Geometry no place is special; It must look

Geometry of the Universe

Geometry was of great importance to the ancient Greeks.
Intensive study for thousands of years lead to deep understanding and firm
foundations in mathematics. Among many influential mathematicians stands Euclid,
whose five postulates, which stood tall for thousands of years, gave rise to a
whole new outlook on the universe years later 12. A single postulate shaped
a big chunk of how we interpret the universe today. It sparked debates on the
true shape of the universe; from Einstein to Hubble, physicists pondered
whether the universe had a specific large scale ‘shape’ in which geometry works
a little different. With 3 possible geometries deciding the fate of the
universe, it was no surprise physicists wanted answers.

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 Euclid’s five
postulates are as follows 1:

I.                    
“There
is a straight line between any two points.

II.                  
A line
can be continued in the same direction.

III.                
A
circle can be constructed around any point with any radius.

IV.               
All
right angles are equal to each other.

V.                 
Lines
which are not parallel will, if continued to infinity intersect somewhere.”

The first four were basic truths, accepted with little
thought. However, the fifth postulate sparked much interest and discussion even
in ancient times 1. It caused discomfort as there was no real way to prove
what would occur at infinity. So, mathematicians decided to assume the
postulate was false and, using the other four axioms, would try to find
contradictions which would prove it to be true 1. However, after intensive
searching they couldn not find any contradictions 1. Lack of contradictions
caused them to develop non-Euclidian geometries which proved or disproved the 5th
postulate 12. These topologies were equal, no geometry was favoured which
forced physicists and mathematicians alike to wonder whether the universe
follows a specific geometry 2.

There are three total topologies that the universe can
follow, restricted by the Cosmic Principle 3. The Cosmic Principle states
that, by observation, the universe must be isotropic and homogeneous 7. The
universe must have a constant density of galaxies throughout so that no place
is special; It must look the same (isotropic) and be the same (homogeneous) in
all directions 7.

The first geometry which follows Euclid’s fifth postulate is
follows Euclidean geometry, known as flat geometry 35. Inside this
geometry, parallel lines will always remain a fixed distance apart, the sum of
the angles in a triangle will always be 180°, the square of a triangles sides
would equal the square of its hypotenuse and finally the circumference of a
circle will always equal 2*pi 2356. For our universe to follow this
geometry it must be infinite, an edge of the universe would violate the Cosmic
Principle as the universe would no longer be the same in all directions. It
would also have a specific point which is different than all the others, the
center. A flat universe is a universe which follows the above geometry 3.

The second geometry doesn’t follow Euclid’s fifth postulate,
it is non-Euclidean, called spherical geometry 346. In the 19th
century Bernhard Riemann demonstrated that Euclid’s fifth axiom was noting but
a choice, other postulates could be chosen and would hold as much meaning 3.
In doing so he founded non-Euclidean geometry. In spherical geometry, angles in
triangles add to more than 180°, the circumference of a circle is less than
2*pi*r and parallel lines do meet 36. This can only be possible if the
universe is curved like a sphere. In this case it is said that space is curved,
it has positive curvature 1. A universe that follows this curvature is finite
and is called a closed universe 36.  It is exactly like a sphere with no edge and
no center. This does mean that if you were to continually move in a single
direction you would eventually end up where you started 3. Of course, the
universe is too big, and continually expanding, so it would take an infinite
amount of time to accomplish this 3.  It
is common to think that on a sphere; parallel lines will also never meet.
However, if you were to take the Earth, for example, you will find that all the
lines of longitude meet at the north and south pole 8. These start an equal
distance from each other, perpendicular to the equator but end up meeting. In
fact, if you take any sphere, starting at the equator and draw straight lines
up perpendicular to it, these will always meet creating a triangle. What is
interesting is that this triangle will have three angles each 90°, proving that
the sum of a triangles angles in curved space is greater than 180° 12356.

The last geometry the universe could follow is also
non-Euclidean. Just like before space must be curved for this geometry to work.
In this case, the curvature of space is negative and the geometry is called
Hyperbolic 3. In this geometry the angles in a triangle add up to less than
180° and the circumference of a circle is greater than 2*pi*r 36. Interestingly,
parallel lines in this curvature do not meet yet they still disprove Euclid’s
fifth postulate as they do not remain a constant distance apart as they diverge
from each other 3. The shape of the universe can be represented using a
saddle 3. A universe which follows this geometry is said to be open; it must
also be infinite so that it doesn’t violate the Cosmological Principle 36.

