Geometry no place is special; It must look

Geometry of the UniverseGeometry was of great importance to the ancient Greeks.Intensive study for thousands of years lead to deep understanding and firmfoundations in mathematics. Among many influential mathematicians stands Euclid,whose five postulates, which stood tall for thousands of years, gave rise to awhole new outlook on the universe years later 12. A single postulate shapeda big chunk of how we interpret the universe today. It sparked debates on thetrue shape of the universe; from Einstein to Hubble, physicists ponderedwhether the universe had a specific large scale ‘shape’ in which geometry worksa little different.

With 3 possible geometries deciding the fate of theuniverse, it was no surprise physicists wanted answers.  Euclid’s fivepostulates are as follows 1:I.                    “Thereis a straight line between any two points.II.                  A linecan be continued in the same direction.III.

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                Acircle can be constructed around any point with any radius.IV.               Allright angles are equal to each other.V.                 Lineswhich are not parallel will, if continued to infinity intersect somewhere.”The first four were basic truths, accepted with littlethought. However, the fifth postulate sparked much interest and discussion evenin ancient times 1. It caused discomfort as there was no real way to provewhat would occur at infinity.

So, mathematicians decided to assume thepostulate was false and, using the other four axioms, would try to findcontradictions which would prove it to be true 1. However, after intensivesearching they couldn not find any contradictions 1. Lack of contradictionscaused them to develop non-Euclidian geometries which proved or disproved the 5thpostulate 12. These topologies were equal, no geometry was favoured whichforced physicists and mathematicians alike to wonder whether the universefollows a specific geometry 2.There are three total topologies that the universe canfollow, restricted by the Cosmic Principle 3. The Cosmic Principle statesthat, by observation, the universe must be isotropic and homogeneous 7. Theuniverse must have a constant density of galaxies throughout so that no placeis special; It must look the same (isotropic) and be the same (homogeneous) inall directions 7. The first geometry which follows Euclid’s fifth postulate isfollows Euclidean geometry, known as flat geometry 35.

Inside thisgeometry, parallel lines will always remain a fixed distance apart, the sum ofthe angles in a triangle will always be 180°, the square of a triangles sideswould equal the square of its hypotenuse and finally the circumference of acircle will always equal 2*pi 2356. For our universe to follow thisgeometry it must be infinite, an edge of the universe would violate the CosmicPrinciple as the universe would no longer be the same in all directions. Itwould also have a specific point which is different than all the others, thecenter. 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 19thcentury Bernhard Riemann demonstrated that Euclid’s fifth axiom was noting buta 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 intriangles add to more than 180°, the circumference of a circle is less than2*pi*r and parallel lines do meet 36. This can only be possible if theuniverse 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 finiteand is called a closed universe 36.  It is exactly like a sphere with no edge andno center. This does mean that if you were to continually move in a singledirection you would eventually end up where you started 3. Of course, theuniverse is too big, and continually expanding, so it would take an infiniteamount of time to accomplish this 3.  Itis 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 thelines of longitude meet at the north and south pole 8. These start an equaldistance from each other, perpendicular to the equator but end up meeting.

Infact, if you take any sphere, starting at the equator and draw straight linesup perpendicular to it, these will always meet creating a triangle. What isinteresting is that this triangle will have three angles each 90°, proving thatthe sum of a triangles angles in curved space is greater than 180° 12356.The last geometry the universe could follow is alsonon-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 calledHyperbolic 3. In this geometry the angles in a triangle add up to less than180° 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’sfifth postulate as they do not remain a constant distance apart as they divergefrom each other 3. The shape of the universe can be represented using asaddle 3.

A universe which follows this geometry is said to be open; it mustalso be infinite so that it doesn’t violate the Cosmological Principle 36.It became obvious that the fate of the universe depended onwhich geometry our universe resides in. A closed universe would end in the “BigCrunch”, while an open and flat universe would continue forever with galaxiesdrifting away from each other 3. Naturally, physicists and mathematiciansalike tried figuring out whether our universe was flat, open or closed.

