Rational Tangles

I know I promised to do more physics and less mathematics, but this game caught my attention and I couldn't resist adding at least a link. I would give the details here, but the explanation at John Baez's website is far superior to what I could write.

The Many Dimensions of Dimensions II

Now that we know from the previous posting why the usual coordinate based definition of dimension is not always valid, we need to explore the alternative.

One possibility is to consider the scaling of length/area/volume. For example, consider first a line that is one unit in size. How do I double its size - by gluing a second unit length line to one end. So a factor of two increase in size requires two units. To increase by a factor of three takes three unit lines. We define all objects with this property to be one-dimensional.

What about a square? Start with a 1x1 square, and then to create a 2x2 square requires four of the 1x1 squares, or 2^2 squares(note that here and in all posts, 2^n represents 2 multiplied by itself n-times). A 3x3 square requires 9 or 3^2 squares. So we define this to be 2-dimensional.

You can do this for any simple objects (with the obvious requirement that you can tile the small pieces to make larger copies) and define dimension to be the log n/log m where an m-fold increase in the size of the object requires m^n tiles (or in non-mathematical terms, if an m-fold increase in size requires m^n tiles, the space is n-dimensional)

And now the fun begins!

Consider the Koch snowflake. It doesn't tile in the right way to define a dimension, but try taking 1/3 of the snowflake - or one side of it. If you make four copies, and line them up with the center two meeting at an angle as shown, you would triple the size of the object. So the dimension must satisfy 4 = 3^n, or n = 1.262 ! So here we have an object with a well-defined dimension which is not a whole number!

Consider the Siepinski triangle. and the Sierpinski carpet. In the first case three copies are needed to double the size, so n = 1.585. In the second case eight copies triple the size, so n = 1.893.

Which leads one to ask - with all of the recent discussion of extra dimensions in physics, could some of them have non-integer dimensions? (Before anyone gets too excited about this possibility, remember that such spaces either have (1) no easy way of defining derivatives and therefore no particle momenta or (2) require an infinite number of singularities in spacetime itself )

The Many Dimensions of Dimensions - Part I

Practically everyone knows how to define dimension - it is the minimum number of numbers needed to uniquely define position. For example along a line every point can be described by its distances from the origin, or in a cube every point is given by three numbers labelled x,y,z. It is so simple, and so obvious, and yet wrong. (DISCLAIMER: The statement that it is wrong is partly satirical. There are special examples in which it doesn't work, but the reader should not conclude that this definition is not useful. It is still the definition the is used in almost every field of math and science, and is perfectly acceptable in nearly every situation,)

It is wrong because this definition implies that every compact (or non-infinite) surface or solid object has dimension 0 or dimensions 1. That is because there are known curves (which are too complicated to ever actually be drawn) in which a single number (such as the distance along the curve) can be used to define every point in a solid square, cube, or any other shape.

Although it can never be drawn, there are some nice approximations. (A link to some pictures of these will follow as soon as I find some). For example, take a square. The first approximation - simple draw a line from one corner to the diagonally opposite corner. For the second approximation, draw the same diagonal line, except at the 3/4 distance turn 90 degrees right, go to the edge of the square, turn 90 degrees again, and continue to the next edge, and keep doing this until you return to the original point and then finish the diagonal. For the third approximation redraw the second approximation, except now every time there is a straight line add a square as before. If you were to continue this process forever, you would get closer to the curve that actually goes through every point in the plane, and therefore length along this curve provides a description of every point in the plane, making the plane 1-dimensional!

The Banach-Tarski Paradox

This is one of my favourite mathematical topics. There are many ways to state it, but I will give the simplest and most general:

In theory, it is possible to take any object (usually a solid ball) and cut it into a finite number of pieces, apply a rigid transformation (so no stretching is allowed) to each piece, and reassemble them into any other object of any size, or multiple objects of any size!

So you could take a walnut, cut into a few pieces, and reassemble it into a copy of the planet Earth! Of course this only works for a theoretical walnut in which there is no discrete atomic structure, but still the fact that it could be done even in theory is astonishing!

(For more detailed information, try the sites here, here, and here. )

This also leads to a very interesting question in physics. Since normal objects are only excluded because they have discrete structure on the small scale (atoms and molecules), what allows volume to be defined in spacetime? If spacetime is continuous as most scientist believe, then it is possible to apply the Banach-Tarski paradox to a piece of empty space and change its volume arbitrarily. The only way out would be if spacetime itself is discrete on some scale (and in fact this has been proposed in numerous different models of quantum gravity). But does the BT paradox actually require discrete spacetime?

Koide Formula

In my opinion this is one of the most interesting but simple mysteries in modern physics. Anyone can understand it, but so far no one can explain it!

If you take the masses of the three known leptons, (the electron, the muon, and the tau) which are the seemingly random numbers 0.511 MeV, 105.7 MeV, and 1777 MeV, sum the masses , and then divide by the square of the sum of the square roots of the masses, the result is equal to 2/3 accurate to less than ~0.01% !

And the mystery gets more interesting still. Consider the most extreme cases: If all three lepton masses are exactly equal, the result would be exactly 1/3. If one lepton is very heavy and the other two are very light, then the result is very close to 1. The measured value of 2/3 falls exactly halfway between the two!

This clearly suggests some underlying theory, but in 25 years no one has found it. And this relation does not seem to apply to any other set of particles.

First Link

Every blog needs a first link, so I will start with a link to my own homepage.
It may be out of date, and really not very useful, but there are a few articles I wrote that some people claim to like. Such as a super-simple (but very accurate) introduction to the theory of relativity, and a series of "Made-Simple" articles in which I outline the ideas of modern physics without all the mathematics behind it, so that average people can understand what we are studying now and how the universe works. Enjoy!

Intoducing...

Random Thoughts on the Universe

I know you are say "what the hell is this?". What it is is my random thoughts on hard science (chemistry and physics) and mathematics. Any interesting sites I find, or interesting little theories or ideas. Anything that catches my interest, and deserves sharing with a broader audience.

If you like it, enjoy it. If you don't like it, go somewhere else.