Math Pulse

Mathematics, rightly viewed, possesses not only truth, but supreme beauty.

[Math Pulse @themathpulse]

Nobody on this board wants a segregated neighbourhood. Each person only asks that at least 3 in 10 of their neighbours look like them. If not, they jump to an empty spot where that is true.

Then the colours pull apart into big blocks, and the average person ends up with about 3 in 4 neighbours like them, far more than anyone asked for.

Thomas Schelling showed this in 1971 with pennies and dimes on graph paper, and he shared the 2005 economics Nobel for work on how individual choices add up to patterns nobody chose. The lesson is that a pattern in the whole does not prove a wish in the parts.

In the second run each person asks for 5 in 10, and the blocks are even sharper.

[Math Pulse @themathpulse]

The integral sign is a stretched letter S, for the Latin word summa, a sum. Leibniz wrote it down in 1675.

Every kind of integral is the same idea in a different room, add up many tiny pieces. What changes is what a piece is.

Under a curve a piece is a thin strip of area. Along a curve it is a tiny step of a path, and you add up how much a field pushes along it. Over a surface it is a small patch, and you add up how much flows through. Through a solid it is a tiny block of volume.

These four are tied together. The divergence theorem says the flow out through a closed surface equals the total spreading out inside it, which is how physicists get Gauss's law in electrostatics.

[Math Pulse @themathpulse]

This is really crazy, btw.

What if you’re the absolute best, but your hiring manager is just a math geek?

[Math Pulse @themathpulse]

You must hire the best of 100 applicants. They arrive one at a time in random order, you must say yes or no on the spot, and you can never go back. What should you do?

Reject the first 37 without exception, then take the first applicant who is better than everyone you have seen so far.

This wins you the very best person 37 times out of 100, far above the 1 in 100 you get by choosing blindly.

[Math Pulse @themathpulse]

Pluck a long string and the bump does not stay where you made it. It splits into two smaller bumps of half the height that run away in opposite directions at the same speed. That is the whole content of the wave equation.

Jean le Rond d'Alembert solved it in 1747 while studying vibrating strings, and the answer is one line: any shape moving right plus any shape moving left. Euler and Daniel Bernoulli argued for years about which shapes were allowed, an argument that helped push mathematicians to define what a function really is.

The speed depends only on the tension and the mass of the string, which is why tightening a guitar string raises its pitch.

[Math Pulse @themathpulse]

Slope tells you where a hill is rising. The Hessian tells you what shape the ground has right where the slope is zero. Collect every second derivative into a small grid, and its signs say whether you are standing at the bottom of a bowl, on a summit, or on a mountain pass.

The matrix is named after Ludwig Otto Hesse, a German mathematician who introduced it in 1844 while studying curves. Today it sits inside every optimisation routine. When a neural network is trained, a flat valley direction and a steep direction are the reason some steps work well and others crawl.

A saddle is the interesting case. It looks like a minimum from one side and a maximum from the other, so simple downhill methods can stall there.

[Math Pulse @themathpulse]

You must hire the best of 100 applicants. They arrive one at a time in random order, you must say yes or no on the spot, and you can never go back. What should you do?

Reject the first 37 without exception, then take the first applicant who is better than everyone you have seen so far.

This wins you the very best person 37 times out of 100, far above the 1 in 100 you get by choosing blindly.

[Math Pulse @themathpulse]

A quadcopter has no tail, no flaps and no rudder. It steers by changing only the speeds of four propellers, and everything comes from simple sums of force and torque.

For hover the four thrusts add up to the weight. To tip forward or sideways, two motors speed up while the two opposite ones slow down by the same amount, so the total lift stays the same but a twisting force appears. To turn, the clockwise pair and the counter-clockwise pair are changed in opposite ways, because each spinning propeller pushes back on the frame in the opposite direction.

A small computer repeats this arithmetic hundreds of times a second to keep the drone steady.

[Math Pulse @themathpulse]

Archimedes found the volume of a sphere around 250 BC by comparing it with a cylinder and a cone, and he was so proud of the result that he asked for it to be carved on his tomb.

Two thousand years later calculus gives the same answer in a few lines.

The trick is to use the natural coordinates of a ball: how far out, how far down from the pole, and how far around. A tiny block in these coordinates is not a cube. It gets wider as you move away from the centre and near the equator, and the factor that measures this is what makes the integral come out exactly.

[Math Pulse @themathpulse]

The correct answer: Red.

After 1 year the total is only $3,678, which is where most people quit. After 4.4 years (day 1,620) it passes $1 billion, and that single day pays $9.9 million. After 5 years the total is about $7.8 billion, and after 10 years about 6 × 10^17

[Math Pulse @themathpulse]

Which one do you choose?

The correct answer in the reply:

[Math Pulse @themathpulse]

Pick up a book and turn it a quarter turn about a vertical line, then a quarter turn about a line pointing at you. Put it back and do the same two turns in the opposite order. The book ends up in a different position.

That is what mathematicians mean when they say rotations in three dimensions do not commute. Flat rotations on a table never show it, because there is only one axis to turn about. The moment there are three, the order carries information.

This is why aircraft, robots and phone sensors store orientation carefully, and why quaternions, invented by Hamilton in 1843, became useful for the job.

