The beauty of the Riemann sphere


renering of the graph of the Sphere of Rieman
rendering of the graph of the Sphere of Riemann (Photo credit: Wikipedia)

Reading Elliptic Tales: Curves, Counting, and Number Theory today, I came across the Riemann Sphere – a thing of beauty: a closed surface of infinite extent.

I will explain the maths of the sphere in a moment, but I am left wondering if one of two things apply:

(a) We are terrible at teaching maths in this country because although it produces such beautiful concepts too many children – perhaps a majority – think it is a thing of drudgery; or

(b) The appeal of maths is inherently limited and the number of people fascinated by a concept as seemingly contradictory as a closed surface of infinite extent is always going to be limited.

Anyway, here’s an explanation of the sphere…

We want to think about infinity but we will, at first, restrict our thinking to a single, straight line. We can say how far along this line we are by measuring the distance from an arbitrary starting point. So, we could be 1 unit along, \pi units along or -\pi units along and so on.

In fact we can represent where we are on the line by a ratio \frac{a}{b} and represent each point in the form a:b . So \pi can be \pi:1 for instance, so we are not restricted to the rationals in this form.

Now , think of this as one way to get further and further from our arbitrary starting point – first we go 1 unit and so are at 1:1 then we go to 1:\frac{1}{2} then \frac{1}{3} and so on – and we approach the limit 1:0 which is the furthest we can possibly be and which we call infinitely far away from our starting point.

But what if we went in the other (negative) direction? Then we’d go to 1:-1 , 1:-\frac{1}{2} and so on … but at the limit we’d go to 1:0 – exactly the same place we’d go to if we started in the opposite direction: hence the “negative” and “positive” infinities are the same and this line is actually a loop.

To go from this loop to the sphere we need to consider complex numbers. Here every number is of the form a + b\Im, where \Im is the square root of -1.

Then we have to plot numbers on a plane, not a line and so every number has two co-ordinates, one of the form we discussed above in the form of the reals – eg 1:\frac{1}{7} and one along the imaginary axis of the form \Im:\frac{1}{7} – we thus we get an infinite set of infinite closed loops – a sphere.

The maths of the spirograph… with the drawings


Well, I sat down and thought this was going to be easy, but it has taken me three hours to work the maths of a smaller inner wheel rolling around inside a large outer wheel: mainly because for the first two of those I neglected the basic insight that the inner wheel rolls in the opposite direction to its direction of travel (think of it this way – as a car wheel moves forward the point at the top of the wheel moves backwards – relative to the centre of the wheel).

And instead of using MetaPost I resorted to a spread sheet – though I might do a MetaPost drawing still.

Anyway – assume you have a big wheel of unit radius and a small wheel inside it of radius \frac{1}{R} .

At any given time the centre of this small wheel will be at cartesian co-ordinates (assuming the big wheel is centred on (0,0):

(cos(\theta)(1 -\frac{1}{R}) , sin(\theta)(1 -\frac{1}{R})) (1)

where \theta is the angle of rotation of the small wheel relative to the centre of the big wheel.

But if the small wheel has moved through angle \theta relative to the centre of the big wheel, then it will have itself rotated through the angle R\theta – in the opposite direction to its rotation around the centre of the big wheel.

This means a fixed point on the surface of the small wheel will now be, compared to the centre of the smaller circle, at cartesian co-ordinates:

(\frac{cos(\theta - R\theta)}{R}, \frac{sin(\theta - R\theta)}{R}) (2)

And we add (1) and (2) together to get the co-ordinates relative to the origin (ie the centre of the bigger circle).

Looking at the above it should be relatively obvious that if R is an integer then the pattern will represent R cusps – and not much less obvious is the fact that if \frac{1}{R} can be expressed as a rational number then the pattern will repeat. But if \frac{1}{R} cannot be expressed as a rational then it turns out there are a countably infinite number i.e., \aleph_0 , number of cusps. In a way this is just a graphical way of representing an irrational number – it is a number that cannot be made to divide up unity (the circle) into equal proportions.

