## The Geometry of The Universe

Rao Vemuri

A paradigm shift is said to have occurred when the prevailing way of thinking takes a radical change. One such shift occurred in ancient Greece when it was first understood that the Earth is not flat as it appears, but shaped like a ball. Another shift occurred when Galileo turned his telescope toward the heavens and opened a new-era of ever expanding horizons. Yet another shift occurred when we realized that it is not just our horizons, but the universe itself is expanding. If the Earth, planets and the Sun are like gigantic balls, what is the shape of the universe? Another huge ball?

This is not an easy question to answer. So let us go back to the original question, "what is the shape of the Earth?" The Greeks wondered about this too. To answer this they invented a tool and called it "geometry", which means, literally, "measuring the Earth." Development of geometry occurred over a period of hundreds of years. First they discovered a wealth of results from experiments and experience. Then they organized this knowledge into a systematic theoretical framework, which in turn, consisted of very few unproved statements, called "axioms" followed by a large number of statements called "propositions" or "theorems." Then they developed a systematic logical procedure for proving the validity of these theorems assuming that the validity of the axioms is self-evident. The culmination of this effort is Euclid's Elements, written some time around 300 B. C.

The first developments in geometry came with the flowering of three ancient civilizations - the Babylonian, the Egyptian and the Chinese - during the second millennium B. C. All three seemed to be aware of what we now call the Pythagorean theorem, which states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. Despite considerable sophistication in mathematics and astronomy, none of these civilizations understood that the Earth was round, not flat. The clues were there, but they could not make that intellectual leap of logic, that paradigm shift.

The early civilizations knew that as one travels south, the North Star appeared lower and the noon-time Sun appeared higher. The Chinese went one step further and estimated the distance of the Sun from Earth. For this calculation they extended the method they used to estimate the height of a mountain on an inaccessible island. They measured the angle of elevation of the mountain peak from two different points, one point closer to the mountain than the other, along a base line. Using the length of the baseline, the two angles, and some ingenious combination of algebra and geometry they deduced the height of the mountain. When they extended this method to estimate the height of the Sun, using a north-south baseline for their experiment, they got a ridiculously low value for the distance to the Sun. They never even questioned their assumption that the Earth was flat. Eratosthenes repeated a similar experiment in the third century B. C. By not accepting a flat Earth, he correctly deduced the size and shape of the Earth.

In the seventeenth century, after Galileo, our vision began to change drastically. Newton accepted the universe to be infinite in size and wrote down his laws in terms of Euclidean geometrical concepts. Little did Newton suspect that not only his laws of motion but also the geometrical edifice constructed by Euclid, in which his laws operate, would be called into question during the eighteenth and nineteenth centuries. This attack came on two fronts. One questioned whether the axioms of Euclidean geometry are as self-evident as they are touted to be; other equally "self-evident" axioms lead to several non-Euclidean geometries. The leaders of this flank were the Russian Nicolai Lobachevsky and the Hungarian Janos Bolyai. The other asked whether the classical Euclidean geometry provided a true description of the physical world. The leader of this front is the shy and retiring German mathematician, Bernhard Riemann.

Crediting Riemann for his seminal role in shaping his own views, Einstein once wrote, "Only the genius of Riemann, solitary and uncomprehended, had already won its way ... to a new conception of space." The work that Einstein referred to was a lecture delivered by the 28-year old Riemann, "On the hypotheses that underlie geometry," on the occasion of his attempt to secure a job at the University of Gottingen. In this lecture Riemann introduced the ideas of curved space, curvature of space, the notion of curved spaces in four or more dimensions, and examples of curved spaces. He described the old-fashioned Euclidean plane geometry and solid geometry, respectively, as two-, and three-dimensional examples of what we now call Riemann spaces with zero curvature. It turns out that the Lobachevsky and Bolayi spaces are also special cases of Riemann spaces. Although many of these ideas are deeply rooted in esoteric mathematics, it turns out that the shape of the universe is closer to the geometrical vision of Riemann than the centuries-old conventional wisdom handed over from Euclid. Riemann's suggestion that the entire universe may be modeled by a finite, but unbounded (i.e., spherical) space, "is one of the greatest ideas about the nature of the world which ever has been conceived," according to Max Born. What Born did not mention is the notorious difficulty of comprehending and visualizing a spherical universe and the attendant curvature of space.

