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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