So here are the pretty pictures:

Let R = 2 and we have a degenerate case

2 cusp figure from 'spirograph'Then the 3 cusps of the ‘deltoid’:

Three cusped deltoidThe four cusped astroid:

Four cusped astroidAnd, here is the spreadsheet:
spirograph spreadsheet

Cosmologists’ problems with aleph-null and the multiverse


Cyclic progressions of the universe
Cyclic progressions of the universe (Photo credit: Wikipedia)

This is another insight from Brian Greene’s book The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos – well worth reading.

Aleph-null (\aleph_0) is the order (size) of the set of countably infinite objects. The counting numbers are the obvious example: one can start from one and keep on going. But any infinite set where one can number the members has the order of \aleph_0. (There are other infinities – eg that of the continuum, which have a different size of infinity.)

It is the nature of \aleph_0 that proportions of it are also infinite with the same order. So 1% of a set with the order \aleph_0 is also of order \aleph_0. To understand why, think of the counting numbers. If we took a set of 1%, then the first member would be 1, the second 101, the third 201 and so on. It would seem this set is \frac{1}{100}^{th} of the size of the counting numbers, but it is also the case that because the counting number set is infinite with order \aleph_0, the 1% set must also be infinite and have the same order. In other words, if paradoxically, the sets are in fact of the same order (size) – \aleph_0.

The problem for cosmologists comes when considering the whether we can use observations of our “universe” to point to the experimental provability of theories of an infinite number of universes – the multiverse.

The argument runs like this: we have a theory that predicts a multiverse. Such a theory also predicts that certain types of universe are more typical, perhaps much more typical than others. Applying the Copernican argument we would expect that we, bog-standard observers of the universe – nothing special in other words – are likely to be in one of those typical universes. If we were in a universe that was atypical it would weaken the case for the theory of the multiverse.

But what if there were an infinite number of universes in the multiverse? Then, no matter how atypical any particular universe was (as measured by the value of various physical constants) then there would be an infinite number of such a typical universes. It would hardly weaken the case of the multiverse theory if it turned out we were stuck inside one of these highly atypical universes: because there were an infinite number of them.

This “measure problem” is a big difficulty for cosmologists who, assuming we cannot build particle accelerators much bigger than the Large Hadron Collider, are stuck with only one other “experiment” to observe – the universe. If all results of that experiment are as likely as any other, it is quite difficult to draw conclusions.

Greene seems quite confident that the measure problem can be overcome. I am not qualified to pass judgement on that, though it is not going to stop me from saying it seems quite difficult to imagine how.

Goedel’s Incompeteness theorem surpassed?


Portrait of Kurt Gödel, one of the most signif...
Image via Wikipedia

Gödel’s Incompleteness Theorems are one of the cornerstone’s of modern mathematical thought but it is also a major blot on the mathematical landscape – as it establishes an inherent limit on the ability of mathematicians to describe the mathematical world: the first theorem (often thought of as the theorem) states that no consistent (ie self-contained) axiomatic system is capable to describing all the facts about natural numbers.

To today’s physical scientists – used to concepts such as relativity and quantum uncertainty – the broad idea that there could be an uncertainty at the heart of mathematics is maybe not so difficult to take, but it is fair to say it broke a lot of mathematical hearts in the 1930s when first promulgated. (This book – Godel’s Proof – offers an excellent introduction for the non-mathematician who is mathematically competent – ie like me!).

Gödel thought at the time that this kink in mathematical reality could be smoothed out by a better understanding of infinities in mathematics – and, according to the cover article in this week’s New Scientist (seemingly only available online to subscribers), by Richard Elwes, it is now being claimed by Hugh Woodin of UC Berkeley that just that has been shown.

Along the way, this new hypothesis of “Ultimate L” also demonstrates that Cantor’s continuum hypothesis is correct. I do not claim to understand “Ultimate L”, and in any case, as is their style, the New Scientist don’t print the proof, they just describe it in layman’s terms. I do have a basic understanding of the continuum hypothesis, though, and so can show the essential points that “Ultimate L” claims to have found.

Georg Cantor showed that there were multiple infinities, the first of which, so-called \aleph_0 (aleph null) is the infinity of the countable numbers – eg 1, 2, 3… and so on. Any infinite set that can be paired to a countable number in this way has a cardinality of \aleph_0 . (And, as the New Scientist point out, this is the smallest infinity – eg if you thought that, say, there must be half as many even numbers as there are natural numbers, you are wrong – the set of both is of cardinality \aleph_0 – 2 is element 1 of the set, 4, is element 2 and so on: ie a natural number can be assigned to every member of the set of even numbers and so the set is of cardinality \aleph_0 .)

