Matpic
mathematics, in pictures
Concept

Infinite descent

An argument that from any solution in whole numbers a strictly smaller one can be produced, which is impossible because the whole numbers do not descend forever. It proves an impossibility by exhibiting a procedure, and it survives into settings where no polynomial is available to argue about.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

number · Irrationality Every candidate for a rational square root, tried. A column for each of 2, 3, 4, 5, 6, 7, 8, 9, listing the whole numbers that divide it with their squares, and the verdict the search returns.

Which roots refuse to be fractions

The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.

number · Irrationality A pentagon, its pentagram, and the pentagon inside. A regular pentagon with its diagonals drawn as a pentagram, enclosing a smaller pentagon, repeated 3 times inward; every pentagon has diagonal-to-side ratio φ.

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.

geometry · Golden ratio Right triangles of area five, six and seven. area 5: 3/2, 20/3, 41/6; area 6: 3, 4, 5; area 7: 24/5, 35/12, 337/60.

The areas a right triangle can have

A right triangle with sides 3, 4 and 5 has area 6; one with sides 3/2, 20/3 and 41/6 has area 5. No right triangle with rational sides has area 1, 2 or 3. Which whole numbers are areas is a thousand-year-old question, and its best answer — two counts of solutions to quadratic equations — rests on one of the great unproved conjectures.

geometry · Pythagoras

Named alongside it

The objects these essays reach for when they reach for this one.

IrrationalityContinued fractionsIncommensurabilityAlgebraic numberCongruent numberConjectureCounting two waysDescentDivisorElliptic curveExhaustive searchGolden ratio

All concepts