Saying that the space is curved, rather than flat or Euclidean, is another way saying that the familiar properties of Euclidean geometry - such as the Pythagorean theorem - do not hold. Suppose we start from Earth and go to Galaxy A, 3 million light years away, turn right and go to Galaxy B 4 million light-years away, then the shortest distance between us and Galaxy B would be 5 million light years if the space is Euclidean or flat; if not, we say that space is curved or non-flat or non-Euclidean. If the distance is less than 5 million light years, then we say that space has a positive curvature. Just as the ancient Greeks were able to deduce that the Earth is roughly spherical, we can also deduce, in broad strokes, that the shape of the universe is - spherical, in the Riemanian sense!

The best way to visualize such a space is by analogy to familiar things. Start with the definition of a circle as the curve that describes the outer edge of a round disk. In common usage we interchangeably use the word "circle" to refer to the "disk" as well the "loop." Here our definition is more precise. A sphere is the shape of a perfectly round soap bubble or the outer surface of a ball. So a circle is a one-dimensional curve and a sphere is a two-dimensional surface. Higher-dimensional spheres are called hyperspheres. A three-dimensional hypersphere "looks" like a circle and a sphere, but has one more dimension. That is, it is three dimensional, it has no edges, and it is finite in size.

Visualization of hypersheres is difficult. The closest example from everyday life is a toy figurine that contains a smaller figurine within that, in turn, contains a yet smaller figurine within, and so on. Or, think of a spherical onion. As you peel the outer spherical layer, there is another spherical layer. An infinite succession of these concentric spheres is an example of hyphershere.

Now let us attempt to visualize the observable universe using this onion model. There are two versions. In the geocentric version, as seen from the Earth, this model of universe can be thought of as a huge onion with the Earth represented by the innermost spherical layer. Each observed object, be it a planet, star, or galaxy is placed at a proper distance in the proper direction. Modern astronomy tells us that the farther we go from the Earth, the older the universe. The furthest layers of this onion then represent the background radiation, the remnant from the very early stages of creation. The outermost layer of the onion, the skin, represents a single point - the Big Bang. Alternatively, we can think of a model where the singular point that has exploded is placed at the center of the concentric spheres. Indeed both these hypersphere models are eversions ( a sphere pulled inside out) of each other.

The model of the cosmos constructed by the 13th century Italian poet Dante, in his Divine Comedy, bears a resemblance to the geocentric model described in the preceding paragraph. In the section called Paradiso, Dante describes a complex structure of two inter penetrating hyperspheres - a structure that is very difficult to visualize. In the center of this model is Earth. A traveler ascends through successive planetary spheres and reaches the sphere represented by the stars. At this point Beatrice, the guide, shows the traveler a bright light surrounded by angels orbiting in nine concentric circles. This is Dane's Empyrean Paradise. A projection of these concentric circles on the earthly sphere results in a pyramidal structure whose apex is the Earthly Paradise and whose base is the Purgatory. If one descends - a direction opposite to the planetary ascent - one eventually reaches Hell.

Hindus of the ancient past also constructed several models of the Earth and the universe. Some of these are utterly ridiculous. Some, with certain modifications, are fairly close to the accepted models of contemporary times. In one model, presumably of the planet Earth, there is Mount Meru at the center of a huge ocean. In that ocean there are four islands, around the Mount. One of these is Jambudvipa. The northern portion of this island is separated from the southern portion by the Himalayan range. South of this range is Bharatavarsha. From this we can infer that Mount Meru probably represents the North Pole and Jambudvipa is probably the Eurasian continent. Given the fact that it is much easier to observe the sky above than to infer the geography of a planet, the above model is not a bad model for the planet Earth. Unfortunately, a model described in some Puranas has Mount Meru, once again at the center. Surrounding Meru is Jumbudvipa, a ring-shaped island, which in turn is separated from an outer ring-shaped island - Plakshadvipa - by an ocean of salt. These concentric islands and concentric oceans continue. Starting with the ocean of salt after Jambudvipa, the other oceans are treacle, wine, ghee, milk, curds, and fresh water.

Ancient Hindus believed that our universe itself was shaped like an egg - the Brahmanda. This was divided into twenty one zones, or lokas, of concentric spheres. The Earth is located on the seventh sphere from the top. Above the Earth there are six spheres representing heavens of increasing beatitude: Bhuvarloka, Swargaloka, Maharloka, Janaloka, Tapoloka and Satyaloka. Below the sphere of the Earth are seven stages representing nether worlds or patalalokas: Atala, Vitala, Sutala, Talatala, Mahatala, Rasatala and Patala. There is no suggestion that Patala lokas are bad. Below Patala lokas lay seven Naraka lokas (the hell) of increasing misery. This onion model is not too far from the Reimannian model described earlier. Furthermore, to their credit, ancient Hindus believed that our universe hung in the void and speared from other universes.

rvemuri@ucdavis.edu
Tuesday the 3rd September, 1996