The same thing twice — page 9

Probability

The bound is the answer to a search

Chebyshev's inequality is not a clever estimate that happens to be sharp. It is the exact answer to a maximisation over all distributions with a stated mean and variance, and the polynomial that proves nothing beats it is the certificate a search of that kind always produces.

4 surfaces with an edge, and the three numbers they need. Polygons whose gluing words leave some edges unpaired, each with its Euler characteristic, its sidedness and the number of boundary circles the unpaired edges form.

Topology

The third number a surface needs

Leave an edge unpaired in a gluing word and the surface acquires an edge of its own. Two numbers no longer name it — a count of boundary circles is needed as well — and with that third number the list is complete again, every triple occurring exactly once.

The nine-point circle, touching four others. A triangle with its nine-point circle, its inscribed circle and its three escribed circles, each of the four tangent to the first — with the distances between centres compared against the radii.

Geometry

One circle touching four

The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.

The classical centres as three weights each. A table of triangle centres with the weights on the three corners that produce each, and the determinants that decide which triples of them are collinear.

Geometry

A centre is three weights

Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.

4 axioms against 4 finite algebras. A table of candidate axioms against finite Heyting algebras built from small orders, marking which algebras validate which axiom at every valuation.

Logic

Not one step but a continuum

Classical logic is the constructive system plus one axiom, which makes it sound as though there are two logics and one gap. There are uncountably many logics in that gap, each one a class of algebras, and the smallest separations between them fit in five elements.

Which polygons two instrument sets reach, up to 24. A strip of the polygons from 3 to 24 sides, each marked according to whether compass and straightedge reach it and whether a conic or a trisector does, with the degree of its cosine beneath.

Computation

Two instruments with one reach

Allow every conic to be drawn at will, or allow an angle to be cut in three. The two permissions look nothing alike and reach exactly the same numbers — because what an operation buys is a degree, and both of these buy three.

A matching of 4 and a cover of 4. A bipartite graph with its largest matching drawn thick and a smallest set of vertices meeting every edge ringed, the two having the same size.

Discrete

What the search has when it fails

A largest matching is easy to find and hard to certify: the claim that nothing larger exists is a claim about every arrangement not tried. The certificate turns out to be free — it is the wreckage of the search that failed.

The payoffs repetition makes available in the prisoner's dilemma. A plot of the two choosers' average payoffs, with the stage game's four cells marked, their convex hull drawn, the two minmax values shown as lines, and the region above both shaded.

Applied

Patience instead of a contract

Commitment had to assume an announcement binds. Play the same game again tomorrow and the assumption is unnecessary — the future does the binding. What it costs is that nearly every outcome becomes an equilibrium, so a theory that could not choose between two now cannot choose between infinitely many.

n / 2ⁿ held under a geometric series. A bar for each term of the series n / 2ⁿ with a decaying geometric curve above them, the curve lying above every bar from term 2 onwards.

Analysis

The series everything else is measured against

A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.

The two squares of 97, produced by division. A table of the division chain on 97 and a square root of minus one modulo it, with each row's quotient and remainder, the point at which the remainder falls below the square root marked, and the two squares that add to 97.

Number

The two squares actually produced

Three proofs say a prime one more than a multiple of four is a sum of two squares, and not one of them hands over the squares. Running the Euclidean algorithm half-way does — and where to stop is the whole of the correctness argument.

How long 4 equally likely patterns take to appear. A bar for each of 4 patterns of 3 coin tosses giving the expected number of tosses before it first appears, with the lengths at which each pattern overlaps itself listed.

Probability

Two patterns, one chance, different waits

HTH and HTT are equally likely in any given window of three tosses. Waiting for HTH takes ten tosses on average and waiting for HTT takes eight, and the difference is not about probability at all — it is about what a failed attempt leaves behind.

A rule that meets the diagonal, and the same rule with a hole in it. Two panels, each the graph of a rule assigning a set of values to every point of the unit interval, drawn against the diagonal: the first meets the diagonal at a filled-in jump, the second has the jump left open and misses it.

Topology

A map that offers a choice

Brouwer's theorem needs a function, and the object it was most wanted for is not one — a best reply is a whole set whenever a chooser is indifferent. Allow a point to be sent to a set and the fixed point survives, provided the sets are convex, and the convexity is the entire hypothesis.

