Thursday, 24 April 2014

Astrophysics Corner, Part 8 - The Shape of Space, with a Particular Emphasis on the Dodecahedron

In Book 1 of the Witches Stones series, a form of space travel takes advantage of a previously unknown peculiarity of the shape and dimensionality of space, called Omega-space:

“No human being understood exactly what happened when a spaceship slipped through "omega-space", instantaneously passing from one portion of the galaxy to another.”

The subject of the shape of space has long fascinated people.  The more primitive ancients tended to see the universe as a flat surface (the Earth), surrounded by water (the ocean), and with a dome-like structure above (the heavens) which were the domain of the bodies of the sky - the sun, moon, planets and stars.  Some theories said all of this was held up by a giant (Atlas), who was standing on the back of an equally enormous turtle.  What the turtle was standing on wasn’t considered a fair question - the famously facetious phrase “it’s turtles all the way down” was invented as a humorous evasion of the issue and has now become a standard joking response to skirt around any question involving an inconvenient  infinite regression.  I don’t think what was outside the dome was considered a fair question either.  Perhaps “it’s domes all the way up” might have been a good response.
Plato, in the Timaeus, looked at the universe both in its physical form and in an idealized form.  The former is subject to change and decay, the latter is eternal.  The former is also apprehended by the senses while the latter could be apprehended by pure reason.  In overall form, the universe was thought of as a sphere, the most perfect form, which the divine would naturally choose.
 It was also conceived to be related to one of the Platonic solids, the dodecahedron.  There are five platonic solids - the other four were thought to be what the matter of the physical universe was made of.  These are tetrahedron (fire), the octahedron (air), icosahedron (water) and cube (earth).   The shapes were matched with the four elements as then conceived, on the basis of such physical characteristics as spikiness, resistance to rolling and so forth.   The fifth, the dodecahedron was thought to represent the shape of the universe itself, perhaps partly because it is the platonic solid that looks most like a sphere.  You might think of a football (in North American terms, a soccer ball) to help you visualize the dodecahedron.   A dodecahedron is composed of 12 equal sized pentagons, however, while a soccer ball is technically a spherical truncated icosahedron, which is a combination of dodecahedron and icosahedron.
 Eventually other views of the universe came to be dominant.  Perhaps the one that we are all most familiar with is what might be described as the Newtonian view - that there are three dimensions that all go on forever, along with an absolute time that clicks along uniformly and relentlessly.  That’s what the average person probably thinks of when he or she hears the phrase “the shape of space”.
It might surprise you to know that mathematicians and astrophysicists (cosmologists) don’t see things that simply.  Mathematicians have devised a whole branch of mathematics (topology) that concerns itself with what spaces are logically possible and what characteristics those various spaces would have - dimensionality, distance metrics, axiomatic foundations and so forth.   Astrophysicists are actively seeking ways to measure the actual observable universe on the large scale to attempt to determine the shape of the space that we live in.  While doing so, they incorporate the complexities of Einstein’s theories of general and special relativity, which involve a much more elastic notion of space-time than Newton’s absolute space and time.
What is meant by measuring the large scale properties of the universe in order to infer its shape?  A useful analogy is that of intelligent creatures that live on a sphere, in only two dimensions.  How would they conceive of the space that they inhabit?  Well, since they can’t jump out into the third dimension, they have to do experiments to infer the large scale properties of their space.  If you find this example to be too contrived, you might think of people living on a planet with a very flat surface and a very opaque, cloudy atmosphere, tidally locked to its sun, such that all they saw when they looked up was a flat white sky and all that they saw when they looked around was a horizon stretching away.
One experiment these spherical surface bound creatures might try, is to simply walk in a straight line for a long time.  As we know, if you walked straight forward on an infinite plane, you would never come back to the same point, regardless of how long you walked.  But if you were walking in a straight line on the surface of a sphere, you would eventually come back to your starting point.  For example, if one of these creatures started at his equivalent of our north pole, he would eventually pass the equator, then the south pole, then the equator again, then back to the north pole.   Assuming that he could recognize some sort of unique landmark (or created one himself), he would know that he was back where he started.  From that, he could tell his space was unbounded (he didn’t come to a barrier he couldn’t cross) but still closed (he didn’t go on forever without coming back to his starting point).
