1.
Activity 2-19: Could p be 3?

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 compass
it 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 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 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 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 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 have
Spherical Geometry, where pS is 3 some of the time.
Is it possible to have a geometry where p is constant at 3?
Euclidean Geometry
There is just one line through a point
parallel to a line
not through that point
pE = …
Triangle angle sum = 180°
Spherical Geometry
There are no lines through a point
parallel to a line
not through the point
2 < 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 start
with spherical geometry on the globe?

10. There are infinitely many lines through a point
Hyperbolic Geometry
There are infinitely many lines through a point
parallel to a line not through the point
pH = ?
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 between
the complex numbers z1 and z2 in the circle is
Task: 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 B
by moving along gridlines.
Cutting corners
is not allowed.
The distance from
(a, b) to (c, d) is
a  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 of
a  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, then
diameter = 2, circumference = 8,
So pM = 4.
A circle with the Chessboard metric looks like this:
If our radius is 1, then
diameter = 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 metric
that works on an isometric grid
rather than a square one.
This metric obeys
the laws for distance
that we stipulated
earlier.

17. What does a circle look like in this geometry?
If we have radius 1 here, then
diameter = 2, circumference = 6,
so pI = 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(!)’

18. With thanks to:
Bob Burn.
Carom is written by Jonny Griffiths,