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The same thing twice — page 21

Two constructions that look unrelated and turn out to be the same object wearing different clothes.

The curve whose bending grows with its length. Euler's spiral, curvature equal to arc length, for s from −7 to 7, winding into ±(0.88623, 0.88623).

The curve that bends as far as it has gone

Say how sharply a curve bends at each distance along it, and the curve is fixed. Bending in proportion to distance gives Euler's spiral, whose arms are infinitely long and end a finite distance away. Bending that repeats decides at once whether the curve ever closes, and taken in steps the spiral is a Gauss sum.

Three widows and the Mishna's table. The Mishna's division of estates of 100, 200 and 300 among claims of 100, 200 and 300: 33⅓ each; 50, 75, 75; 50, 100, 150.

Applied

The debts the Mishna divided

A man dies owing his three wives 100, 200 and 300. The Mishna divides an estate of 100 equally, of 300 in proportion, and of 200 as 50, 75 and 75, and for nearly two thousand years nobody could say why. The table is the nucleolus of a cooperative game, and the only division in which every pair splits its share as two people split a contested garment.

Two blocks and a wall count to 31. Space-time diagram of a heavy block of mass 100 driving a block of mass 1 against a wall: 31 elastic collisions in all.

Dynamics

Two blocks and a wall count π

Slide a heavy block towards a light one resting by a wall, and count every collision until they part for good. A heavy block 100 times the mass gives 31; ten thousand times, 314; a million, 3,141. The digits are π because the two blocks are secretly one billiard ball in a wedge, and a ball in a wedge bounces as many times as the wedge's angle fits into half a turn.

Four perpendiculars from inside, two from outside. Normals to the ellipse x²/4 + y²/1.44 = 1 from (0.55, 0.18) inside its evolute (4) and from (2.3, 0.9) outside it (2), with the evolute drawn.

Geometry

Four perpendiculars to an ellipse

From a point near the centre of an ellipse, four lines meet the curve at right angles; from a point far away, only two. Apollonius found the boundary between the cases twenty-two centuries ago. It is the curve traced by the ellipse's centres of curvature, a four-cusped star, and it is also the envelope of all the normals, the fold in a family of lines where two perpendiculars merge and vanish.

Four rods, hinged, enclosing the most. Quadrilaterals with sides 3, 4, 5, 6 as one corner angle varies; the largest has area 18.97367, equal to Brahmagupta's √360 = 18.97367, with its corners on a circle of radius 3.2873.

Geometry

The most four hinged rods can enclose

Four rods of fixed lengths, hinged at their ends, can be pushed into a whole family of quadrilaterals. The one enclosing the most area has its corners on a circle, and Brahmagupta gave its area in 628 as a product of four terms. Order the rods any way round the circle and the area is the same; add a fifth rod and the circle still wins, but the formula is gone.

$Five powers of two that are seven more than a square. How close 2ⁿ − 7 comes to a square for n from 3 to 40, as a fraction of the gap between neighbouring squares: squares at n = 3, 4, 5, 7, 15 (1, 3, 5, 11, 181 squared), and at no other n up to 10000.$ Number

Five powers of two, seven more than a square

2³ − 7 = 1², 2⁴ − 7 = 3², 2⁵ − 7 = 5², 2⁷ − 7 = 11², 2¹⁵ − 7 = 181². Ramanujan asked in 1913 whether there are any more. Testing every exponent up to 10,000 finds none, and Nagell proved in 1948 that none exist, by moving to a number system where 2 splits into two primes and factorisation is still unique. The same five solutions decide when a two-error-correcting code can be perfect.

$A knot written as a fraction. The two-bridge knot K(17/5) drawn as a plat on four strings from the continued fraction 17/5 = 3 + 1/(2 + 1/2), with 7 crossings.$ Topology

A tangle that is a fraction

Two ropes, four people, two moves: twist the right-hand pair, or turn everyone a quarter of the way round. Conway noticed that every tangle made this way carries a fraction, that the moves act on it as x + 1 and −1/x, and that the fraction is a complete invariant: two such tangles are the same exactly when their fractions are. Close the tangle and the fraction names a knot.

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