But there was one number missing, and it was the Indians who would introduce it to the world ... zero. ibid.
The power of trigonometry is that it acts like a dictionary translating geometry into numbers and back. ibid.
The Muslim scholars collected and translated many ancient texts, effectively saving them for posterity. ibid.
Al Khwarizmi was to create a whole new mathematical language. It was called Algebra. ibid.
Algebra is the grammar that underlines the way that numbers work. ibid.
Wherever you find growth in Nature you find the Fibonacci numbers. ibid.
The problem of perspective is how to represent the three-dimensional world in a two-dimensional canvas. To get a sense of depth, a sense of a third dimension, Piero [della Francesca] used mathematics. Marcus du Sautoy, The Story of Maths III: The Frontiers of Space
How can you actually know anything at all? Then he [Descartes] slips into a dream. And in the dream he understood that the key was to build philosophy on the indisputable facts of mathematics. Numbers he realised could brush away the cobwebs of uncertainty. ibid.
Suddenly geometry has turned into algebra. ibid.
[Pierre] Fermat managed to find several new patterns in numbers that defeated mathematicians for centuries. ibid.
In two miraculous years here [Grantham] he [Newton] developed a new theory of light, discovered gravitation, and scribbled out a revolutionary approach to maths: the calculus. ibid.
[Gottfried] Leibniz also discovered the miracle of calculus shortly after Newton ... Contributions to philosophy and logic which are still highly rated today. But Leibniz was not just a man of words. He was also one of the first people to invent practical calculating machines that worked on the binary system: true forerunners of the computer. ibid.
There is one great dynasty of mathematicians: the Bernoullis. ibid.
I think of him [Leonhard Euler] as the Mozart of Maths ... It concerns calculating infinite sums, the discovery that shot Euler to the top of the mathematical pops when it was announced in 1735. ibid.
In Germany they got, at least in my opinion, the greatest mathematician ever ... The Prince of Mathematics: Carl Friedrich Gauss. ibid.
Imaginary numbers have helped us to understand radio waves, to build bridges and aeroplanes; they’re even the key to quantum physics, the science of the sub-atomic world. They provided a map to see how things really are. ibid.
Riemann’s mathematics changed how we see the world: suddenly higher dimensional geometry appeared. Multi-dimensional space is at the heart of so much mathematics done today. ibid.
Hilbert, a young German mathematician, boldly set out what he believed are the twenty-three most important problems for mathematicians to crack. Professor Marcus du Sautoy, The Story of Maths IV: To Infinity & Beyond
Before Cantor no-one had really understood infinity ... There are different infinities, some bigger than others. ibid.
[Kurt] Godel lived in Vienna in the 1920s and 1930s ... The Incompleteness Theorum. Godel proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove. ibid.
With the help of colleagues, Julia [Robinson] developed what became known as the Robinson hypothesis. ibid.
The Americans were right: Nicolas Bourbaki does not exist at all and never has; Boubaki is in fact the nom de plume for a group of French mathematicians led by Andre Weil. ibid.
Mathematics now pervades every aspect of our lives. ibid.
Until we can prove it [Reimann Hypothesis] there will still be doubt. ibid.
Hidden in this cathedral are clues to a mystery. Something that could help answer one of humanity’s most enduring questions. Why is the world the way it is? Marcus du Sautoy, The Code I: Numbers BBC 2011
The precise proportions of this magnificent cathedral: to the medieval clergy these divine numbers were created by God. But to me they are evidence of something else: a hidden code that underpins the world around us, a code that has the power to unlock the doors that govern the universe. ibid.
The code: an abstract world of numbers that has given us the most detailed description of our world we’ve ever had. For centuries, people had seen significant numbers everywhere. An obsession that has left its mark on the stones of this medieval cathedral. ibid.
For Saint Augustine, numbers had to come from God. ibid.
I don’t share their religious beliefs; I can’t help feeling something in common with the people who built this place. I share their awe and wonder at the beauty of numbers. For them those numbers brought them closer to God. But I think they’re important for another reason: because I believe they are the key to making sense of our world. ibid.
Numbers have given us an unparalleled ability to understand our universe. And in places this code emerges from the ground. ibid.
Unlike other numbers primes can only be divided by themselves and one. ibid.
Pi is the essence of circle-ness distilled into a language, a code. ibid.
Just as the circle appears everywhere in nature, so Pi crops up again and again in the mathematical world. ibid.
Some numbers are so strange they don’t seem to make sense as numbers. ibid.
We’ve given this number a name and it’s called i – part of a whole class of new numbers called imaginary numbers. ibid.
1.08 ... The Nautilus is growing at a constant rate. So every time the Nautilus builds a new room the dimensions of that room are 1.08 times the dimension of the previous one. ibid.
Behind the world we inhabit there’s a strange and wonderful mathematical realm. ibid.
It’s only through the code that we can understand the laws that govern the universe. ibid.
Gravity ... It will accelerate to land at a constant rate of 0.98 meters per second per second. This is a fundamental law on our planet. ibid.
This is the Giant’s Causeway at the northern tip of northern Ireland. And it’s famed for these strange angular rocks. There are forty thousand of them ... These stones tell of a hidden geometric force that underpins and pervades all Nature. And if we can uncover that force it will tell us the shape of everything. Professor Marcus du Sautoy, The Code II: Shapes
The bees’ honeycomb is a marvel of natural engineering. The bees have made an identical pattern to the columns on the Giant’s Causeway. Each cell is exactly like the others ... It’s as if the hexagon is built into the bees’ DNA. ibid.
It’s the hexagons that use the least amount of wax. ibid.
This is Nature’s code at work and the bees are in tune with it. ibid.
The soap bubble reveals something fundamental about Nature – it’s lazy; it tries to find the most efficient shape, the one using the least energy, the least amount of space. ibid.
Frei Otto started something of a revolution in architecture. The sweeping curves of the Olympic Stadium are echoed in countless modern structures. ibid.
The mainstay of Greek geometry was the discovery of five perfect shapes now called the Platonic solids ... The tetrahedron with its four faces, the cube with its six faces, the octahedron with its eight faces, the dodecahedron twelve faces, and the most complicated shape of all the icosahedron with its twenty faces. Today these are more commonly known as dice ... These are the only five shapes like this that can possibly exist; they are the only perfectly symmetrical solids. ibid.
The world clearly isn’t just built from simple geometric shapes. ibid.
We now know that the movement of the planets is incredibly predictable. By understanding the code we can model their orbits far back into the past and thousands of years into the future. Marcus du Sautoy, The Code III: Prediction
We are so addicted to patterns that we let them seep into almost everything we do. ibid.
Very few of us are aware of the patterns we leave behind. ibid.
The future rarely turns out exactly as we planned. ibid.
Our lives are controlled by the strangest code of all – the code of chaos. ibid.
The average guess to the nearest bean is 4,515 [4,510] ... Collectively we get something which is just 0.1% away from the real number of beans in there. So as individuals the guesses are just that – guesses, but when you take them collectively they become something else entirely. ibid.
The accuracy of the group is far greater than the individual – we call this the wisdom of the crowd. ibid.