Proof without words — page 5
Algebra
Random roots crowd onto the circle
The roots of a random polynomial are not scattered across the plane. They gather within about 1/n of the unit circle, and their directions spread evenly round it. Paul Erdős and Pál Turán proved in 1950 that this is not a fact about randomness at all: any polynomial whose coefficients are all of roughly one size has roots whose directions are nearly even, and the inequality says exactly how nearly, in terms of nothing but the coefficients.
Geometry
Wings that cut a chord equally
Take a chord of a circle and its midpoint. Draw any two more chords through the midpoint, join their ends crosswise, and the two crossing lines — the butterfly's wings — cut the first chord at equal distances from the middle. The proof is the inscribed angle and the power of a point working together; move the point off the middle and what survives is a law about reciprocals; replace the circle by any conic and the theorem does not notice.
Geometry
The point nearest in total to three corners
Three towns want one depot with the least total road to all of them. The answer is not the centroid or any of the classical centres: it is the point from which every side of the triangle is seen at 120°, found by building equilateral triangles on the sides. Three straight lines through it have the same length, and that length is the least total. When an angle of the triangle reaches 120°, the answer jumps to that corner — and the same 120° rule shapes soap films and the shortest networks joining many points.
Topology
Torus knots that fill the three-sphere
Hopf filled the three-sphere with circles, every two linked once. Let the circles turn at two different speeds instead — p turns one way while they make q the other — and the three-sphere is filled with (p, q) torus knots, trefoils when p and q are 2 and 3, every two of them linked exactly pq times. Two exceptional fibres remain plain circles; every knot links them q and p times, and they link each other once. Computed from the drawn curves for ten fibrations, every pair comes out at pq to six decimal places.
Topology
An instrument that computes with circles
An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.
Topology
Lines that leave half on each side
Scatter ten points on a page and draw every line through two of them that leaves four on each side. There are at least five such halving lines and at most thirteen, and every point lies on an odd number of them. Both facts come from turning a line through half a revolution, the argument that halves two shapes with one cut. How many halving lines n points can have is the k-set problem, and its answer lies somewhere between a little more than n and n to the four-thirds.
Computation
Dudeney's four pieces are the fewest
In 1902 Henry Dudeney cut an equilateral triangle into four pieces that rearrange into a square, and hinged them so the triangle folds into the square in one motion. For a hundred and twenty-two years nobody could say whether three pieces would do. In December 2024 Erik Demaine, Tonan Kamata and Ryuhei Uehara proved they would not. The construction, built exactly here, shows why four is natural; the proof shows why it is necessary.
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.