Deeply Nested Divine Pattern

Also not sure what this game is called, but it contains some interesting mathematical properties. Can you see the oblong numbers (2,6,12,20,30...) in this representation?

Per Mathgym:

Readers who are familiar with the theory of music will recognise the list of oblongs as the intervals in decreasing order of consonance: Octave (1:2), Perfect Fifth (2:3), Perfect Fourth (3:4), Major Third (4:5), Minor Third (5:6), etc. It is Pythagoras who is credited with discovering this mathematical relationship between music and numbers.

This discovery, that the pitch of a note is related to the length of the string which produced it, is credited as being the spark which ignited Pythagoras' imagination and philosophy. It allowed Pythagoras a glimpse of a whole new order in the Universe, one governed by intellect and logic and capable of the sublimest of pleasures. And a glimpse was all that he needed.

With this discovery, Pythagoras and the Pythagoreans set in train a way of investigation which has proved to be one of the most productive ideas in human history - that mathematics can be used to unravel the mysteries of the Universe.

Ulam spiral of prime/composite numbers, starting with 1 at white square, continuing 2,3,4 to the right and then going up and spiraling counterclockwise. Primes are solid. Composites are made up of their prime components.

Prime Numbers plotted on Ulam Spiral as wood grain beads. The spiral starts at 1 in the center of the image and incorrectly labels '1' as a prime (lighter colored bead). The composites are shown in a darker color.

The Twin Prime Conjecture

A teenager stands at the edge of a lake skipping stones. Notice that the stones follow a pattern on the water- they come in pairs separated by one line, but those pairs can be many lines apart. The stone skips far off in the distance, disappearing on the horizon.

Twin primes are prime numbers that come in pairs on the number line, separated only by one number. One example is the pair 41 and 43, another is the pair 59 and 61. No one knows if there are infinitely many of these or if there is a greatest twin-prime pair. The farther out you go, the rarer they become.. Does the rock stop skipping somewhere?

woodcut

One of my colleague PhD students wrote a whole lot of magic on the blackboard in our room the other day. He says he understands it...