In Pursuit of the Unknown summary

In geometry, a polyhedron is a three-dimensional solid consisting of a collection of faces joined at their edges – the cube, for example. Euler’s formula for polyhedra is a simple but remarkable statement:

If we count the numbers of faces (F), edges (E) and vertices (V), or the corners of the solid, it holds that F-E+V=2.

In the nineteenth century, mathematicians started seeing statistical patterns in chance events. They found that the probability of observing a particular data value is greatest near the mean value, i.e., the average, and dies away rapidly as the difference from the mean value increases. The formula for this phenomenon is called the normal distribution and, since its curve looks like a bell, it’s also called the bell curve.

In 1835, Adolphe Quetelet, a Belgian mathematician and sociologist, collected large quantities of data on crime, births, deaths, height, weight and so on. When he plotted the proportions of people with any of those given social variables, he started to see patterns, obtaining beautiful bell curves.

Earth is a water world. Its surface is covered in oceans, seas, rivers and lakes that flow. And water’s patterns attracted the attention of mathematicians who wanted to try to predict the flow.

Predicting the flow became possible in the first half of the nineteenth century, thanks to the Navier-Stokes equation. A world of possibilities opened due to the equation of flow.

Legend has it that Newton came up with his law of gravitation one day, while sitting in the grass and an apple fell from a tree and hit him on the head. Whether the story is true or not, the apple was important for Newton because it made him realize that the same law of forces could explain the motion of the apple as well as the moon.

Newton’s law enables accurate predictions of the motion of the solar system. His law of gravitation determines the force of gravitational attraction between two bodies in terms of their masses and the distance between them. And the most important aspect of Newton’s law of gravitation is the assertion that gravitation acts universally: any two bodies anywhere in the universe attract each other.

Einstein’s theory of relativity has given us the best understanding of the origins of the universe and the structure of the cosmos to date. It has also been at the root of radical new physics.

In Newton’s day, time was independent of space, and the mass and size of a body didn’t change when it moved. When Einstein introduced his theory of relativity, these statements turned out to be wrong. Moreover, classical mechanics had taught every physicist to ask: speed relative to what? But the speed of light turned out to be a constant, not relative to any physical quantity.

Schrödinger’s equation is a statement in quantum theory, a branch of physics that deals with phenomena at atomic- and subatomic-length scales.

The quantum world is notoriously weird. To give you an idea of its weirdness, it’s a world in which light is both a particle and a wave. It is so different from our comfortable human-scale world that even the simplest concepts change beyond recognition.

Chaos theory has grown from a minor mathematical curiosity into a basic feature of science. It shows that even simple theories can generate deterministic chaos – that is, random behavior with no random cause.

Picture a food processor. Its blades follow a very simple rule: they spin around really fast. And when the food interacts with the blade, the result is chaotic: the food doesn’t spin around with the blades, it’s mixed up.

Financial instruments known as derivatives are the greatest source of growth in the financial sector. Derivatives are not money; they’re investments in investments, promises about promises – and the standard of the world’s banking system.

The Black-Scholes equation made trading with derivatives reasonable and facilitated the financial sector’s growth.