Have we ever lost mathematics?

May 9, 2012

If you study the history of modern mathematics one of the recurring themes is the collapse of the foundations. A realisation that the assumptions underlying a topic were not as strong as might be hoped. There are three classics, which (with a broad brush) might be described as:

  • The collapse of Analysis
    The problems with infinity and infinitesimals had been known to the Greeks who discussed their paradoxes. In the seventeenth century, however, arguments and proofs involving infinitesimals became more acceptable. A powerful system started to emerge developed most clearly by Newton and Liebniz: the Calculus. Some still had issues, Berkeley famously mocked the “ghosts of departed quantities”, but most persisted and were greatly rewarded. The problems did eventually arise in mathematical settings, for example in Fourier’s study of heat, and it required most of the nineteenth century for Cauchy and Weierstrass to give a rigorous version based on limits.
  • The collapse of Geometry
    With issues arising in the basic assumptions of Analysis, mathematicians looked for firmer ground. Many felt that Geometry and, in particular the great work of Euclid’s Elements might provide this. Unfortunately there was a wrinkle in the Elements, often known as the Parallel Postulate, a statement that seemed far too complicated to be an assumption. Many had tried to show that the parallel postulate could be proved from the other axioms, and failed. Some had actually glimpsed something further, notably Khayyam and Saccheri. In fact, as Bolyai and Lobachevsky would eventually show, there are perfectly good geometries that obey all the axioms apart from the Parallel postulate. With the zoo of examples that started to be considered, could geometry did not look like such a good foundation.
  • The collapse of Arithmetic
    Instead of geometry, therefore other systems were considered, ideas from logic, and Cantor’s set theory were brought into play. Hilbert hoped that a system could be created that was consistent, complete and decidable. Many took on the challenge, Russell and Whitehead created the magnificent Principia Mathematica that famously does not prove that 1+1=2 until page 362. As the ideas started to become clearer, however, the way was left clear for a deeper issue. The results of Kurt Gödel showed that, if we want to include arithmetic, we cannot hope for a system for mathematics that is both complete and consistent. Furthermore,  we cannot even prove the consistency of our systems without resorting to a more powerful one (who’s own consistency cannot be proved without a more powerful system still). Some even wonder whether arithmetic is consistent!

As these collapses hit the ideas that fields rest on, one would expect there to be some consequences. Some areas that turn out to be fallacies. Yet this does not seem to be the case.  The fundamental ideas of calculus remained the same, although one had to be careful about the exact functions you were talking about. The discovery of non-Euclidean geometries simply revealed additional worlds, all the old results held but some now needed to note an additional assumption. Even the work on undecidability leads, most obviously through Alan Turing, to the theoretical underpinnings of computers. In fact studying the deeper issues seems to open up new areas but not harm those that have been established.

I therefore have a question:

Have we ever lost any mathematics?

Are there mathematical areas that have simple collapsed, having been accepted widely as true, even rigorous? I would like to rule out the case where an area has been rendered unimportant by the development of different techniques. In that case the results still hold, but are no longer as interesting.

Edit: 10/5/12

Some great discussion on math overflow, including one serious candidate, Italian algebraic geometry.

6 Comments | Mathematics | Tagged: , , , , , , , , , , , , , , , , , | Permalink
Posted by gelada

The 2×1 rectangle and Domes

March 26, 2012

Next week I am going to be at the Gathering for Gardner, an exciting meeting of mathematicians, magicians, puzzlers and others inspired by the life and work of Martin Gardner. This post is a version of the talk I will be giving.

The 2×1 rectangle is not one of mathematics most celebrated shapes.

Yet it is so much more flexible than the more common square.

Even better you can cut it in half on the diagonal to make a 2×1 right triangle,

which has the beautiful property that it is a 5-reptile. Five copies of it come together to make a larger version. Repeating this gives the Conway Pinwheel tiling, which has triangles occurring in an infinite number of directions.

Yet the 2×1 rectangle is a lot more common in life, just go into your local hardware store:

Using the diagonal cut triangle and uncut rectangles, Vinay Gupta designed the hexayurt,

a small house that can be built from 12 sheets, without waste. In contrast to geodesic domes, that cannot be made from sheet materials without making many cuts or wasting material. Here is one:

and a plywood one:

Hexayurts have become one of the standard accommodations at Burning Man:

or look at this map, the red dots show the location of the hundreds of hexayurts at last years event.

Vinay set me the challenge of making larger domes using these shapes. The hexayurt itself suggests that hexagons will be important, and we can put two 2×1 rectangles together to make a square. Squares and hexagons come together to form the truncated octahedron.