It became obvious that the fate of the universe depended on
which geometry our universe resides in. A closed universe would end in the “Big
Crunch”, while an open and flat universe would continue forever with galaxies
drifting away from each other 3. Naturally, physicists and mathematicians
alike tried figuring out whether our universe was flat, open or closed.
However, this is incredibly difficult. You cannot draw a triangle and sum its
angles to find out the curvature of space, the triangle simply isn’t big enough
for the curvature of space to be noticeable 3. We are far too small, so our
geometry approximates to Euclidean geometry just like the Earth appears flat
until you zoom out far enough where you can truly see it as a sphere 3.  So, mathematicians such as Carl Friedrich Gauss
tried constructing triangles big enough for the effect of curvature (if there
was one) to emerge 1.

Gauss armed with flashlights and a telescope, travelled to 3
mountains in Germany (Hohenhagen, Inselberg,
and Brocken 9) placing flashlights on two of them and the
telescope on the third, creating an imaginary triangle of light 1. Carefully
measuring the angles between the mountain tops, they calculated the sum of all
3 angles 1. The result came very close to 180° within experimental error.
Defeated, Gauss deduced that the universe is described by Euclidean geometry
1. However, as it turns out, the scale of Gauss experiment was too small to
validate anything 4.

Edwin Hubble led a different praise worthy experiment meant
to decide the curvature of space 5. Instead of creating triangles, Hubble
decided to work with spheres 5. Just like how the circumference of a circle
changes depending on the curvature of space, so does the volume 5. If the
radius of a sphere is doubled in Euclidean geometry, the volume increases 8
times. If the sphere is in closed space, the volume will increase by a little
less than 8 times, and if the sphere is in an open universe the increase in
volume will be over 8 times 5. If the universe is homogeneous as assumed,
then the number of galaxies should increase as you create a larger sphere. The
number of galaxies can be thought of as the volume of the sphere 5. By
counting the number of increasing galaxies, Hubble hoped to discover whether
they increased more, or less, than predicted by the Euclidean model 5. This
model predicts that if the number of galaxies recorded up to say 50 million
light years, was 50, then in a flat universe (follows Euclidean geometry) the
number of galaxies at 100 million light years should be around 400. If the
number of galaxies is only 300 up to that distance the universe is closed and
if the number were as high as, let’s say, 500 then the universe is open 5.

Unfortunately, the distance range Hubble worked at was still
too small to produce noticeable effects of curvature 5. With the technology
he had available to him, Hubble could only scan galaxies to a distance of a few
hundred million light years 5. Even as impressive as this is, a noticeable difference
of galaxies is predicted to occur at a distance of 3000 million light years
5. Two problems emerged from this: one, the number of galaxies which needed
counting rose dramatically from hundreds or thousands to millions, and two, the
light from galaxies so far away was too fait to be registered on the telescopes
available to Hubble 5. During Hubble’s quest the technology which allowed
autonomous galaxy counting was not available, nor were telescopes capable
seeing at such high distances and so he never managed to find the answer 5.

Even to this day we do not know the shape of the universe
356. It is important to note that not every physicist agreed on the
concept of a fixed curvature of the universe. In a scientific paper published
in 1889, Auguste Calinon suggested that the curvature of space could vary in
time, changing from Euclidean to non-Euclidean geometry 2. Meaning resources
spent finding the geometry of the universe are ultimately wasted; whether it is
worth the trouble is still debated today. With the James Webb Space Telescope
planned launch in 2019 and the steady increase in sophisticated technology, it is
possible to continue Hubble’s work and find out the curvature of the universe
in the future.

References

1 Kennedy, J. (2003). “Space, Time
and Einstein: An Introduction”. 1st ed. Chesham: Acumen, pp.149-158.

2 Kragh, H. (2013). Conceptions of cosmos.
1st ed. Oxford: Oxford University Press, pp.125-128.

3 Liddle, A. (1999). An introduction to
modern cosmology. 1st ed. Chichester: John Wiley & Sons Ltd, pp.39-44.

4 Narlikar, J. (1983). Introduction to
cosmology. 1st ed. Boston: Jones and Bartlett, p.47.

5 Narlikar, J. and Burbidge, G. (2008). Facts
and speculations in cosmology. 1st ed. Cambridge: Cambridge University
Press, pp.106-110.

6 Shu, F. (1982). The physical universe.
1st ed. Mill Valley: University Science Books, pp.368-375.

7 Heckmaier, E. and Lii, P. (2005). The
Universe Adventure – The Homogenous and Isotropic Universe. online
Universeadventure.org. Available at: http://www.universeadventure.org/big_bang/expand-balance.htm
Accessed 11 Jan. 2018.

8 Stern, D. (2018). Latitude and
Longitude. online Www-istp.gsfc.nasa.gov. Available at:
https://www-istp.gsfc.nasa.gov/stargaze/Slatlong.htm Accessed 12 Jan. 2018.

9 Brown, K. (2017). Reflections on
relativity. MathPages, p.565.

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