Squares are often thought to be the most boring of all shapes. Yet every square holds a stunning secret – something that in legend prompted a mathematical cult to murder a traitor. If each side of a square is one unit long, how long is the square’s diagonal, that is, the line from one corner to the opposite corner?

By Pythagoras’ theorem, the answer is this:

• x^2 = 1^2 + 1^2

• x^2 = 2

• x = √2

But what is √2? Pythagoras and his followers thought that all numbers could be represented as either whole numbers or ratios of whole numbers. To their dismay, so it’s said, they discovered that they were wrong. √2 is an irrational number – it can’t be represented as a ratio. In modern notation, it’s an infinitely decimal that never repeats:

• √2 = 1·414213562373095048801688724209698…

A modern story, unattested in ancient records, says that the irrationality of √2 was a closely guarded secret in the Pythagorean cult. When Hippasus of Metapontum betrayed the secret, he was drowned at sea by enraged fellow cultists. Apocryphal or not, the story shows that squares aren’t so boring after all.

Nor are they boring when they’re caught in the fract. Divide one square into nine smaller copies of itself:

Discard three of the copies like this:

Stage 1

Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)

Then do the same to each of the sub-squares:

Stage 1

And repeat:

Stage 3

Stage 4

Stage 5

Stage 6

The result is a fractal of endlessly subdividing contingent hexagons:

Animated vesion

Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)

Here are a few more of the fractals you can create by squaring and paring:

Retain squares 1, 3, 5, 7, 9 (reading left-to-right, bottom-to-top)

Retain squares 2, 4, 5, 6, 8

Retain squares 1, 2, 4, 5, 6, 8, 9

Retain squares 1, 4, 6, 7, 10, 11, 13, 16

Retain squares 1, 3, 6, 7, 8, 9, 10, 11, 14, 16

Retain squares 2, 3, 5, 6, 8, 9, 11, 12, 14, 15

Retain squares 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25

Retain squares 1, 3, 7, 8, 11, 12, 14, 15, 18, 19, 23, 25

Retain squares 1, 5, 7, 8, 9, 12, 14, 17, 18, 19, 21, 25

Retain squares 2, 3, 4, 6, 7, 9, 10, 11, 15, 16, 17, 19, 20, 22, 23, 24

Retain squares 1, 2, 5, 6, 7, 9, 13, 17, 19, 20, 21, 24, 25

Previously pre-posted (please peruse):