This obviously would not work as a dome, so we must cut it. There are two natural cuts that can be made. One perpendicular to the 4-fold axis, and one perpendicular to the 3-fold:

So we have two new larger domes, the tri-dome and the quad-dome:

What is really cool is that both of these domes were made for Burning Man last year:

Tri-dome:

Quad-dome:

One neat thing about the truncated Octahedron is that it is a space-filler. You can use them to tile 3d space. We can therefore bring quad-domes together to make even larger structures, like this one:

4 Comments | Main Article, Mathematics | Tagged: , , , , , , , , , , , , | Permalink
Posted by gelada

Prime Phyllotaxis Spirals

March 18, 2012

The phyllotaxis spiral is one of the classical forms of mathematics, and there is a wonderland of resources available online both images and explanations. The basic idea is to put points round in a spiral with the same angle between each point. This gives a family of forms:

Note that, as the angle changes the dots sometimes pack in better than others, this can actually be studied and the best packing is related to the golden ratio. The points in this spiral are placed down in order, so we can associate each to a whole number:

Now when I see a lit of numbers like this, I want to pull out the primes, see what pattern they make:

There are some hints at patterns, lets expand out, and look just at dots:

There seem to be spiral arms which are richer in prime numbers than others. We can analyse things further by colouring each number depending on its prime factors. The more prime factors the lighter the number, giving the image for the start of this post:

Now there is a clear pattern, light and dark arms spiraling out. Can we understand this pattern?

Think about the construction of a phyllotaxis pattern we turn the same angle every time, that means within a particular pattern we can find other phyllotaxis patterns. The one at twice the speed, three times the speed and so on.  For example we could dive our pattern into two patterns each with twice the rotation angle. This gives:

All prime number (other than 2) are odd, so they must lie on the subspiral corresponding to the odd numbers. In addition it turns out that the spiral arms that we see are related to the Fibonacci numbers (themselves closely linked to the Golden Ratio). The particular curves we see relate to 144. Here is the spiral given by multiples of 144, pulling out just one such curve:

Note that in the prime factor picture this curve gives a very light line as every number in it is a multiple of 144, and 144 itself has 6 prime factors (three twice and two four times). Taking the multiples of 6 instead of 144 (which gives us several of these curves as 6 divides 144) we see another pattern of lines that are light in the image:

More importantly the curves next to these ones give numbers one more or less than a multiple of six. Every prime number has this form (all other numbers are multiples of 2 or 3 or both). This gives the curves of prime numbers we saw.

So by considering the construction of the initial image it begins to reveal its secrets. Yet, just as with the primes on their own there seems to be plenty of mystery left for investigation…

6 Comments | Main Article, Mathematics | Tagged: , , , , , , , | Permalink
Posted by gelada

2+2 = 1? Patterns in Modular arithmetic

November 20, 2011

When someone is talking about the absolute truth of mathematics and declares that once you have defined 2 and +, then 2+2 must equal 4, there is a slightly glib response:

but 2+2 = 1…Mod 3

Despite this surprise, we actually all use modular arithmetic regularly, quite literally on a daily basis. When we consider six hours after 8am, the answer is not 14, but 2pm. Well you could argue for using a 24 hour clock, but no one would claim that 3am on a Tuesday morning is really 27:00 on Monday (well apparently some do, thanks to kuromagi on reddit for ref) In these cases we are not counting as we usually do, but counting on a circle mod 12 or 24. It is not hard to see that we could do this with other numbers. if we do decide that 2+1 is 0, and not 3 we are now working mod 3. In this case 2+2 is 1, as is 2*2. We can put together a small table:

+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

Showing what happens when the values for the column and row are added together. We can make the same table for multiplication:

x 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1

I have to admit these table are a little boring, we can make things more interesting by replacing the numbers by colours. As we are working with modular arithmetic we know that the range of numbers we will come across, lies between 0 and the value we are using for modulus, so we can map these onto some circle of colours. So work mod 151 we get a new table for addition:

Using the same system of colours we can do the same thing for multiplication:

Which is starting to get interesting. We do not need to stop there, we can produce an image where the row number is taken to the power of the column:

This looks a little jumbled, in fact it seems to have very little structure at all. This is not very useful if our goal is to make pretty images, and on this blog that is normally the goal, but it other areas it turns out to be incredibly useful. The process of modular exponentiation is an essential part of public key cryptography, one of the technologies that allows secure communication over the internet. The jumble and lack of pattern that we can see is a sign that modular exponentiation is a good method to use to jumble things up. if there were structure that could be used to help decrypt the messages!

Returning to images, lets make a big version of the multiplication image, mod 1583 (you need to click it to get the full effect, scaling the image down blurs out a lot of structure):

Another option is to make an animation. what happens as we move the modulus value:

There is plenty to study in these images, for example, the curves that can be seen are approximately hyperbolae as they occur when x*y is some fixed value. The central star point occurs in the middle of the image, and there are further stars at 1/3, 2/3, 1/4, 3/4 etc. Can you work out why?

The appearance of hyperbolae perhaps implies that other curves might be possible. What happens if we consider x^2 + y^2? An obvious guess from this formula would be circles and we indeed get (for 151):

Playing around a little further this image comes from x^2 - y^2 +3 x y:

These images are certainly worth repeating for 1583 (again the details get blurred out, so click the images to see the full detail):

To finish let us consider something even simpler. Taking the value of a square to be \frac{x \mod y}{y} this will always give a value between 0 and 1. We can then colour again, and animate with \frac{x \mod Q y}{y} and Q going from 5 to 0:

I first came across these patterns in the December Issue of notices of the AMS, I have always been surprised how little they have been explored. This post is my attempt to do a little to correct that.

13 Comments | Art, General, Mathematics | Tagged: , , , , , | Permalink
Posted by gelada

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