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By D.N. Seppala-Holtzman St. Joseph’s College faculty.sjcny.edu/~holtzman What is the Shape of the Universe ? Nobody knows There are many, many theories Here, we present a simple argument, from basic principles, that the universe is a hypersphere A Hypersphere??!!! Yes, a hypersphere Specifically, a 3-dimensional sphere in Euclidean 4- dimensional space Don’t panic Some Basic Background Higher dimensional space Points Distance Spheres Balls Dimensions Most people feel comfortable with dimensions 1, 2 and 3 Mathematicians are comfortable in higher dimensions because they don’t try to visual them; they reason by analogy Dimension Zero Dimension zero consists of a single point There is no notion of distance because the notion is vacuous in this dimension Dimensions: A 1st Attempt D=1 D=3 D=2 D=4 ???? Dimension One: 1 R Can be thought of as an infinite line: There is a one-to-one correspondence between each point on the line and a real number Dimension Two: 2 R Consists of two, mutually orthogonal infinite lines called axes: Dimension Two Each point corresponds to an ordered pair of real numbers: (x1 , x2) The first real number, x1 , corresponds to a point on the horizontal axis while the second number, x2 , corresponds to a point on the vertical axis These values are the (signed) distances from the origin (point where the axes meet) on each axis Dimension Three: 3 R Consists of 3 mutually orthogonal axes Dimension Three Each point corresponds to an ordered triple of real numbers: (x1 , x2, x3) These values correspond to the (signed) distances from the origin on each of the three axes Higher Dimensions We continue by analogy Each point in Rn is just an ordered n-tuple of real numbers:(x1, x2, x3, x4, …. , xn) Thus, your Social Security Number could be considered to be a point in R9 Your phone number (with area code) could be considered to be a point in R10 Distance in 1 R The distance between two points, x and y, is defined to be d(x , y) = | x y | ( x y ) 2 Distance in 2 R The distance between two points, x=(x1,x2) and y =(y1,y2) is defined to be d(x , y) = ( x1 y1 ) ( x2 y2 ) 2 2 Distance in 3 R For two points, x and y, in R3 we define d(x , y) = ( x1 y1 ) ( x2 y2 ) ( x3 y3 ) 2 2 2 Distance in n R Generalizing, we define the distance between two points in Rn to be the square root of the sum of the squares of the differences of the coordinates. Thus d(x , y) = ( x1 y1 ) .... ( xn yn ) 2 2 Spheres Now that we have, in each dimension, an understanding of what points are and a formula for the distance between them, we have the necessary ingredients for spheres n-Spheres An n-sphere, denoted by Sn, is an n-dimensional subset of Rn+1 An n-sphere of radius r with center at a fixed point, c, is just the set of points, x, in Rn+1 which are distance r from c. Thus: Sn = {x in Rn+1 | d(x , c) = r} The 0-sphere: 0 S S0 is a subset of R1 S0 consists of just 2 points on the real line, namely the point r units to the left of c and the point r units to the right of c The 1-sphere: 1 S S1 is a subset of R2 It is precisely those points in the plane some fixed distance, r, from a given center, c S1 is just the familiar circle: The 2-sphere: 2 S S2 is a subset of R3 It is precisely those points in space some fixed distance, r, from a given center, c S2 is just the familiar sphere The 3-Sphere: 3 S How are we to picture this? Once again, we resort to reasoning by analogy This will require us to compile a short list of properties of all spheres. This will facilitate generalization We will also need to introduce the notion of an n-Ball The n-Ball: n B The n-ball is an n-dimensional subset of Rn consisting of all points within distance r of some fixed point, c Bn = {x in Rn | d(x , c) ≤ r} ( Compare Sn = {x in Rn+1 | d(x , c) = r} ) The 1-Ball: 1 B B1 is just a line segment connecting the 2 points that make up S0 Thus, the boundary of B1 is S0 The 2-Ball: 2 B B2 is just the familiar disk The boundary of B2 is S1, the circle The 3-Ball: 3 B B3 is just a solid 3-dimensional ball Its boundary is the 2-sphere, S2 Balls and Spheres Thus, we see that the 0-Sphere is the boundary of the 1-Ball, the 1-Sphere is the boundary of the 2-Ball and the 2-Sphere is the boundary of the 3-Ball In general, in each dimension, Sn is the boundary of Bn+1 Bn+1 consists of Sn together with its interior A Property of 1-Spheres S1 can be thought of as infinitely many copies of S0 with the pair of points (starting at the north pole as a single point – a degenerate S0) getting farther and farther apart until they reach a maximum distance (at the equator), thereafter getting increasingly close until they come together at the south pole (again a degenerate S0) 1 S A Property of 2-Spheres S2 can be thought of as infinitely many copies of S1 with the circles (starting at the north pole) getting bigger and bigger until they reach a maximum radius (at the equator), thereafter getting increasingly small until they shrink to a point at the south pole 2 S Other Properties of 2 S The circumference of a circle with radius R on the surface of S2 is less than 2πR The maximal circumference for the circle is reached when its radius is ¼ of the circumference of the 2sphere (i.e. at the equator) When the radius of the circle reaches ½ the circumference of S2, its circumference becomes zero Circle in Space vs. Circle on Sphere The Earth as an Example Approx. Dist. From N.P. (R) 70 1645 6300 10955 12,600 Lat. Approx. Circum. 