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mathematics, in pictures
The collection

Every essay — page 17

Page 17 of 45, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Discrete

Counting, graphs, and things that come in whole pieces.

A ball counted slice by slice. Ball of radius 7.5: slices 0:177, 1:177, 2:169, 3:145, 4:129, 5:97, 6:69, 7:21; total 1791, volume 1767.146.

The dots a ball catches

6 figures The bipartite share falls before it rises. n=1: 1 of 1 (100.00%); n=2: 2 of 2 (100.00%); n=3: 7 of 7 (100.00%); n=4: 41 of 41 (100.00%); n=5: 376 of 388 (96.91%); n=6: 5177 of 5789 (89.43%); n=7: 103237 of 133501 (77.33%); n=8: 2922446 of 4682270 (62.42%); n=9: 116011231 of 246348115 (47.09%).

Triangle-free, and two-sided eventually

6 figures The twenty-three trees on eight points. 23 shapes on 8 points by Prüfer search; diameters 2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,7.

How many shapes a tree can take

6 figures The forests of a complete graph on four points. K4 forests by size: 1, 6, 15, 16; ratios 2.4000, 2.3438.

The forests a graph can hold

6 figures Two arrows in and two out at every point. 9×9 patch from a random height function, seed 3; heights range 1..6.

Two arrows in and two out

6 figures No graph has fewer four-cycles than chance. random p=0.092 t(C4)=0.00141 p^4=0.00007; random p=0.143 t(C4)=0.00344 p^4=0.00042; random p=0.112 t(C4)=0.00203 p^4=0.00016; random p=0.112 t(C4)=0.00276 p^4=0.00016; random p=0.112 t(C4)=0.00266 p^4=0.00016; random p=0.112 t(C4)=0.00266 p^4=0.00016; random p=0.306 t(C4)=0.01739 p^4=0.00878; random p=0.378 t(C4)=0.03223 p^4=0.02032; random p=0.316 t(C4)=0.02140 p^4=0.01001; random p=0.276 t(C4)=0.01400 p^4=0.00576; random p=0.245 t(C4)=0.01249 p^4=0.00360; random p=0.235 t(C4)=0.01182 p^4=0.00303; random p=0.490 t(C4)=0.07476 p^4=0.05755; random p=0.469 t(C4)=0.06862 p^4=0.04854; random p=0.449 t(C4)=0.05685 p^4=0.04064; random p=0.520 t(C4)=0.09126 p^4=0.07335; random p=0.480 t(C4)=0.07648 p^4=0.05290; random p=0.490 t(C4)=0.08080 p^4=0.05755; random p=0.673 t(C4)=0.23074 p^4=0.20572; random p=0.724 t(C4)=0.29628 p^4=0.27550; random p=0.673 t(C4)=0.23511 p^4=0.20572; random p=0.612 t(C4)=0.16150 p^4=0.14051; random p=0.633 t(C4)=0.19013 p^4=0.16020; random p=0.694 t(C4)=0.25750 p^4=0.23181; bipartite p=0.500 t(C4)=0.12500 p^4=0.06250; bipartite p=0.408 t(C4)=0.08330 p^4=0.02775; bipartite p=0.337 t(C4)=0.05670 p^4=0.01286; structured p=0.462 t(C4)=0.05189 p^4=0.04538; structured p=0.214 t(C4)=0.03389 p^4=0.00211; structured p=0.429 t(C4)=0.06778 p^4=0.03374; structured p=0.071 t(C4)=0.00036 p^4=0.00003

No graph is poorer in squares than chance

6 figures Powers of 5 and 6 until the digits settle. five-adic tail 0896109004106619977392256259918212890625; six tail 9103890995893380022607743740081787109376; settled after 8: 10, 9.

A number that is its own square

6 figures · new A flow that climbs for ever and falls short. 0:p0 +1.000000 → 1.000000; 1:p1 +0.618034 → 1.618034; 2:p2 +0.618034 → 2.236068; 3:p1 +0.381966 → 2.618034; 4:p3 +0.381966 → 3.000000; 5:p1 +0.236068 → 3.236068; 6:p2 +0.236068 → 3.472136; 7:p1 +0.145898 → 3.618034; 8:p3 +0.145898 → 3.763932; maximum 5.

A search for room that never ends

6 figures · new

Topology

What survives bending, and what does not.

A Möbius band. A strip joined end to end after a half twist, so it has one side and one edge.

The surface with one side, and what happens when it is cut

7 figures One point on the sphere for every point of the plane. Lines from the north pole of a sphere through each of its points land on a plane below, matching the sphere minus one point with the whole plane.

A sphere is a plane plus one point

7 figures V − E + F = 2, five times. Vertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row.

Every corner pays for itself

7 figures A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944.

Something always stays put

6 figures the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

Three moves, and what they cannot undo

6 figures The field that cannot be combed. A tangent field on the sphere, flowing along the meridians. Every arrow is tangent to the surface, and at the two poles there is no direction for an arrow to take — the field is zero there, and no rearrangement removes both zeros.

Nothing on a sphere can be combed flat

6 figures A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Which side of the line is inside

7 figures The four ways to glue a square's edges in pairs. Squares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.

Every surface is a sphere with handles

7 figures One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

One line that halves them both

7 figures 3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

A loop that cannot be pulled tight

7 figures How many colourings each knot allows. Three knots, and the number of ways their arcs can be coloured with three, five and seven colours under the crossing rule, beside the determinant computed separately from the same crossings.

Colours that count more than three

6 figures Two trees, sharing every edge between them. The cube flattened into a planar graph, with a spanning tree of its corners drawn solid and the leftover edges drawn dashed; the leftover edges join the faces into a second tree, and the two counts add to the number of edges.

Two trees, and every edge in exactly one of them

7 figures The gap at a corner of the cube. The 3 faces meeting at one corner of the cube, unfolded onto the page. They leave a gap of 90.0 degrees, and the 8 gaps come to 720 degrees in total.

Seven hundred and twenty degrees of gap

7 figures A solid where V − E + F is 0. a slab with one hole through it, drawn as a wireframe. Its 32 vertices, 64 edges and 32 faces give an alternating sum of 0 rather than 2.

The solid where the answer is not two

7 figures Three sets where a fixed point escapes, and one where it cannot. A ring turned about its centre, an open disc halved toward a point of its rim, the plane shifted sideways, and the closed disc turned and shrunk. Only the last has a point that its map leaves where it is.

Where the fixed point escapes

7 figures the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Two loops and one number

7 figures