Another experiment that these creatures could do involves measuring triangles.  On a flat Euclidian plane, we know that a triangle’s angles sum to 180 degrees (or pi radians).  But on a sphere, a triangle sums to more than 180 degrees, while on a saddle shaped surface it  sums to less than 180 degrees.  To see this for a sphere, just think of starting at the north pole, travelling straight down 0 longitude, turning right at the equator (i.e. a 90 degree turn), following the equator one fourth of the way, turning right again at the 90th longitude (another 90 degree turn), then ending up at the north pole. You would have done two right turns (180 degrees) and still had a 90 degree angle between the path you started on from the north pole, and the path you returned on.  That’s 270 degrees in all, more than the 180 degrees in a triangle on a plane.  So, once again, these creatures would know that the space that they inhabit is very different from a flat, infinite plane.
Similarly, human astrophysicists would like to measure certain large scale features of our observable universe to see what sort of space we live in.  One experiment would be to head out in a rocket for a long, long time and see if eventually you seemed to be passing the same bunch of galaxies that you started from.  Obviously that’s not exactly an achievable experiment for a number of reasons - for one thing, we lack such a rocket and for another the galaxies are constantly changing so you might not recognize them when you passed them again.  Plus, by the time you got back to let people know your result, they would probably have evolved into something else and you wouldn’t know how to communicate with them anyway.
But there are ways to attempt to measure the universe.  One is to examine large scale maps of the observable universe (usually based on satellite observations in various wavelengths, but especially microwave) to see whether the universe looks to be homogeneous and isotropic on all scales (i.e. it appears to be the same wherever you look in the sky, and on every scale).
This is generally done using a technique called spectral analysis, which is a mathematical algorithm for finding patterns or regularities in data of all sorts, based on math first developed by Fourier in the 19th century and elaborated by many others since then.   In simple cases, these patterns can be detected by humans without sophisticated mathematics or computer analysis, but in more complex cases, spectral analysis is used to discover where spikes of the “power” in a signal are located, which indicate some kind of regularity in the data, which could be in the time or the spatial domains.
For instance, in music we know there are various harmonics along with the fundamental note.  These can be uncovered using spectral analysis.  Extra-solar planets are often discovered via this technique - regularities in the dips in a stars light curve can reveal the presence of a planet or multiple planets.   Geophysical exploration makes wide use of these methods, to find interesting and possibly profitable anomalies in magnetic or gravitational data.  Electrical engineering makes wide use of it to examine the frequency responses of circuits - that’s where the expression “power spectra” comes from.  Many other examples of the use of spectral analysis could be listed.
In 2003, Luminet, Weeks et al produced a paper that examined the WMAP data (Wilkinson Microwave Anisotropy Probe), to examine the spherical harmonics in that dataset.  Basically, it is a temperature map of the sky, produced by a satellite telescope.  Much of the data maps quite well to infinite flat space, especially the higher harmonics.  But there are problems at some levels of the spherical harmonics that can’t be well explained by such a model.    Essentially some of the lower harmonics imply a finite universe, where the size of space itself cuts off some of the expected wavelengths in the spectrum.  In their paper they use the analogy of a bell, where some overtones are impossible because the wavelengths would be bigger than the bell itself.
They argue that something called a Poincare dodecahedral space fits the power spectra of the data very nicely.  Essentially, in this space any object “leaving the universe” (including light) goes out one face of the dodecahedron and returns from the opposite face, with a twist.  In this case, space would be unbounded but closed, rather like the sphere is for two dimensional creatures.  In this case it would be akin to a hyper-sphere and the Poincare dodecahedral space would bear a similar relationship to it that the dodecahedron bears to the sphere in our “normal” space.
So, amusingly enough, Plato might have been (more or less) right all along.  At least I think that’s one way of looking at it.  Read the paper “Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background” and judge for yourself.  It was in Nature and is also on the physics arxiv site (there was another paper on the subject in 2008 as well, using more data).
And here are a few visualizations just for fund (not strictly mathematically the same as what is described above, which is pretty hard to visualize J).


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