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Activity 2-19: Could be 3? www.carom-maths.co.uk
What exactly do we mean by ? Given a circle, =
Now there is a famous verse in the Bible, that describes Solomon building the Temple in Jerusalem: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, […] and a line of thirty cubits did compass it round about.“ 2 Chronicles, chapter 4 verse 2. A ‘molten sea’ is a basin of water.
So this seems to suggest that here, circumference / diameter = 3. The most likely thing is that there has been some mis-measuring, but then... Some people say that every word in the Bible is literally true... The state of Indiana once almost passed a law to the effect that was rational.
In the spirit of play, let’s try to imagine a world where might be 3 exactly. After all, depends on our idea of distance, and there are notions of distance that differ from the usual one that we use. Our day-to-day model for geometry is the Euclidean one (as given to us by Euclid) and I’m going to call E = Euclidean = 3.1415... What other geometries could we use instead?
How about - Spherical Geometry? Here we move not on a plane, but on the surface of a sphere. Given two points A and B, we say AB is the shortest distance between A and B moving on the sphere, which means moving on a GREAT circle (a circle with the centre of the sphere as its centre).
Now let’s draw a circle with AB as diameter. Notice that a circle on a sphere looks exactly like a circle in Euclidean geometry. Question: if S is in spherical geometry, what values can S take? Circumference = 2 E R sin x. So S depends on x, and in particular, if x = E /6, then S = 3.
Is it possible to have a geometry where is constant at 3? So we have Euclidean Geometry, where E is constant, and we have Spherical Geometry, where S is 3 some of the time. Euclidean Geometry There is just one line through a point parallel to a line not through that point E = 3.14159… Triangle angle sum = 180° Spherical Geometry There are no lines through a point parallel to a line not through the point 2 < S < 3.14159… Triangle angle sum > 180°
Some questions: Why do we emphasise so much that is a constant at 3.14159...? Why start with plane geometry? Why not start with spherical geometry on the globe?
Hyperbolic Geometry There are infinitely many lines through a point parallel to a line not through the point H = ? Triangle angle sum < 180° Task: experiment with the Hyperbolic Geometry website – what values of H are possible? This site is based at the University of New Mexico, and is run by Joel Castellanos. Click here...
So it seems that in Hyperbolic Geometry, E < H < . What is the idea of distance used here? It’s fairly complicated: the distance between the complex numbers z 1 and z 2 in the circle is Task: what should we insist upon for some function to be a sensible measure of distance?
Classic definition of a distance function (or metric) 1. d(z 1, z 2 ) 0 (any distance is non-negative) 2. d(z 1, z 2 ) = 0 z 1 = z 2 (the distance from A to B is zero A = B) 3. d(z 1, z 2 ) = d(z 2, z 1 ) (the distance from A to B is the distance from B to A) 4. d(z 1, z 2 ) + d(z 2, z 3 ) d(z 1, z 3 ) (the distance from A to C via B the direct distance from A to C)
The Manhattan metric (here called d M ) Travel from A to B by moving along gridlines. Cutting corners is not allowed. The distance from (a, b) to (c, d) is a c + b d Also known as the Taxi-cab metric.
The Chessboard metric (here called d C ) The distance from one ‘rook’ to the other is the length of the single move required to bring them as close together as possible. The distance from (a, b) to (c, d) is the larger of a c and b d
Task: what are M and C ? A circle with the Manhattan metric looks like this: If our radius is 1, then diameter = 2, circumference = 8, So M = 4. A circle with the Chessboard metric looks like this: If our radius is 1, then diameter = 2, circumference = 8, So C = 4.
So if Solomon had given his measurements as forty around and ten across... we would have a choice! Now there is a version of the Manhattan metric that works on an isometric grid rather than a square one. This metric obeys the laws for distance that we stipulated earlier.
What does a circle look like in this geometry? If we have radius 1 here, then diameter = 2, circumference = 6, so I = 3. This will be true for all circles with this metric. Could Solomon have been using this idea of distance as he created his Temple? He was building a long time before Euclid… If artefacts from the original Temple emerge that are hexagonal in shape (to our eyes), then I shall say, ‘I told you so(!)’
With thanks to: Bob Burn. Carom is written by Jonny Griffiths, email@example.com