Introducing Infinity: A Graphic Guide (Introducing...)
Brian Clegg, Oliver Pugh
Format: PDF / Kindle (mobi) / ePub
Infinity is a profoundly counter-intuitive and brain-twisting subject that has inspired some great thinkers – and provoked and shocked others.
The ancient Greeks were so horrified by the implications of an endless number that they drowned the man who gave away the secret. And a German mathematician was driven mad by the repercussions of his discovery of transfinite numbers.
Brian Clegg and Oliver Pugh’s brilliant graphic tour of infinity features a cast of characters ranging from Archimedes and Pythagoras to al-Khwarizmi, Fibonacci, Galileo, Newton, Leibniz, Cantor, Venn, Gödel and Mandelbrot, and shows how infinity has challenged the finest minds of science and mathematics. Prepare to enter a world of paradox.
functions were “fractured”. Mandelbrot is probably best known for the Mandelbrot set, a particular fractal form that became a poster image of the 1980s. THE BOUNDARY OF THE SET THAT PRODUCES THIS STRIKING IMAGE GETS MORE AND MORE COMPLEX AS MORE DETAIL IS ADDED. Recursion A fractal is generated by repeatedly applying what is usually a (relatively) simple equation to build a structure that becomes more and more complex. It’s no coincidence that their popularity arose in the 1970s and 80s as
cut out a three-dimensional object. The result is a shape like a straight hunting horn, but the pointy bit heads off to infinity. BUT HOWEVER MUCH PAINT WE HAVE, WE CAN NEVER COVER THE HORN WITH IT, BECAUSE IT HAS AN INFINITE SURFACE AREA. SPOOKY. Now the volume of Gabriel’s horn can be calculated – it’s pi: 3.14159 and so forth. If you’re wondering how something can have a volume of π, remember the shape is for every x greater than 1. If that’s 1 metre, the volume is pi cubic metres – 1 mile
Newton, Isaac ref 1, ref 2, ref 3 Nicholas of Cusa ref 1, ref 2 non-standard analysis ref 1 “NOT” ref 1 notation ref 1 number line ref 1, ref 2 hyperreal ref 1, ref 2 number sequences ref 1 o ref 1, ref 2, ref 3, ref 4 odd numbers ref 1 Olympic Games ref 1 Omega (Ω) ref 1 omega (ω) ref 1 one (1), infinity of ref 1 opinion ref 1 “OR” ref 1 ordinal infinity ref 1 ordinal numbers ref 1 Orr, Adam C. ref 1 Paradoxes of the Infinite ref 1 particles ref 1 parts ref 1 Peano,
The most dramatic demonstration of Galileo’s ideas on infinity involved wheels. He imagined a pair of multi-sided wheels, one stuck to the face of the other, running on rails. Say they’re hexagons. We give them a turn until they move from one face on the rails to the next face. The bigger wheel will have moved forward by the length of one of its sides. The smaller wheel has to move this distance too, even though its sides are shorter, because the wheels are fixed together. It manages to do this
response is rueful. That’s just the way it is with infinity – a problem, Galileo reckons, of dealing with infinite quantities using finite minds. And he goes on to show how this is perfectly normal behaviour for the infinite. REMEMBER THAT WE ARE DEALING WITH INFINITIES AND INDIVISIBLES, BOTH OF WHICH TRANSCEND OUR FINITE UNDERSTANDING … IN SPITE OF THIS, MEN CANNOT REFRAIN FROM DISCUSSING THEM. Back to geometry One way Galileo demonstrates the odd mathematics of infinity is to use geometry,