#mathexplainsworld

I think the universe is pure geometry - basically, a beautiful shape twisting around and dancing over space-time. - Antony Garrett Lisi

Last time we discussed the shape of the universe as a Torus. There could be another type of shape: the 3-Sphere. How can we imagine such a shape? Well, it's just like the 3-Torus, our mathematical object is 4-dimensional. To get some intuition, we will start with lower dimensions.

We imagine a 1-Sphere, i.e. just a circle. We call it S^1. This is the 1-dimensional universe of a 1-dimensional creature which might be a dot. Our one dimensional friend is called "Karl" (obviously Karl is a German guy).

If he moves around in his 1-dimensional universe, he can just go forward or backwards. If he goes in the direction of the green arrows, he will get back, where he started. This phenomenon will also occur in higher dimensions.

Now we go one dimension higher. In the following step we will construct a 2-sphere. This construction will help us getting an intuition of the 3-sphere and construct a new universe for Karl who has grown into a 2-dimensional creature.

We take two copies of a disc (see picture). In the blue and the red disc D now the 1-dimensional sphere has become the boundary of a 2-dimensional ball. If we buckle the 2-dimensional balls to hemi-spheres and glue them at their boundaries, we get a 2-dimensional sphere. Now we look at Karl. He now goes along the green line (Greenwich) starting at the north pole, changing the hemi-sphere, going to the south pole, one more time changing the hemi-sphere and gets back, where he started, if he goes long enough straight forward. Recognize that the north and the south pole are the middle points of the previous 2-dimensional balls. Let's consider Karl's field of view. He only knows his 2-dimensional world and might think that he lives on a plane and not on the surface of something 3-dimensional, a 3-ball (this is, what we called locally Euclidian last time). Indeed this is the way mankind imagined for a long time the shape of the earth we are living on. Sailors even feared to fall down at the end of the earth, because they thought it was a plane.

Now we do the same one further dimension higher. This is an operation in the 4-th dimension, so we can't really imagine it. But we have to do the same. We just glue the boundaries of two 3-dimensional balls together. The boundaries are 2-Spheres. How to buckle the 3-dimensional balls is not imaginable, but we can call one ball the northern hemi-sphere and one the southern hemi-sphere, like before. The north-pole and south-pole is the middle point of the corresponding hemi-sphere, like in the 3-dimensional case.

Karl has grown into a 3-dimensional creature and learned to drive a rocket. He starts at the north pole of the northern hemi-sphere and goes up to a point at the "equator", i.e. the boundary of the northern hemi-sphere (1). In the southern hemi-sphere he goes to the south pole (2) and arrives at the next point of the "equator" (3). He changes the hemi-sphere and gets back, where he started (4). Like in the 2-dimensional case, he would also recognize his universe as something 3-dimensional, although the shape of the universe is 4-dimensional.*

Many scientists don't prefer this type of universe. They think the universe is flat, i.e. that the parallel axiom of Euclid holds, because the so called inflation-theory postulates it. But the parallel axiom doesn’t hold for a spherical universe, because two parallel lines going straight forward will intersect each other at some point. in such a universe. There are a lot of scientists who also prefer a infinite universe. But is this logical? Here are some questions, you have to wonder: How can such a universe grow infinite in a finite amount of time? Was it finite at the time of the Big Bang? How can an infinite universe expand?

I think this is the last post of our series #mathexplainsworld. There are a lot more topics about this theme, but I think next time I will start some interesting inner-mathematical topics and how mathematics influences parts of our everyday life. Be surprised! ;-)

Source: *Spektrum der Wissenschaft, Sept. 2004, pp. 90-91. More informations: http://www.avgoe.de/StarChild/DOCS/STARCH00/questions/question35.html Last questions: http://www.thenakedscientists.com/forum/index.php?topic=52438.0

"Space: the final frontier. These are the voyages of the starship Enterprise. Its continuing mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no one has gone before."

This is the intro of many episodes of Star Trek. But is the universe infinitely big or is it finite? This question is not really clarified. We will discuss two different topological structures of the universe, the first one in the following post.

Stephen Hawking to Homer: "Your theory of a doughnut shaped universe intrigues me. I may have to steal it." (The Simpsons, episode: "They Saved Lisa's Brain")

A finite space (better: compact topological space) could be a doughnut, in other words a torus. This is the word a mathematician prefers. In particular a doughnut is a 2-Torus. A 2-Torus T = S^1 x S^1 is described as the Cartesian product (this is the "x") of two 1-Spheres S^1, i.e. two circles. We can imagine forming the Cartesian product as the three dimensional figure which is created by going around the first circle with the second one. This is exactly the shape of a doughnut.

The universe is not exactly a doughnut, but it's the 4-dimensional analogon of it, a 3-Torus which is described as S^1 x S^1 x S^1. So it has some properties in common with the 2-Torus, but this comes later...

