What I like doing is taking something that other people thought was
complicated and difficult to understand, and finding a simple idea, so that
any fool – and, in this case, you – can understand the complicated thing.
These simple ideas can be astonishingly powerful, and they are also
astonishingly difficult to find. Many times it has taken a century or more
for someone to have the simple idea; in fact it has often taken two thousand
years, because often the Greeks could have had that idea, and they didn’t.
People often have the misconception that what someone like Einstein
did is complicated. No, the truly earthshattering ideas are simple ones.
But these ideas often have a subtlety of some sort, which stops people from
thinking of them. The simple idea involves a question nobody had thought
of asking.
Consider for example the question of whether the earth is a sphere or
a plane. Did the ancients sit down and think “now lets see – which is it, a
sphere or a plane?” No, I think the true situation was that no-one could
conceive the idea that the earth was spherical – until someone, noticing
that the stars seemed to go down in the West and then twelve hours later
come up in the East, had the idea that everything might be going round –
which is difficult to reconcile with the accepted idea of a flat earth.
Another funny idea is the idea of ‘up’. Is ‘up’ an absolute concept?
It was, in Aristotelian physics. Only in Newtonian physics was it realised
that ‘up’ is a local concept – that one person’s ‘up’ can be another person’s
‘down’ (if the first is in Cambridge and the second is in Australia, say).
Einstein’s discovery of relativity depended on a similar realization about
the nature of time: that one person’s time can be another person’s sideways.
Well, let’s get back to basics. I’d like to take you through some simple
ideas relating to squares, to triangles, and to knots.
Squares
Let’s start with a new proof of an old theorem. The question is “is the
diagonal of a square commensurable with the side?” Or to put it in modern
terminology, “is the square root of 2 a ratio of two whole numbers?”
This question led to a great discovery, credited to the Pythagoreans, the
discovery of irrational numbers.
Let’s put the question another way. Could there be two squares with
side equal to a whole number, n, whose total area is identical to that of a
single square with side equal to another whole number, m?
m
m
n
n
n
n
Figure 1. If m and n are whole numbers, can the two grey n × n squares have the
same area as the white m × m square?
This damn nearly happens for 12 by 12 squares: 12 times 12 is 144;
and 144 plus 144 equals 288, which does not actually equal 289, which is
17 times 17. So 17/12 = 1.41666 . . . is very close to √
2 = 1.41421 . . . – it’s
only out by two parts in a thousand.
But we’re not asking whether you can find whole numbers m and n that
roughly satisfy m2 = 2n
2
.
Origin paper: http://thewe.net/math/conway.pdf
Thanks
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