし
言語: 中文 · English

章

ナレーション原稿 · 88

Prologue
IOnly one side
IICounting to two
IIIA sphere you can't comb flat
IVLinked
VA sphere made of circles
Coda

図版の解説 · 5

I The Möbius strip

Topology asks what survives bending and stretching, as long as nothing is torn or glued. A band joined after a half twist, described in 1858 by Listing and by Möbius, has one side: a painter who never crosses its edge covers all of it, and an ant walking once down the middle comes back upside down. In the corner the sides, the edges and the pieces are counted, and how often curves are linked is integrated; every answer is a whole number.

One side: X(u+2π,v)=X(u,−v)Paint starts at one point and spreads along the band without ever crossing the edge. After one lap it arrives at the back of the place it started from, and after two it has covered both faces of every point: there is only one side. The ant at the front of the paint comes back after one lap with its normal pointing the other way, and after two the right way up.
Half twists: |Lk|=k/2(k even)Join the ends of a strip after $k$ half twists. With none it is a flat ring, two sides and two edges; with one it is the Möbius strip, with one of each; with two it has two sides again, but its two edges are linked once, as the corner finds by integrating; with three it has one side and one edge, and the edge is tied in a trefoil knot. Only an even number of half twists leaves two edges whose linking can be measured. Where gold runs into blue is the seam that an odd number of half twists leaves behind.
Cutting it: halftwists=2|Lk(edges)|Cut down the middle, the Möbius strip does not fall in two: it becomes one band, twice as long, with two sides and four half twists, which the corner reads off as twice the number of times its two edges link. Cut a third of its width from the edge, it falls into two pieces, a thinner Möbius strip and a long band with four half twists, and the two are linked.
The Klein bottle: χ=0Sew two Möbius strips together along their edges and the result is a Klein bottle, described by Felix Klein in 1882. In ordinary space it has to pass through itself, but it has no edge and, like the strip, only one side: the paint again covers everything. Gold and blue then pick out the two Möbius strips it is made of.

II Euler's formula

Euler noticed in 1750 that the vertices, edges and faces of a convex polyhedron always satisfy $V - E + F = 2$ . The number does not depend on the shape, only on the surface the polyhedron covers: 2 for a sphere, 0 for a torus, and, as L'Huilier found in 1813, 2 less for every hole. It is now called the Euler characteristic $χ$ , and a closed surface with an inside and an outside and $g$ holes right through has $χ = 2 - 2 g$ . In the corner the vertices, edges and faces are counted from the mesh on screen, up to tens of thousands of them.

V − E + F: V−E+F=2The five Platonic solids: the tetrahedron, $4 - 6 + 4$ ; the cube, $8 - 12 + 6$ ; the octahedron, $6 - 12 + 8$ ; the dodecahedron, $20 - 30 + 12$ ; the icosahedron, $12 - 30 + 20$ . Every one comes to 2.
Refining: V−E+F=2Split every triangle of the icosahedron into four and push the new vertices out onto the sphere. Each split adds a vertex on every edge, cuts every edge in two, and adds three edges and three faces inside every old face, so the new vertices and faces exactly pay for the new edges: 12, 42, 162 and 642 vertices, and still 2.
A handle: V−E+F=0Draw the grid on a torus instead and $V - E + F$ comes to 0 however fine the grid: 24 squares, then 96, 384 and 1,536. The number belongs to the surface, not to the drawing.
More handles: χ=2−2gSurfaces with two and three handles, cut into tens of thousands of triangles by marching tetrahedra through a field that vanishes on them. The corner counts every vertex, edge and triangle of the mesh and gets $- 2$ and $- 4$ : each handle costs 2.

III The hairy ball

A tangent vector field puts an arrow on every point of a surface, like hair combed over it or wind blowing over the Earth. Where the arrow shrinks to nothing the field has a zero, and the index of an isolated zero counts, with sign, how many times the arrow turns as you go once round it: $+ 1$ at a source, a sink or a swirl, $- 1$ at a saddle, and other whole numbers, such as $2$ , at more complicated zeros. Poincaré proved in 1885 for surfaces, and Hopf in 1926 in every dimension, that when the zeros are isolated their indices add up to the Euler characteristic: 2 on a sphere, which therefore can never be combed flat, and 0 on a torus, which can. In the corner every zero is found and its index counted on a small loop round it; zeros too close together to tell apart are shown as one cluster with their summed index, and the total is counted again, cell by cell, and agrees.

