I think although space appears to be Euclidean at small scales, has hyperbolic geometry when you take time into account as well, may be roughly speaking a 3 sphere (plus time) when you look at the whole universe, we may be able to show it is not fractal. It all comes down to making measurements. You basically determine the geometry by making measurements. A great example is making measurements on the Earth's surface and finding they are inconsistent with the Earth being flat. If you were on a hyperbolic plane rather than on a sphere, in the small scale it would look the same as a plane or a sphere, but as you made larger scale measurements the discrepancies would tell you what the geometry was.
You can do a similar thing with fractals. For instance, suppose you wanted to find out the dimension of the coast of Britain. You might believe it was 1-dimensional, but someone else says "No - it is fractal! You can check like this. First you make a very crude measurement of the perimeter length, using steps of 1 mile between points on the coast. Then you repeat with points half a mile apart. Then again with points a quarter of a mile apart (good Imperial units - after all, we invented them). In each case, you form the sum of the distances between adjacent points. If the coast is 1-dimensional, the lengths converge to some length as the points get nearer to each other. If they do not, then various things may happen. One thing is that each time you halve the distance between the points, the total length goes up by a similar multiple (say k). In this case, the fractal dimension is log(k)/log(2) [The 2 is because we halved the distance between the points each time].
The same thing would happen if space was fractal. Except here, it would affect all the physical laws. I wave my hands here due to my own incomplete knowledge, but I am pretty sure this is true. For example, you might expect the speed of light to vary depending on the scale! Gamma rays would have to go further along the jagged space-time than a radio wave (which only "sees" space at a very crude resolution of length), because they are more localised (higher energy ~= higher frequency ~= smaller length scale).
By co-incidence, I was very interested to find something closely related in the surprising area of financial markets recently, with a connection to thermodynamics (specifically entropy) as well. The question of interest was how random markets are.People who believe in a theory called the Efficient Market Hypothesis or EMH (specifically the strong version of this) believe that the movements of a market are indistinguishable from random movements. A corollary of this hypothesis is that you cannot make money by trading except with inside information or blind luck. Other people try to prove them wrong, usually by ignoring what they say and making lots of money despite them. I read a study where someone tried to show this result in a less lucrative way. What they did was look at the sequences of prices that occur and show statistically that they are not random.
Each ratio of price from one day to the next is a number (usually not too far from 1). The sequence of ratios on N consecutive days can be viewed as a point in an N-dimensional space. One study looked at such sequences for 15 consecutive days in the American stock market. The EMH would suggest these points would form an amorphous blob in 15 dimensional space, clustered around the point (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) [Or (0,0,...0) if you take logs].
Now here's the interesting bit. You can measure the dimension of this blob of data in the same way as you can for the coast of Britain. What you do is you try to cover the whole set of points with sets of 15 dimensional hypercubes of different sizes. Now if the vectors were truly random, each time you halved the size of the cubes (i.e. their edge length), you would need about 2^15 times as many hypercubes to cover them all. But when they tried this, they found they only needed 2^4 times as many. (actually I assume they reduced the size of the cubes less than this, or they wouldn't have had enough data, but anyhow the exponent was indeed 4).
What this means is that the sequences of prices are near a 4-dimensional hypersurface in the 15-dimensional space of all possible sequences. Which means that it is not entirely random, but there is some connection between the movements that occur over a period of a few weeks, as one might expect.
I've been wondering for a while, what if space were shaped like a fractal? The evidence we have at the moment says that space is shaped like a sphere, so everything's even, but how different would living in a fractal be? I looked at these two videos:
http://www.youtube.com/watch?v=AGLPbSMxSUM
http://www.youtube.com/watch?v=MKwAS5omW_w
And they talk about the space that is "not a knot", and the way light bends to create strange patterns in a space without a certain locus of points. So is there any way to apply this to, say, a point living in the whitespace of the Sierpinski triangle? Is there any way to distort a 2-D infinite space to fit inside only the whitespace of the triangle?
Alternatively, is there a function f(x,y) with domain real values of x and y, such that the range is a mapping onto and into the whitespace of the Sierpinski triangle?
Or, a simpler question -- picture a 3-D sierpinski triangle, the whitespace of it. It's a bunch of tetrahedrons. Imagine if each tetrahedron was a planet, with sufficient gravity to hold mass onto it, despite lacking mass. The center of each tetrahedron has a heat sorce to sustain life. They're all connected, so it's easy to move between planets, but the direction of gravity gets confusing. Also, planets get smaller as you go farther out.
In other words, the blue one: http://en.wikipedia.org/wiki/File:Sierpinski_pyramid.png
How would technology evolve on this kind of world, specifically regarding the unique structure? Purely theoretical. I'm just looking for ideas on writing something regarding life on/in a fractal. Dazzle me with imagination!