2 independent rows and 2 independent columns. An array of 3 rows and 4 columns beside its transpose, with the independent rows of each shaded, showing the same count on both.

Algebra

Counted across and counted down

A rectangular array has a number of independent rows and a number of independent columns. The two are counted in different spaces, from different objects, by computations that share nothing — and they are always the same number, which is why 'rank' is one word.

2 independent cycles and 3 independent cuts, on 5 edges. A small graph beside its incidence matrix, with the matrix's rank and nullity given and shown to be the number of independent cuts and the number of independent cycles.

Algebra

The cycles and the cuts

The count that splits a map's source into what dies and what survives has nothing to do with graphs. Apply it to a matrix built from a graph's edges and points and it says that a graph's independent cycles and its independent cuts add to its number of edges — a theorem about drawings, obtained from an array.

A design on 7 points cannot have fewer than 7 blocks. The incidence matrix of a design on 7 points and 7 blocks beside the product of it with its own transpose, which has a constant off the diagonal and a determinant computed exactly.

Computation

More blocks than points

A schedule in which every pair meets once cannot use fewer groups than it has people. Nothing about the counting conditions says so, and the proof is not combinatorial at all — it is a determinant, computed over a field the schedules have nothing to do with.

A plane of 13 points from a list of 4 numbers. A ring of 13 points with one block of 4 of them drawn as a closed path, beside the table of the 13 blocks its shifts produce.

Computation

A plane in a list of numbers

A projective plane of order three has thirteen points and thirteen lines and fifty-two incidences. All of it is in the four numbers 0, 1, 3, 9 — because their pairwise differences hit every non-zero residue modulo thirteen exactly once, and the plane is that list's thirteen shifts.

Probability

Any unevenness brings the match sooner

Real birthdays are not spread evenly across the year, and every such departure pushes the famous twenty-three down rather than up. The proof is one move on two days at a time, and what it leaves behind is a single number — the one ecologists use to count species.

The walk x² + 1 modulo 101, drawn as the letter ρ. Starting at 2 and squaring and adding 1 modulo 101, the walk visits 8 values once on a tail and then runs round a cycle of 9 values for ever.

Probability

A collision that finds a factor

A walk through the remainders modulo a number must eventually repeat, and it repeats modulo each hidden prime factor long before it repeats modulo the number. Pollard saw that the earlier repeat can be detected without knowing the prime — and that its timing is the birthday problem, so the cost is the square root of the factor.

The Alexander matrix of the trefoil. The trefoil with its 3 arcs numbered and its 3 crossings lettered, beside the 3 by 3 matrix they give. A minor of the matrix is the Alexander polynomial t − 1 + t⁻¹, whose value at −1 is the determinant 3.

Topology

A polynomial behind the colourings

The figure-eight knot and the cinquefoil both have determinant five, so they admit exactly the same colourings, and every counting argument treats them as one. Put a variable where the colouring rule has a two and the determinant becomes a polynomial — and the two knots come apart.

The turning point's orbit over the bifurcation diagram, 3.55 to 4. The bifurcation diagram of the logistic map from 3.55 to 4 with the curves f(1/2), f²(1/2), … up to the 6th image drawn over it. The first two bound the attractor and the rest trace the dark lines inside it.

Dynamics

The dark lines are one point's orbit

Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.

A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

Analysis

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

Analysis

When two circular motions come home

Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.

100 as three triangular numbers. 100 drawn as three triangles of dots with 36, 36 and 28 dots. There are 6 such decompositions.

Geometry

Three triangular numbers, and no fewer

On 10 July 1796 Gauss wrote in his diary: ΕΥΡΗΚΑ — num = Δ + Δ + Δ. Every whole number is a sum of three triangular numbers. Two are not enough, and not by a little: the numbers that are sums of two thin out to a share of nought. Both facts are statements about squares in disguise, and one picture translates them.

Maximising the product of two people's values. The frontier of value pairs from dividing 4 goods between two people, with the points maximising the product, the sum and the smaller value. The product's maximum is (65.0, 54.2) and is envy-free.

Applied

The product that makes a division fair

Divide goods to make the total happiness as large as possible and the result can be monstrously unfair; make the least happy person as happy as possible and it can waste. Multiply the people's values together and maximise the product instead, and something unexpected happens — nobody envies anybody when goods can be split, and nobody envies by more than one item when they cannot.