A calm spot: V=∇zComb every hair uphill, towards the north pole. The hair springs out in every direction from the south pole and gathers at the north pole: two calm spots, a source and a sink, each of index $+ 1$ . The corner finds each calm spot and counts how many times the hair turns on a small loop round it; the sum it counts again round the edge of each of the 1,350 cells that tile the sphere.
Swirls: V=cosα∇z+sinαe_z×pTurn every hair gradually sideways and the flow becomes a swirl round the axis, then a spiral. The calm spots stay at the poles and their indices stay $+ 1$ : a source, a sink and a whirlpool all turn once as you go round them.
Every sum is 2: 1+1+1−1=2A dipole field, a steady wind on the plane carried onto the sphere by stereographic projection, has a single calm spot, at the north pole, of index 2. Bend it into the gradient of a landscape with two peaks, a pass between them and a valley, and the zero breaks at once into four, at first too close to tell apart: a pass of index $- 1$ that stays at the pole and three zeros of index $+ 1$ . One moves out to the first peak. The other two swing round to the far side and meet on its meridian, where a new pair, $+ 1$ and $- 1$ , appears for a moment, making six; then the two original zeros and the new $- 1$ merge into a single $+ 1$ , leaving four, and the two $+ 1$ zeros on the meridian leave along it, one to the second peak and one to the valley at the south pole. The count changes; the sum never leaves 2.
A torus: χ(torus)=0Stand a torus on its edge and comb it uphill: a peak on top, a valley at the bottom and two passes, above and below the hole, with indices $+ 1$ , $+ 1$ , $- 1$ and $- 1$ . Now comb it round the axis: the peak and the pass above the hole slide together over the near face of the tube and cancel, and a moment later the valley and the pass below do the same on the far face. No zero is left: a torus, unlike a sphere, can be combed flat.

IV Knots and links

A knot is a closed loop in space, and two knots are the same if one can be moved into the other without cutting it or passing strands through one another. Telling them apart takes invariants, numbers that survive every such move. The linking number of two loops, which Gauss wrote down as a double integral in 1833, is one; the number of valid ways to colour the arcs of a diagram in three colours, after Fox, is another. In the corner the crossings are counted in a view from above, the colourings are tried one by one, and the integral is evaluated along the curves.

Linking number: Lk=signedcrossings/2Two rings, each passing once through the other: the Hopf link. Count the crossings between them in a view from above and add up their signs: half the sum, −1, is the linking number, its sign only recording which way round each ring is followed. Gauss's integral over the two rings, an entirely different calculation, gives the same number to four decimals, and it stays there while one ring is bent about, since no bending that avoids cutting and never passes one strand through another can change it. The Solomon link, whose rings wind twice round each other, gives 2.
Three colours: colourings(trefoil)=9Seen from above, the trefoil falls into three arcs, each ending where it passes under another. Colour every arc in one of three colours so that at each crossing the three arcs that meet are all alike or all different: of the 27 ways, 9 work, and the 6 that use all three colours prove the trefoil is knotted, since a plain loop allows only the 3 in a single colour. The figure-eight knot allows only 3 as well: it is knotted too, but this test cannot tell.
Torus knots: c=min(p(q−1),q(p−1))A torus knot winds $p$ times round the axis of a torus and $q$ times round its tube. $T (2, 3)$ is the trefoil; then come $T (2, 5)$ , $T (3, 4)$ and $T (3, 5)$ . The crossings counted from above, 3, 5, 8 and 10, are already the fewest that any drawing of these knots can have, as Murasugi proved in 1991.
Borromean rings: Lk=0,0,0Three rings of which no two are linked: each pair has linking number 0, and any two on their own could be pulled apart. Yet the three together cannot be separated. Take one away and the other two slide apart.

V The Hopf fibration

In 1931 Heinz Hopf found a map from the three-dimensional sphere $S 3$ , the unit sphere of four-dimensional space, onto the ordinary sphere $S 2$ , under which the points over any one point form a circle. He showed that the map cannot be deformed into a constant one, and the number that proves it, now called the Hopf invariant, is the linking number of any two of the circles: 1. Stereographic projection carries all of $S 3$ but one point into ordinary space, where the circles stay circles, except the one through that point, which becomes a straight line. In the corner the circles are counted and their linking numbers integrated, pair by pair.

Linked circles: Lk=1Over six points of the sphere, six circles appear one after another. Each passes once through every other: the corner integrates the linking number of every pair, and every pair gives 1. The small sphere at the top shows the points they lie over, in the same colours.
A torus of circles: |z₁|=cos(α/2)Seventy-two points spaced round a line of latitude give seventy-two circles that lie on one torus and weave it, each winding once round it both ways, like the Villarceau circles of an ordinary torus. As the latitude moves south the torus swells from a thin ring round one circle to a wide one round the axis; at the equator it is the Clifford torus. Any two of the circles, measured here on opposite sides, stay linked once.
The fibration: S¹→S³→S²Four lines of latitude, forty-two points on each: 168 circles on four nested tori. The whole fibration fills space, every point lying on exactly one fibre; the fibres are circles, except the one through the point at infinity, which is a straight line, the axis. Pairs chosen from across the tori are measured, and every one is linked once.
Turning S³: q↦q(cosω+ksinω)Turn $S 3$ by multiplying every point on the right by a unit quaternion. Circles go to circles and the fibration is unchanged, but projected into space it flows: the tori tilt and swell, and a circle that swings through the point at infinity grows past the edge of the picture. Throughout, every pair measured stays linked once.