First we want to see why we should assume such a strange shape. And the answer is: It has something to do with the cosmic microwave background (CMB). There are several reasons to assume, that in the early time of our universe after the so called Big Bang, when it cooled down, protons and electrons reacted to hydrogen atoms which induced "de-coupling" of radiation and matter. So the universe was penetrable for radiation, because there was no scattering at electrons. By the growth of matter, in some parts of the agglomerated matter became more weighty and had a higher force of gravity. This led to a fluctuation of density in the early universe. But now photons had to face gravity in the more dense regions. This is the reason for fluctuation of temperature in the CMB. Normally the average spectrum of the microwave radiation is that of a black body, i.e. 2.725 Kalvin. But because of the reasons before, the radiation became the more colder the more dense the parts of the universe were. The following picture shows the fluctuation of temperature of the CMB:

Frank Steiner and his team of the University of Ulm showed that the torus-universe is compatible with the CMB (for more information: Spektrum der Wissenschaft Jan. 2009, pp. 24-29).*

(Mathematically) reasonable implications of this result are the following. We look at our doughnut, the 2-Torus. If a little 2-dimensional creature would life on the surface of the torus and it would go straight in a forward direction, then it would get to the place, where it started. The reason is described in a simulation I posted some month before (#thefundamentalgroupofatorusisabelian). The simulation shows how to glue a paper, s.t. it forms a torus. In the following picture the opposite sites of the rectangle are colored green and purple.

Our two-dimensional friend could go the red or the blue way to get back where it started. Furthermore our friend would think that his universe might be two dimensional (locally euclidan, in the sense, that it is homeomorphic to the IR^2). Analogously we think our universe is 3-dimensional, because we only have a local point of view. But in reality it could be 4-dimensional, like the 3-Torus. In addition: If our universe was a 3-Torus, we could go by a rocket in one direction and get back where we started! :)

Next time we will discuss the second interesting structure of the universe. See you then!

Source: *Spektrum der Wissenschaft Jan. 2009, pp. 24-29.

In George Orwell's distopian novel "Nineteen Eighty-Four" the author describes a world divided into three pieces by totalitarian or authoritarian states which are always fighting against each other. The people living in the state "Oceania" are under constant surveillance by televisions and have to worship the great "Big Brother" whose existence is not even clear. The protagonist tries to escape from the system to get privatcy, but in the end he becomes a part of the omnipresent system after a brain-washing and loves the "Big Brother". Well, that's a nice story, but why is it a part of the series "math explains world"? Last time we had the Borsuk-Ulam-theorem. It gives an awsome application in this context. :) Let's consider the map of Orwell's distopian world.

There are three countries: Oceania, Eurasia and Eastasia. Now we wounder, if there is a pair of antipodal points, s.t. one of the teritories of the countries includes this pair of points! Further more we wounder, if this is always the case. So is it possible, that there exists a pair of antipodal points, s.t. in any constellation of divisions of the world into three teritories this pair is included in one of the teritories?

The answer is YES!!! :D You can use any division into three pieces, but there is always a pair of antipodal points in one of the three teritories. OK, to be mathematically correct, the division of the 2-Sphere (notation: S^2), which is in our case just the surface of the earth, into three sets has to be, s.t. this sets are closed. This means: If we consider the boundaries of the sets, then their boundaries have to be in each set they're belonging to (see: https://en.wikipedia.org/wiki/Closed_set). Further more there have to be exactly three teritories (not 4 like in the picture). The lemma is the following:

Whenever S^2 is expressed as the union of three closed sets C1 , C2 and C3, then at least one of these sets must contain a pair of antipodal points {x,−x}.*

The question is: Why?

Proof: Let's consider S^2 as a subspace of the IR^3 and let d be the distance from one point of the 2-sphere to an other point on this sphere by going a path on the sphere. This is just the distance you know in real life. If you go from your home to an other city by car and you can drive just the linear distance, then you are not going the direct way, but a way which is determined by the curvature of the earth which we can approximate by a 2-sphere. We now define three distance functions, D_i: S^2 --> IR for i = 1,2,3 mapping a point x to the distance of this point to C_i, i.e. D_i(x) = inf_{y in C_i}(d(x,y)). What is this cruel map describing? The inf (read: Infimum) of something searches for the smallest value of d(x,y) by varying the point y in C_i. These D_i are continuous. Now consider the map f: S^2 --> IR^2 with f(x) = (D_1(x), D_2(x)). We sayed that this lemma comes from the Borsuk-Ulam-theorem. So we will use it here. By the theorem there exists a point x, s.t. f(x) = f(-x), i.e. D_1(x) = D_1(-x) and D_2(x) = D_2(-x). If D_1(x) = D_1(-x) = 0, then x and -x are in C_1. Similarly for D_2. But if they are both D_1(x) and D_2(x) not 0, then x and -x must belong to C_3.

That's the proof. Q.E.D.

