2 What exactly do we mean by p? Given a circle,p =
3 2 Chronicles, chapter 4 verse 2. 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 compassit round about.“2 Chronicles, chapter 4 verse 2.A ‘molten sea’ is a basin of water.
4 So this seems to suggest that here, circumference / diameter = 3. The most likely thingis that there has been some mis-measuring,but then...Some people saythat every word in the Bibleis literally true...The state of Indiana once almost passed a lawto the effect that p was rational.
5 After all, p depends on our idea of distance, In the spirit of play, let’s try to imagine a world where p might be 3 exactly.After all, p depends on our idea of distance,and there are notions of distancethat 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 pE = Euclidean p =What other geometries could we use instead?
6 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).
7 Circumference = 2 pER sin x. 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 pS is p in spherical geometry,what values can pS take?Circumference = 2 pER sin x.So pS depends on x, and in particular,if x = pE/6, then pS = 3.
8 Euclidean Geometry Spherical Geometry So we have Euclidean Geometry,where pE is constant, and we haveSpherical Geometry, where pS is 3 some of the time.Is it possible to have a geometry where p is constant at 3?Euclidean GeometryThere is just one line through a pointparallel to a linenot through that pointpE = …Triangle angle sum = 180°Spherical GeometryThere are no lines through a pointparallel to a linenot through the point2 < pS < …Triangle angle sum > 180°
9 Some questions: Why do we emphasise so much that is a constant at ?Why start with plane geometry?Why not startwith spherical geometry on the globe?
10 There are infinitely many lines through a point Hyperbolic GeometryThere are infinitely many lines through a pointparallel to a line not through the pointpH = ?Triangle angle sum < 180°Task: experiment with the Hyperbolic Geometry website –what values of pH are possible?This site is based at the University of New Mexico,and is run by Joel Castellanos.Click here...
11 So it seems that in Hyperbolic Geometry, pE < pH < . What is the idea of distance used here? It’s fairly complicated:the distance betweenthe complex numbers z1 and z2 in the circle isTask: what should we insist upon for some function to be a sensible measure of distance?
12 Classic definition of a distance function (or metric) 1. d(z1, z2) 0(any distance is non-negative)2. d(z1, z2) = 0 z1 = z2(the distance from A to B is zero A = B)3. d(z1, z2) = d(z2, z1)(the distance from A to B is the distance from B to A)4. d(z1, z2) + d(z2, z3) d(z1, z3)(the distance from A to C via B the direct distance from A to C)
13 The Manhattan metric (here called dM) Also known as the Taxi-cab metric.Travel from A to Bby moving along gridlines.Cutting cornersis not allowed.The distance from(a, b) to (c, d) isa c +b d
14 The Chessboard metric (here called dC) 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 ofa c and b d
15 A circle with the Manhattan metric looks like this: Task: what are pM and pC?A circle with the Manhattan metric looks like this:If our radius is 1, thendiameter = 2, circumference = 8,So pM = 4.A circle with the Chessboard metric looks like this:If our radius is 1, thendiameter = 2, circumference = 8,So pC = 4.
16 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 metricthat works on an isometric gridrather than a square one.This metric obeysthe laws for distancethat we stipulatedearlier.
17 What does a circle look like in this geometry? If we have radius 1 here, thendiameter = 2, circumference = 6,so pI = 3.This will be true for all circles with this metric.Could Solomon have been usingthis idea of distance as he created his Temple?He was building a long time before Euclid…If artefacts from the original Temple emergethat are hexagonal in shape (to our eyes),then I shall say, ‘I told you so(!)’
18 With thanks to:Bob Burn.Carom is written by Jonny Griffiths,