Geometry — page 3

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.

Three ordinary lines from a count

Kelly's proof finds one line through exactly two of the points by minimising a distance. Melchior, seven years earlier, had found three — by turning every point into a line and counting the corners, edges and regions of the picture that results. Euler's formula for the projective plane does the rest, and it says exactly which configurations have no more than three.

The fewest ordinary lines a polygon allows

Take the corners of a regular polygon and add the points at infinity where its parallel chords meet. Every chord then carries three points, the line at infinity carries all the new ones, and the only lines left with exactly two points are the tangents at the corners — half as many as there are points. Dirac guessed in 1951 that nothing does better, and Green and Tao proved it in 2013.

One number for every chord through a point

Draw any line through a point and let it cut a circle twice. The two distances from the point to the circle multiply to the same number whichever line is drawn — inside, outside, or grazing as a tangent. The number belongs to the point, and the reason it does not depend on the line is the inscribed angle: two chords through a point cut out two triangles with the same angles.

Every side measured by one diameter

In any triangle, divide each side by the sine of the angle opposite it: the three answers are equal. That much is the law of sines, and it is usually left there. The common answer has a name — it is the diameter of the circle through the triangle's corners — and the reason is the inscribed angle once more. It is also why the sine was first a half-chord, why Ptolemy's theorem is the addition formula, and why a circle can hold infinitely many points all at rational distances from each other.

Seven pieces and an equilateral middle

Morley's theorem has two proofs worth knowing, and they run in opposite directions. The trigonometric one starts from the triangle and computes each side of the inner one as 8R sin α sin β sin γ, symmetric in the three angles. Conway's starts from an equilateral triangle, builds six pieces round it from their angles alone, and shows they fit — so the triangle they make is whatever triangle was wanted, and its middle is equilateral because it was built that way.

Eighteen equilateral triangles

Every angle of a triangle has three trisectors, not one, once the angle and its outside are both counted. Choosing one at each corner gives twenty-seven ways to cut out a triangle, and eighteen of them give an equilateral one. The nine that fail are exactly the choices whose labels add to 2, 5 or 8 — and all eighteen equilateral triangles have their sides in the same three directions, fixed by a third of the difference between two angles.

The player who meets the first long run

Turn Euclid's algorithm into a game: two players take turns cutting squares off the rectangle, any number from the current run, and whoever cuts the last one wins. The whole game is decided before it starts — by whether the ratio of the sides is more or less than the golden ratio, which is the same thing as how many runs of length one come first.

The remainders that count the roots

Run Euclid's algorithm on a polynomial and its derivative, flipping the sign of each remainder, and write down the signs of the whole chain at any point. The number of sign changes drops by exactly one each time the point passes a real root — so the roots in any interval can be counted, exactly, without finding a single one.

The diagonal no unit measures

Draw the five diagonals of a regular pentagon and they make a star with a smaller pentagon at its centre. Subtract the side from the diagonal and what is left is the smaller pentagon's diagonal; subtract that from the side and what is left is its side. The pentagon has handed back a smaller copy of itself, and it will do so for ever — which means no unit, however small, measures both the side and the diagonal an exact whole number of times.

Tiles that never repeat

Two rhombs with angles taken from the pentagon, cut each into halves, and cut each half into smaller copies of the two halves by a fixed rule. Repeat, and the pieces fill the plane with no gaps and no overlaps, in a pattern with five-fold stars everywhere and no period anywhere. The reason it cannot repeat is a single number: thick tiles outnumber thin ones by φ, and a repeating pattern would make that ratio a fraction.

The least wall for equal rooms

Divide the plane into rooms of equal area using as little wall as possible, and every wall does double duty. Bees settled on hexagons long ago, and the proof that nothing does better — not even rooms with curved walls — came in 1999. The straight-walled half of it is two facts: the rooms of any division average six sides, and more sides never cost more wall.

Every chord slid to the middle

Take a shape, pick a line, and slide every chord that crosses the line at right angles until the line cuts it in half. The area cannot change, the boundary can only get shorter, and the result is symmetric. Do it again about another line, and another, and the shape is squeezed towards a disc — unless the lines are badly chosen, in which case it stops short.

At least as many lines as points

Sylvester's theorem says some line through two of the points misses all the rest. Remove one end of that line and the line itself disappears, taking at least one line away with one point. Run that backwards and it proves that n points not all in a line determine at least n lines — and the only sets that manage exactly n are a line of n − 1 points with one point off it.

Rows of three, planted on a cubic

Nine trees can be planted in ten rows of three, and the arrangement that does it is not a grid or a star but nine points on a cubic curve. On the curve three points are in line exactly when their angles add up to a right angle, so choosing the points as a cyclic group turns collinearity into addition — and the count of rows it produces is the number Green and Tao proved is the most any planting can reach.

The skeleton inside a shape

Take the Voronoi diagram not of a handful of points but of a shape's whole outline: the points inside the shape that have two nearest points on its edge. What comes out is a skeleton — a thin branching tree that describes the shape and, with one number attached to each point, rebuilds it exactly. It is also where fires lit all round the edge would meet, and it is violently unstable: a tooth too small to notice grows a branch several times its own size.

Three mirrors make every solid

Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.

Every point has a partner across the bisectors

Draw the three lines from the corners of a triangle through any point, reflect each in the bisector of its own angle, and the three reflections meet again. The pairing this makes swaps the centroid with the symmedian point and the orthocentre with the circumcentre, bends every straight line into a conic through the corners, and sends the circumcircle to infinity.

A triangle that fits once fits everywhere

Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.

A band of a sphere is a band of its cylinder

Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.

The ball that is largest in five dimensions

A disc of radius one has area π, a ball of radius one volume 4π/3, and in each further dimension the unit ball grows — until five, where its volume is 5.264, after which it shrinks towards nothing. The recursion that shows it is the ring dissection of the disc, done one dimension at a time, and the shrinking is not the ball getting small: it is almost all of a high-dimensional cube lying outside the ball, and almost all of the ball lying in a thin rind at its surface.

Six numbers that need five pentagons

In 1638 Fermat wrote that every whole number is a sum of three triangular numbers, four squares, five pentagonal numbers, six hexagonal numbers, and so on for every polygon — and that he had a proof he would not write down. The claim is true; Cauchy proved it in 1813. What the claim hides is how unequal the cases are. Triangles and squares need their full count infinitely often. For pentagons, only six numbers ever need all five — 9, 21, 31, 43, 55 and 89 — and from hexagons on, two apiece.

Fifteen numbers decide every number

Lagrange proved that x² + y² + z² + w² takes every whole value. Ramanujan asked which other sums of four weighted squares do, and in 1917 listed fifty-five. One of them is wrong: x² + 2y² + 5z² + 5w² misses 15. The mistake points at a theorem found eighty years later — to know whether such a form reaches every number, it is enough to check that it reaches 1, 2, 3, 5, 6, 7, 10, 14 and 15. Nine checks decide infinitely many cases, and the nine numbers come out of an escalation any one can run.

What the signs allow

Descartes' rule reads the signs of a polynomial's coefficients and bounds its positive roots by how often they change — a bound set by the number of terms, not the degree, and proved by nothing more than Rolle's theorem. Each count the rule allows can be had. But not every pair of counts, positive and negative, that the rule allows together can be had together: the first combination that never occurs is at degree four, and the list of impossible ones is still being worked out.