All three pictures come from the same two-line rule. Pick a number c, start at zero, and keep applying z squared plus c. If the result stays bounded, the point is part of the Mandelbrot set. If it escapes, the colour records how many steps it took. The second and third panels are the same edge, magnified 267 and 26,667 times near the seahorse valley. New spirals and copies of the whole set keep appearing at every scale. The set was first plotted by computer in 1978 and 1980, and Benoit Mandelbrot's pictures of it brought fractals to the public.

The curves y = x² and y = 1.1ˣ cross exactly three times: at x = −0.955, at x = 1.051 and at x = 95.717. Between the second and third crossings the square leads, by a lot: 50² = 2500 while 1.1⁵⁰ ≈ 117. The ratio 1.1ˣ/x² bottoms out at x = 2/ln 1.1 ≈ 21 and grows without bound after it, so the third crossing is final. By x = 200, 1.1ˣ ≈ 4 700 · x²

The Sierpinski triangle drawn as a sum of rotating circles. With 1 circle the widest gap to the outline is 73.3% of the shape's height, with 7 it is 30.6%, with 63 it is 5.39% and with 255 it is 1.95%.

In 2013, Francis Su received a letter from Christopher Jackson, a man studying mathematics while serving a prison sentence. The letter began a long correspondence. Jackson wrote about equations, hope, dignity, and what learning could mean inside a cell. Su found himself reconsidering a question larger than any theorem: what is mathematics for? Their exchange helped shape Su's book Mathematics for Human Flourishing. Its central idea grew from those letters: mathematics can feed basic human desires for meaning, play, beauty, freedom, and connection.

Leonhard Euler, from Jakob Emanuel Handmann's 1753 portrait, reduced to two tones, traced into one closed outline and drawn as a sum of circles: z(t) = sum of c_k e^(ikt), the c_k from its discrete Fourier transform. With the N largest circles the widest gap to the outline is 53.1% of the portrait's height at N = 1, 3.87% at 128 and 0.85% at 512, and the circles shrink like k^-1.34.

1910 book “calculus made easy”

Galileo, rebuilt from 384 spinning circles. N = 1: error 53.1% N = 384: error 1.2% Circle sizes fall off like k^-1.24.

The prime numbers become rarer as the integers grow, yet they contain arithmetic progressions of every finite length. For example, 5, 11, 17, 23, 29 is a five-term progression of primes with common difference 6. In 2004, Ben Green and Terence Tao proved that for every positive integer k, there is a k-term arithmetic progression consisting entirely of primes. This was not a direct consequence of Szemerédi’s theorem. Szemerédi had proved that every subset of the integers with positive density contains arbitrarily long arithmetic progressions, but the primes have density zero. Green and Tao’s crucial move was to place the primes inside a larger, carefully constructed set of “almost primes” that behaves pseudorandomly. They then built a transference principle: a theorem designed for dense sets could be transferred into this sparse setting. The conclusion is infinite in scope. No matter how long a finite progression you request, somewhere inside the primes such a progression exists. The theorem guarantees existence. Finding the progressions is another problem entirely.

Gauss's portrait drawn as a sum of circles. One circle is off by up to 37.4% of its height, 128 circles by 3.55%, 512 circles by 0.735%.

Can you reverse the spin of Saturn?

Nikola Tesla, from Napoleon Sarony's c. 1890 portrait, reduced to two tones, traced into one closed outline and drawn by two chains of rotating circles. One chain sets x, the other sets y, and the pen sits where their dashed guides cross. One circle per axis is off by as much as 38% of the portrait's height, 10 circles by 19.9%, and 100 circles by 2.53%. The hair and the coat need the most circles to settle.

George Dantzig arrived late to a statistics class at Berkeley. Two problems were written on the board, so he copied them down and assumed they were the assignment. They seemed unusually difficult. After several days, he handed his solutions to professor Jerzy Neyman and apologized for taking so long. The problems were not homework. They were two famous unsolved problems in mathematical statistics. Dantzig had solved both because nobody had told him they were supposed to be impossible. Years later, he created the simplex method and became one of the founders of linear programming.

Newton's portrait drawn by two chains of rotating circles, one setting x and the other y. One circle per axis is off by up to 48.3% of its height, 10 circles by 17.1%, 100 circles by 3.89%.

An umbrella drawn by two chains of rotating circles, one setting x and the other y. One circle per axis is off by up to 34.4% of its height, 10 circles by 5.17%, 100 circles by 0.422%.

Four graphs that are also music. The audio is computed from the plotted functions. The height of each curve is the pitch, and the dot is the playhead at 144 BPM. Falling exponentials are kicks, a tanh trill doubling its speed each bar is the riser, floor and mod build a 16 step drum machine, and an 8 note motif drops a fourth every bar.

Six polar curves turned into sound: pitch follows the distance from the centre, 100 Hz at r = 0 up to 2100 Hz at the widest point. The rose warbles, the spiral climbs, the heart ends it.

A cube, traced edge by edge as one closed 3D line, drawn by three chains of rotating circles while it turns. Chain one sets x, chain two sets y, chain three sets depth, and the pen sits where the dashed guides meet. One circle per axis is off by up to 56.7% of the shape's diameter, 10 circles by 10%, and 100 circles by 0.958%.

A bat drawn by nothing but rotating circles. One circle per axis is off by as much as 32.7% of the shape's height, 10 circles by 9.89%, and 100 circles by 0.516%. Sharp corners and thin parts need the most circles to settle.

The Eiffel Tower, drawn as one closed 3D line and traced by three chains of rotating circles while it turns. Chain one sets x, chain two sets y, chain three sets depth, and the pen sits where the dashed guides meet. One circle per axis is off by up to 51.8% of the tower's height, 10 circles by 17.3%, and 100 circles by 1.55%.

A pyramid: a square pyramid, traced edge by edge as one closed 3D line, drawn by three chains of rotating circles while it turns. Chain one sets x, chain two sets y, chain three sets depth, and the pen sits where the dashed guides meet. One circle per axis is off by up to 48.3% of the shape's diameter, 10 circles by 5.86%, and 100 circles by 0.666%.