A square wave jumps between two values and stays flat. Fourier showed it is built from sine waves alone: add the odd harmonics 3, 5, 7 and so on, each one that many times faster and that many times smaller. Every circle here is one harmonic and the last tip's height is the wave. With 1, 3, 9, 30 and 100 circles the tops flatten and the edges sharpen, but a 9 percent overshoot never goes away: the Gibbs phenomenon. Pitch follows the height.

Seven bridges, four land masses, and a walk that cannot be done !? Euler replaced the map with dots and lines and counted the lines at each dot. A walk crossing every bridge once can start and end at only two places, so at most two land masses may have an odd count. Königsberg has four, with 5, 3, 3 and 3. He presented this to the St Petersburg Academy on 26 August 1735, and it was published in 1741. Only five bridges remain at the original sites, so a walk is possible today, but only from one island to the other.

Move a point around a closed shape at constant speed and plot only its height against time. A circle gives a pure sine. The heart gives two bumps and one long dip, the astroid gives flat, sharp peaks, the five point star adds five ripples, and the lemniscate, a figure eight, crosses zero four times per turn. All five repeat every 1.4 seconds, so only the shape of the wave changes. A higher point is a higher note.

A tiny change in angular frequency can turn one polar rule into a completely different flower. Circles, roses and figure eights grow from nearly identical notation. For r = cos(nθ) or sin(nθ), odd integer n gives n petals and even n gives 2n. Negative radius values plot in the opposite direction, completing the symmetry. The curves make periodicity visible as geometry instead of a time trace. They are a compact lesson in how frequency and sign control repeated spatial structure.

Repeat one rule, x becomes rx (1 - x), and slowly turn up r. At first x settles on one value, then flips between two, then four, eight, and at about r = 3.57 it never repeats: chaos. Inside the chaos sits a window at r = 3.83 where it snaps back to a cycle of exactly three. Every iterate here is a note, higher x, higher pitch, so you can hear the cascade.

Some infinities are strictly larger than others: the whole numbers, the integers and the fractions can all be put in one list, but the real numbers cannot. Cantor's 1891 diagonal argument builds a number that differs from the n-th entry of any list in its n-th digit, so every list misses one. The same trick shows that a set always has more subsets than elements, so the ladder of sizes never ends. Nobody can say whether any size sits between the whole numbers and the reals. Gödel showed in 1940 that it cannot be disproved and Cohen showed in 1963 that it cannot be proved.

Put 360 points around a circle and join point i to point m times i, wrapping round at 360. With m = 2 the straight lines trace a cardioid, with m = 3 a nephroid, and every whole m gives a curve with m - 1 cusps: the envelope of the lines, an epicycloid. Nothing curved is drawn except the circle. You hear one note for every cusp.

Two identical sources send out waves. Where a crest from one meets a trough from the other, they cancel; where crest meets crest, they double. The cancelling lines sit wherever the two path lengths differ by half a wavelength, so moving the sources apart from 0 to 5 wavelengths opens 10 silent lines. The tone is what a listener at the black dot would hear: loud on the bright bands, silent on the lines.

Transforms answer different questions about the same signal. Fourier exposes global frequencies, Laplace adds growth and decay, wavelets localize features in both time and scale, and the Z-transform handles discrete sequences. None is universally best. The useful transform is the one whose coordinates turn the structure you care about into something simple.

300 identical double pendulums, released almost together: in total they differ by only 0.05 radians, about 3 degrees. At first they swing as one, then the differences grow exponentially and they scatter. The equations are exact, yet the motion cannot be predicted for long.

Martin Gardner printed the puzzle in his February 1960 Scientific American column, where it was known as the game of googol, and it has also been called the marriage problem and the sultan's dowry problem. Candidates arrive one at a time in random order, each must be accepted or rejected on the spot, and only the best counts as a win. Rejecting the first 37 of 100 and then taking the first candidate better than all those seen gives a 37.1% chance, against 1% for a blind pick. For large numbers the best cutoff approaches the total divided by e and the winning chance approaches 1/e, which is 36.8%

A function assigns one output to each allowed input, but its graph need not be smooth or connected. Linear functions change at a constant rate. Exponentials multiply by a fixed factor for each unit increase, while logarithms reverse that relationship. Floor and piecewise functions show why endpoints matter.

Fifteen pendulums, each a little shorter than the last, are released together. In 20 seconds the longest swings 11 times and the shortest 25, so they drift into waves and chaos, then line up flat at 10 s and again at exactly 20 s. A pendulum's period is 2π√(L/g), so lengths from 0.82 m down to 0.16 m do it. Seen from above. A note plays as each one crosses the middle: shorter is higher.

What happens when Galileo, Newton, Zu Chongzhi, Ramanujan, Leibniz, and Gauss are all challenged to calculate 55 × 55? Each mathematician takes a different path, drawing on their own mathematical ingenuity. Different centuries. Different techniques. The same result: 3025.

A moving source squeezes its waves together ahead of it and stretches them behind. On the line of motion you hear f0 / (1 − v/c) as it comes at you and f0 / (1 + v/c) as it leaves, so a 440 Hz source at 0.9 times the speed of sound is heard at 4400 Hz approaching and 232 Hz receding. At the speed of sound the wavefronts pile up, above it they form a Mach cone: 41.8° at 1.5c. Pitch is computed from true wave travel times; the boom is added.

Christian Goldbach wrote to Leonhard Euler on 7 June 1742 that every even number seems to be a sum of two primes. Euler answered on 30 June that he considered it certain but could not prove it, and the conjecture is still open. A distributed search by Tomás Oliveira e Silva had confirmed it for every even number up to 4 x 10^18 by 2013. Above, the number of splits grows with n and never reaches zero, forming a comet that fans into bands. numbers divisible by 3 (green) have about twice as many splits as the rest (orange). The best proven result is Chen Jingrun's 1973 theorem that large even numbers are a prime plus either a prime or a product of two primes, while the odd version with three primes was proven by Harald Helfgott.

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.