Generator

The whirling squares

A generator in the geometry library, called 21 times across 4 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

golden is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly.

A pentagon, its pentagram, and the pentagon inside

Euclid's subtraction on the pentagon's diagonal and side

How many times the smaller fits, step after step

A golden triangle cut into a smaller one, again and again

The pentagon as the fifth roots of one

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

generation 0: the counting rule agrees with the drawn subdivision on thin halves ×7
×1
… and one thick half ×1
… and two thick halves ×1
−2cos 144° = φ ×1
√2's takes it out twice, after the first step ×1
1 to 7 generations ×1
1 to 8 cuts ×1
2cos 72° = 1/φ ×1
2cos 72° is a root of w² + w − 1 ×1
4 to 14 subtraction steps ×1
8/5's stops, because a common unit exists ×1
a thick half becomes one thin half … ×1
a thick half has a 108° apex ×1
a thin half becomes one thin half … ×1
a thin half has a 36° apex ×1
and as long as the piece of the side it cuts off ×1
and it climbs towards φ from below at every cut ×1
and on thick ones ×1
and so is 2cos 144° ×1
diagonal minus side is the smaller pentagon's diagonal ×1
diagonal over side is φ in every pentagon ×1
each inner pentagon is smaller by φ² ×1
each of the first nine Markov values is attained by a repeating pattern of 1s and 2s ×1
each step shrinks the pair by φ ×1
each subtraction leaves something smaller than what it subtracted ×1
every Markov number below 10^30 is the largest member of exactly one triple ×1
every triple generated satisfies x² + y² + z² = 3xyz ×1
every value below 3 is a Markov number's value ×1
in the largest disc the ratio of thick to thin is within 0.05 of φ ×1
leg over base is φ ×1
one square per Fibonacci number ×1
one to five levels ×1
the 1s and 2s come in pairs, read round the block ×1
the bisector is as long as the base ×1
the chord to the second neighbour over the chord to the first is φ ×1
the convergents alternate about φ ×1
the convergents' constants approach the Markov value from above ×1
the diagonal's end piece is s/φ ×1
the first two values are √5 and √8 ×1
the last convergent is close to φ ×1
the long side is the next Fibonacci number ×1
the number of convergents plotted is between 2 and 20 ×1
the number of whirling squares is between 1 and 12 ×1
the pair at every step is in the ratio φ ×1
the pieces fill their parent exactly ×1
the ratio has converged to φ ×1
the rectangle is nearly golden ×1
the short side is the one before it ×1
the thick half built here has its 108° apex ×1
the values increase with the Markov number and stay below 3 ×1
the view is one the family draws ×1
the whirling squares tile their rectangle ×1
φ's subtraction always takes the smaller out exactly once ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Geometry

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

The numbers that come after the golden ratio

The golden ratio is the number worst approximated by fractions: no constant larger than √5 works in Hurwitz's theorem. Set it aside and √2 is worst, with √8. Set that aside and the next is a number whose continued fraction repeats 2, 2, 1, 1, with √221/5. Markov proved in 1879 that the list continues for ever below 3, one entry for each solution of x² + y² + z² = 3xyz.

Geometry

The rectangle that eats itself

Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.

Geometry

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 whole library · What the figures prove