The continuum hypothesis concerns what might be \aleph_1 – the next biggest infinity. Cantor’s hypothesis is that \aleph_1 is the real numbers (the continuum): I discuss why this is infinite and a different infinity from the natural numbers here.

We can show that this set has cardinality 2^{\aleph_0} – a number very much bigger than \aleph_0 . But is there another infinity in between?

Mathematicians have concentrated on looking at whether any projections (a word familiar to me now from relational algebra) of the set of reals has a cardinality between \aleph_0 and 2^{\aleph_0} – if they did then it would be clear the reals could not have the cardinality of \aleph_1 – but some other, higher, \aleph .

No projections with a different cardinality have been found, but that is not the same as a proof they do not exist. But if Woodin’s theory is correct then none exist.

(Just one more chance to plug the brilliant Annotated Turing: if you are interested in computer science you should really read it! This is the book that first got me interested in all this.)

The diagonal proof


This is a pre-1909 image of Georg Cantor (he w...
Image via Wikipedia

I am just reading Computability and Logic and it (or at least the old, 1980, edition I am reading) has a rather laboured explanation of Cantor’s 1891 proof of the non-enumerability of real numbers (the diagonal proof) so to reassure myself and because it is interesting I thought I’d set out a version here (closely modelled on that found in The Annotated Turing).

If you are already lost then – here’s what it’s about: it is a proof that the infinite number of real numbers (that is to say numbers with something after the decimal point eg 3.1) is a bigger infinity that the infinite number of integers – or if you want think in terms of fractions – the rational numbers. One – the infinite number of integers or rationals – is countable, the other is not.

This is a pretty mind-blowing proof and it is also fair to say that a minority of mathematicians have issues with Georg Cantor‘s theory of the continuum, but that is for another blog post, perhaps.

So here’s the example:

Let’s start setting out the real numbers that are less that one:

0.00000000000.....
0.10000000000......
0.20000000000......
0.30000000000......
0.40000000000......
0.50000000000......
0.60000000000......
0.70000000000......
0.80000000000.....
0.90000000000.....
0.01000000000.....
0.02000000000....
.....
0.11000000000.....

And so on. Now this looks like it might be enumerable (countable) by assigning an integer to element in the list- the first in the list is number 1, the second number 2 and so on. Hopefully you can see that this should, in theory, list all the numbers (except it doesn’t, as we will prove)

Bur let’s us take the number formed by the diagonal. In our case this is:

0.00000000000.....
0.10000000000......
0.20000000000......
0.30000000000......
0.40000000000......
0.50000000000......
0.60000000000......
0.70000000000......
0.80000000000.....
0.90000000000.....
0.01000000000.....
0.02000000000....

ie 0.0000000000.......

And let’s add 1 to every digit, so we get 0.111111111111....: is this number in the list? We will now show you that it cannot be by checking off the numbers in the list one by one.

First of all, we can eliminate the first number in the list as a match, because its first digit is 0 and our number has 1 as the first digit. Then we can eliminate the second number in the list in the same way (ie it’s second digit does not match). In fact it is clearly the case that we can eliminate number N from the list because it will not match at digit N.

OK, you say, let’s fix that by adding this ‘new’ number to the list:

0.11111111111......
0.00000000000......
0.10000000000......
0.20000000000......
0.30000000000......
0.40000000000......
0.50000000000......
0.60000000000......
0.70000000000......
0.80000000000......
0.90000000000......
0.01000000000......
0.02000000000......

But let’s diagonalise that one:

0.111111111111.....
0.00000000000......
0.10000000000......
0.20000000000......
0.30000000000......
0.40000000000......
0.50000000000......
0.60000000000......
0.70000000000......
0.80000000000......
0.90000000000......
0.01000000000......
0.02000000000......

And we get 0.1000000000...... Once again adding 1 to all the digits we get a new number 0.2111111111111.... – which we can show in the same way does not exist in the original list (set)