However, this is incredibly difficult. You cannot draw a triangle and sum itsangles to find out the curvature of space, the triangle simply isn’t big enoughfor the curvature of space to be noticeable 3. We are far too small, so ourgeometry approximates to Euclidean geometry just like the Earth appears flatuntil you zoom out far enough where you can truly see it as a sphere 3.  So, mathematicians such as Carl Friedrich Gausstried constructing triangles big enough for the effect of curvature (if therewas one) to emerge 1. Gauss armed with flashlights and a telescope, travelled to 3mountains in Germany (Hohenhagen, Inselberg,and Brocken 9) placing flashlights on two of them and thetelescope on the third, creating an imaginary triangle of light 1. Carefullymeasuring the angles between the mountain tops, they calculated the sum of all3 angles 1. The result came very close to 180° within experimental error.Defeated, Gauss deduced that the universe is described by Euclidean geometry1.

However, as it turns out, the scale of Gauss experiment was too small tovalidate anything 4. Edwin Hubble led a different praise worthy experiment meantto decide the curvature of space 5. Instead of creating triangles, Hubbledecided to work with spheres 5. Just like how the circumference of a circlechanges depending on the curvature of space, so does the volume 5. If theradius of a sphere is doubled in Euclidean geometry, the volume increases 8times. If the sphere is in closed space, the volume will increase by a littleless than 8 times, and if the sphere is in an open universe the increase involume 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. Thenumber of galaxies can be thought of as the volume of the sphere 5.

Bycounting the number of increasing galaxies, Hubble hoped to discover whetherthey increased more, or less, than predicted by the Euclidean model 5. Thismodel predicts that if the number of galaxies recorded up to say 50 millionlight years, was 50, then in a flat universe (follows Euclidean geometry) thenumber of galaxies at 100 million light years should be around 400. If thenumber of galaxies is only 300 up to that distance the universe is closed andif the number were as high as, let’s say, 500 then the universe is open 5.Unfortunately, the distance range Hubble worked at was stilltoo small to produce noticeable effects of curvature 5. With the technologyhe had available to him, Hubble could only scan galaxies to a distance of a fewhundred million light years 5. Even as impressive as this is, a noticeable differenceof galaxies is predicted to occur at a distance of 3000 million light years5.

Two problems emerged from this: one, the number of galaxies which neededcounting rose dramatically from hundreds or thousands to millions, and two, thelight from galaxies so far away was too fait to be registered on the telescopesavailable to Hubble 5. During Hubble’s quest the technology which allowedautonomous galaxy counting was not available, nor were telescopes capableseeing 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 universe356.

It is important to note that not every physicist agreed on theconcept of a fixed curvature of the universe. In a scientific paper publishedin 1889, Auguste Calinon suggested that the curvature of space could vary intime, changing from Euclidean to non-Euclidean geometry 2. Meaning resourcesspent finding the geometry of the universe are ultimately wasted; whether it isworth the trouble is still debated today. With the James Webb Space Telescopeplanned launch in 2019 and the steady increase in sophisticated technology, it ispossible to continue Hubble’s work and find out the curvature of the universein the future. References1 Kennedy, J. (2003). “Space, Timeand 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 tomodern cosmology. 1st ed.

Chichester: John Wiley & Sons Ltd, pp.39-44.4 Narlikar, J. (1983).

 Introduction tocosmology. 1st ed. Boston: Jones and Bartlett, p.47.

5 Narlikar, J. and Burbidge, G. (2008). Factsand speculations in cosmology. 1st ed. Cambridge: Cambridge UniversityPress, 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). TheUniverse Adventure – The Homogenous and Isotropic Universe. onlineUniverseadventure.org. Available at: http://www.

universeadventure.org/big_bang/expand-balance.htmAccessed 11 Jan. 2018.8 Stern, D. (2018). Latitude andLongitude.

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 onrelativity. MathPages, p.565.