 Introduction
The dictionary defines geometry as “the branch of mathematics concerned with the properties and relations of points, lines, surfaces and solids.” The Greeks used geometry extensively and Newton used geometry to describe the motions of heavenly bodies. However, the ancients knew of only one kind of geometry now known as Euclidean geometry. This is the kind of geometry using points and straight lines which work very well at large distances in the universe, but it was Riemann who provided us with a geometric tool to quantify the extent of the curvature of space. Einstein made use of Riemannian geometry to describe the curvature of spacetime near massive bodies. There is, however, another kind of geometry where Euclidean geometry simply cannot be used, quantum geometry, which isthe geometry of the very small, such as at the Planck length of 10 to the power of 35cm. Let’slook at attempts to use quantum geometry to describe the universe at the quantum level. Some might ask why we want to describe something as large as the observable universe in quantum terms. There are some very compelling reasons such as quantum cosmology, a very interesting field of study in cosmology. Before we discuss quantum geometry I have to refer the reader to a very different kind of geometry where lines are not the shortest route between two points and are therefore not straight lines, hyperbolic geometry.
 Hyperbolic geometry
Hyperbolic geometry is sometimes also referred to Lobachevskian (note 1) geometry and provides us with a very accurate example where Euclidean geometry (the parallel postulate) is invalid. To refresh memories, Euclid’s first postulate states that there is a straight line segment connecting two points, his second postulate states that any straight line segment can be extended indefinitely and the third postulate states the existence of a circle with any centre and with any value for its radius and his fourth postulate states that all right angles are equal. Escher (note 2) used a very interesting form of hyperbolic geometry where the entire universe of the hyperbolic plane is squashed into the interior of a circle. Drawing a triangle in the circle so that the lines meet on the line representing the circle shows clearly that the sum of the angles of the triangle does not equal 180o, in fact they always fall short of 180o. This clearly invalidates Euclidean geometry, at least in the case of hyperbolic geometry. We can find another example where the Euclidean postulate of a straight line seems to be false, that is the surface of the earth. A pilot flying an aircraft from point A to point B might insist that he is flying in a straight line, but the statement will clearly be false. The flight path a pilot follows is known as geodesic which is the shortest or the longest route between two points.
What does this say about the universe? The famous British mathematician, Roger Penrose (note 3) stated that we simply do not know. There are observations of the large scale structure of the universe which show that the cosmological principles of homogeneity and isotropy hold and Euclidean geometry is clearly valid. On the other hand some observations of the large scale structures of the universe favour hyperbolic geometry.
 Quantum geometry
What does the universe look like at the level of the very small, far smaller than an atom or even a particle? What kind of geometry can we use to describe the universe at the scale of the Planck length of 10 to the power of 35cm? It is generally accepted that the universe originated from a quantum object far smaller than a particle. The most basic tenet of quantum mechanics, Heisenberg’s uncertainty principle dominates the quantum field to such an extent that geometry in the classical sense does not exist. There are various forms of the uncertainty principle such as the energy of a particle in a certain state cannot be determined within a specified time interval, but the best known form of the uncertainty principle is that the position of a particle and its velocity cannot be determined at the same time. The equation can be written as follows:
(Δpy)(Δy)≥ħ/2π
where Δy = the uncertainty in the particles position moving in y direction; and
Δpy = uncertainty in the particle’s velocity.
This can also be visualized as virtual particles popping in and out of existence. The particle is called virtual because it cannot be detected with a particle detector, but they are real since their influence on other particles can be measured. A pair of virtual particles (one positively charged and the other negatively charged) pops into existence and annihilate each other within the time limit set by the uncertainty principle. This means that the quantum field is in a state of constant flux with the energy levels constantly rising and falling.
This means that the distinction between space and time gets ‘blurred’, time gets spaced out. Points can no longer be farther apart in time than in space. The average speed of light shown as a straight line in a spacetime diagram becomes smeared out at the quantum level and the speed of light fluctuates wildly. There is no before and no after as a result of the blurring effect the uncertainty principle has on time.
The uncertainty principle fulfilled a very important role in the evolution of the universe. Slight temperature variations were detected in the Cosmic Microwave Background Radiation. This was caused by the quantum fluctuations in the very early universe resulting in dense and less dense areas in matter and the denser areas through mutual gravitational attraction formed the first stars and galaxies.
 Quantum cosmology
As we approached the quantum era of the history of the universe, the random appearance and disappearance of particles force us to abandon the idea of a three dimensional space geometry. Instead we have an average over of all three dimensional space geometries that are possible. This is in accordance with the ‘sumoverhistories’ approach of the brilliant American scientist Richard Feynman. Scientists are currently working on the development on a theory of quantum gravity which will enable us to describe gravity at the quantum level and in the early universe. This might be our only hope to calculate with any certainty exactly what the history of the universe was at the Planck scale and before the Planck time of 10 to the power of 43s.
 Penrose, Roger. The Road to Reality, p.33
 Ibid p. 33
 Ibid p. 46
Frikkie de Bruyn
_____________________________________________________
Suggested further reading.
GellMann, M. (1994). The Quark and the Jaguar: Adventures in the Simple and Complex. W.H. Freeman, New York.
Hawking, S. W. and Penrose R. (1996). The Nature of Space and Time. Princeton University Press, Princeton, New Jersey.
Penrose, R. (2007). The Road to Reality. Vintage Books, London.
Smolin, L. (2002). Three Roads to Quantum Gravity. Basic Books,
New York.
Wald, R.M. 1984. General Relativity. University of Chicago Press.
