Don't misunderstand me - I am no professional Mathematician! I just love the beauty of playing with numbers and shapes.
The first time I remember this happening to me was at primary school. We had a maverick head teacher who just wouldn’t be tolerated today – his name was James Denny and he was a rotund, peppery but jolly Scot who would, on a whim, call the whole school together and teach them something en masse! The other teachers would be expected to just stop what they were doing and join him in the assembly hall.
One of these lessons taught us to count up to 31 on one hand using binary (I had no idea what binary meant at the time, but looking back I see that is what it was) the thumb was one, the index finger two, the thumb and index finger was three, the middle finger alone was four, middle finger plus thumb five etc. We all heard a story about a strange woodland elf that Mr Denny had met and who taught him this secret way of counting! Try it – it’s fun!
From this I discovered the rainy day activity that is Pascal’s triangle – my brother and I used to create these for hours. You know – you start at the top with 1, beneath that you write two more 1s and then continue down where underneath every pair of numbers you write the sum of the number above – so row 3 would be “1, 2, 1” and row 4 would be “1,3,3,1” and so on for whatever size the paper is. Fascinating patterns appear even if your writing has to get very small!
But the best fun of all is Geometry. Say the word “Euclid” and you get blank expressions (or ones of abject horror!) most of the time – but he is like the father of Mathematics in so many ways. His theories mean that even today every schoolgirl and boy has a compass and ruler in their Geometry set – and with them can draw almost any shape, angle or plan. Greek mathematics has endured in a way Greek Philosophy and Literature has of course, but arguably in a more prominent role.
What about Pythagoras? The sheer stultifying brilliance of the formula that expresses the relationship between the sides of a right angled triangle is so rewarding – first when you first try it out but then, if you go that far, when you learn how to prove it! The square of the hypotenuse is equal to the sum of the squares of the other two sides. Less well known is if you draw semi circles on each side of the triangle, or pentagons, the same is true! In fact any proportional shape will have the same relationship.
Plato showed how there are only five “solids” that can be generated by regular shapes – the Tetrahedron from a triangle, the Cube from a square, the Octahedron from the triangular pyramid, the Icosahedron from the triangle and the Dodecahedron from the Pentagon. Great for using as die in role playing games!
From here I remember exploring tessellation – basically using a repeating shape to “tile” an area! Some shapes tessellate easily but actually any four sided shape, if inverted exactly, can create a shape which will tessellate! That is pretty amazing in itself. Some of the most beautiful mosaic patterns in the world employ this principle. Interestingly, Islam where graven images were banned still created artistic beauty with mathematics in the tiled floors of mosques from Istanbul to Baghdad – in fact the Islamic nations were well ahead of the west in Mathematics for centuries.
Jack once told me that there are two types of tessellation – essentially with one you can ‘pick up’ a section, move it around and match it exactly somewhere else, and another type where this is impossible. The first is known as periodic tessellation and for years mathematicians tried to discover an example of nonperiodic tessellation. The person who achieved it was known in another field – it was cosmologist Roger Penrose with his “Dart and Kite” pattern shown.
There is no section of this pattern that is ever exactly repeated yet it tiles perfectly! Wonderful… Some even suggest that Islamic mathematicians may have discovered this 1000 years earlier…had religion not poisoned their progress we might have had the Internet in the time of Shakespeare!
Ahh, the beauty of Maths! Hardly even a scratch on the surface…
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