[Math Pulse @themathpulse]

Every furniture mover knows this problem.

The ladder that wins is not the one that fits the corridor, it is the one that fits the corner. As you swing it round, one end slides along the outer wall and the ladder always touches the inner corner, and that moment decides everything.

Calculus finds it by asking which tilt of the ladder gives the shortest line from wall to wall through the corner. Anything longer will jam at that angle, however you twist it.

The same question appears in robot path planning, in moving sofas, and in the famous unsolved moving sofa problem.

[Math Pulse @themathpulse]

Bubble sort swaps neighbours that are in the wrong order, insertion sort slides each bar back to its place, merge sort splits the list in halves, sorts them and zips them together, and quick sort picks a pivot and throws smaller bars to one side.

Each is charged the same price, one array write per hundredth of a second, so the race is fair.

[Math Pulse @themathpulse]

THIS IS THE LONGEST EQUATION IN PHYSICS, AND IT EXPLAINS ALMOST EVERYTHING.

Every collision ever recorded at a particle accelerator has been checked against this one expression.

The Standard Model Lagrangian packs every known elementary particle and three of the four forces into a few lines. Written out in full, with every index and coupling expanded, it fills a page or more, which is why it is called the longest equation in physics. Glashow, Weinberg and Salam shared the 1979 Nobel Prize for its electroweak core, and 't Hooft and Veltman won in 1999 for showing that it gives finite answers. The Higgs boson it required was found in 2012.

It still leaves out gravity, dark matter and the neutrino masses.

[Math Pulse @themathpulse]

Around 240 BC, Eratosthenes heard that at noon on midsummer day the sun shone straight down a well in Syene and left no shadow. In Alexandria, the same day, a stick still threw a short one.

That difference could only mean the ground curves. Sunlight arrives parallel, so the shadow angle in one city equals the angle between the two cities as seen from the Earth's centre. He needed only a stick and a walking measure of the distance.

His answer landed within a few percent of the truth, though nobody knows exactly how long his unit was.

[Math Pulse @themathpulse]

Drop a grain of sand on the middle of a grid. Whenever a square holds 4 grains it gives one grain to each of its 4 neighbours, which can push those over 4 as well, and a chain of collapses runs outward.

That is the whole rule. Nothing in it mentions a shape, yet 36,000 grains drawn one at a time build this.

Per Bak, Chao Tang and Kurt Wiesenfeld introduced the model in 1987 to explain self organised criticality: a pile that tunes itself to the edge, where collapses of every size happen, from one grain to the whole board.

The same statistics show up in earthquakes and forest fires. A second surprise came from Deepak Dhar in 1990: the final pattern does not depend on the order in which squares collapse, so the model has a clean algebra.

[Math Pulse @themathpulse]

In 1643 Descartes described a surprising rule for four circles that all touch each other, in a letter to Princess Elisabeth of Bohemia. The rule connects how curved each circle is.

Three centuries later the chemist Frederick Soddy rediscovered it and published it as a poem called The Kiss Precise. Fill the gaps again and again and you get the Apollonian gasket, a fractal in which an astonishing number of circles have whole number curvature.

Mathematicians still study these packings today, and they connect to number theory in ways nobody expected. Not bad for a rule about touching circles.

[Math Pulse @themathpulse]

In 1899 the mathematician Frank Morley noticed something strange about angle trisectors. Cut each corner of any triangle into three equal angles, and the cuts nearest each side meet at three points that form a perfect equilateral triangle.

For two thousand years geometers had studied triangles without seeing it. The result is so surprising that it was nicknamed the Morley miracle, and later proofs came from trigonometry, algebra and pure geometry. The picture is easy to draw yet hard to explain.

It holds for every triangle, whether fat, thin or lopsided. Try it with a ruler and protractor on a scrap of paper.

[Math Pulse @themathpulse]

In 1822 Joseph Fourier published a theory of how heat moves through a solid, and in doing so invented tools that now run through signal processing and quantum physics. The equation itself is short and strangely general.

It says that a point warms up if its neighbours are hotter on average and cools down if they are cooler. Sharp features smooth out first, which is why a hot spot fades quickly but a gentle warmth lingers. The same law governs the spread of ink in water and the blur in a photo editor.

Over time every rod ends up flat at its average. Heat has no memory of where it started.

[Math Pulse @themathpulse]

In 1952 a graduate student at MIT, David Huffman, was offered a choice: sit an exam or solve an open problem about coding. He found the shortest possible prefix code and never had to take the exam.

The idea is wonderfully simple. Give common symbols short codes and rare ones long codes, and build the codes by repeatedly joining the two rarest items. Because no code starts another, the bits can be read back without any separators.

Variants of it sit inside files you use every day, including JPEG images and ZIP archives. Even a single repetitive word shrinks.

[Math Pulse @themathpulse]

If more pigeons fly into a coop than there are holes, some hole must hold two. It sounds too obvious to be maths, yet it proves things that feel impossible to guess.

The principle is often credited to Dirichlet in the 1830s. It shows up in computer science to prove that no compression method can shrink every file, and in number theory to guarantee patterns in sets of integers. The trick is always in choosing what counts as a box.

Here the boxes are built from numbers with their factors of 2 stripped away. Once you see that, the surprise is no longer a surprise.