89N 399.6 66.5N 10,016 Eqtr 25,120 66.5S 10,016 S.P. 0 2πR 440 10,330 39,564 68,797 79,128 A Property of All Spheres Each Sn is made up of infinitely many copies of Sn-1. They start off as a degenerate sphere (a single point) at the north pole, increase in size until they reach a maximal radius at the equator of the Sn that contains them, thereafter shrinking until they, once again, become degenerate at the south pole Another Property of Spheres S1 can be thought of as two copies of B1 glued together at their common boundary which is just S0 S2 can be thought of as two copies of B2 glued together at their common boundary which is just S1 Sn can be thought of as two copies of Bn glued together at their common boundary which is just Sn-1 Decomposing Spheres Thus we have seen two different decompositions of Sn One consists of infinitely many copies of Sn-1 The other is comprised of two copies of Bn glued together along their common boundary which is Sn-1 The Hypersphere I Using the first of these decompositions, we can view the hypersphere, S3, as infinitely many nested copies of S2 The Hypersphere II Using the second of these decompositions, we can view the hypersphere, S3, as two copies of B3 glued together by identifying corresponding points on their common boundary, S2 More Properties of Spheres Start at any point on a sphere and walk in any direction; you will eventually end up where you started If you are standing at any point on a sphere and I am standing at the antipodal point, then any step you take will be a step in my direction How Does This Relate to the Universe? Putting all this together, leads one to the conclusion (at least naively) that the Universe is a hypersphere To reach this conclusion, we must assemble a few empirically determined and generally accepted facts about the Universe The Big Bang It is generally agreed that the Universe began with the Big Bang some 14 billion years ago The Universe has been expanding ever since Thus, no two galaxies can be farther apart than 14 billion light years (1 light year 6 trillion miles) Sphere of Stars Looking out into space, we are looking back in time If we look out in all directions a distance of r miles, we are looking at a 2-sphere of stars all r miles from us If r miles is 1 light year, for example, we are looking at these stars as they were 1 year ago Hubble’s Law All galaxies are receding from each other at a rate which is proportional to their distance apart Current best estimates say that a galaxy that is 1 billion light years away from us is receding at the rate of 1/14 of a light year each year Thus, all stars 1 billion L.Y. from us now must have been where we were 14 billion years ago Hubble II Since the rate of recession is proportional to the distance, all stars 2 billion L.Y. away are receding at a rate of 1/7 of a L.Y. per year Thus, these stars, too, must have been where we were 14 billion years ago Hubble III Continuing in this way, we see that Hubble’s Law implies that all galaxies were at the same point 14 billion years ago The Universe is a Hypersphere As we look out into space, the sphere of stars starts off (relatively) small and gets larger and larger, the farther out we go At some point, these spheres stop growing and start getting smaller Nothing is farther away than 14 billion light years, so the sphere of stars with radius 14 billion L.Y. is a point Inescapable conclusion: the Universe is a Hypersphere! The Edge of the Universe Note that this description solves a major paradox The Big Bang together with Hubble’s Law imply that the Universe is finite What, then, is outside it? As S3, like all spheres, has no boundary, there is nothing outside What Now? Either the Universe will go on expanding forever or, if gravitational pull becomes sufficient, it will eventually stop expanding and start contracting In the first case, the Universe will grow to be larger and larger hyperspheres In the second case, the Universe must be an S4: i.e. a hyper-hypersphere! Further Reading Poetry of the Universe by Robert Osserman The Shape of Space by Jeffrey Weeks