Next time we will discuss the structure of the universe. See you then!

*Source: Hatcher, Allen (2001): Algebraic Topology, URL: http://www.math.cornell.edu/~hatcher/

Let's start the topic "math explains world" (M.E.W.) with a result from Algebraic Topology. In its mathematical formulation the theorem looks not very interesting:

If f: S^2 ---> IR^2 is continuous, then there exists a pair of antipodal points x and -x, s.t. f(x) = f(-x), where S^2 denotes the 2-sphere (something which we can imagine as the surface of a ball) and IR^2 is the real number plane.

This theorem is the so-called Borsuk-Ulam-theorem. Stanisław Marcin Ulam, who also was involved in the early development of nuclear weapons, assumed that this theorem might be correct. Karol Borsuk, a Polish mathematician, proved 1933 the theorem.

But why should we be interested in such a boring thing from the last century? Well, I can reassure you, it's not that boring. ;-)

The application of this theorem implies a result which is really mindblowing. Let us consider temperature and pressure on earth. We can imagine the S^2 to be the surface of the earth. Antipodal points on earth are just points which lie on opposite sides of the S^2, like the north and south pole. There is a nice antipodal map on http://www.antipodesmap.com/, which helps to find antipodal points on earth. For example an antipodal point to Berlin would be somewhere in the south-west of New Zealand.

The map f could be a function that assigns points of the earth to a pair of two values, temperature and presssure. If we can assume that this map is continuous, then the theorem says that there is always a pair of antipodal points on earth, which has the same temperature and pressure. Think about this! o.O

But why should this statement true? I will explain it.

!!! WARNING: The following section is written for freaks and nerds who are really interested in the idea of the proof. If you are not of this type, then you should skip the following. ;-) !!!

I will give only the idea of the proof by several pictures. This is not really mathematically correct, but intuitive. Proof: In Algebraic Topology the term of covering spaces is connected with this theme (it's also connected with the term of the fundamental group). What is a covering space? We only have to know what is the covering space of the S^1 (unit circle or 1-sphere) to proof the theorem. The so-called universal covering of the S^1 is a map from p: IR ---> S^1 which has the following properties and is shown in the modified picture from Hatcher's Algebraic Topology book (2001, p. 29).

A small (open) neighbourhood U (in red) of an arbitrary point of the S^1 has a bunch of corresponding (open) neighbourhoods in IR (also in red), s.t. we can map them to the first neighbourhood by p and p induces a homeomorphism from one of the neighbourhoods to U, this means that the neighbourhoods are really “similar” to the neighbourhood U. In the following proof we can assume that p(t) = e^(2 pi i t) (cf. https://en.wikipedia.org/wiki/Euler%27s_formula). The proof follows Hatcher’s proof (2001, p. 32,33). Now we are well prepared for the proof. We prove the theorem by assuming the contrary and leading it to a contradiction (https://en.wikipedia.org/wiki/Proof_by_contradiction).

Let's suppose that such a map exists, but has no point x, s.t. f(x) = f(-x). Then we can define the function g: S^2 ---> S^1 with g(x) = (f(x) - f(-x))/|f(x) - f(-x)|. We can calculate just by inserting -x in the formular, that g(-x) = g(x). We call the equator loop (cf. picture down) y: [0,1] ---> S^2, where we choose x_0 as the basepoint of the loop, i.e. 0 and 1 get mapped to x_0 in S^2.

This is indeed a loop, because it starts and ends at the same basepoint x_0. The composition g*y given by g*y(x) = g(y(x)) is a map from [0,1] to S^1 which starts and ends at the same point, hence a loop in S^1.

Since we know that a neighbourhood of a point in S^1 has a bunch of neighbourhoods in IR by the property of the covering space, we can construct a unique path in IR by lifting pieces of the loop g*y in S^1 by this lifting-property of the covering space to IR. This path is unique, if we fix basepoints, i.e. we map 0 in IR to 1 in S^1 by p. (Rmk.: Consider that 1 is in S^1, because we are working in the complex plane. If you are not firm working with complex numbers, just read (1,0) instead of 1 and consider the real number plane IR^2.) We can construct this lifting of the loop in finitely many steps (by compactedness). We call this lifting G*Y which is just a path from [0,1] to IR. So we have G*Y: [0,1] ---> IR. Now we can compare loops in S^1 with paths in IR. An example of the path G*Y in IR (in blue) is given by the following picture:

The corresponding loop at the base S^1 is also coloured blue (modified picture from Hatcher, 2001, p. 29).

Let's consider the difference d: [0,1/2] ---> IR given by

d(t) = G*Y(t) - G*Y(t + 1/2) = G(Y(t)) - G(Y(t+ 1/2)).

This difference is just the difference of antipodal points in S^1 in terms of the difference of the corresponding paths in IR. We know, if we go once around the circle S^1, i.e. we have a loop starting and ending at 1 in S^1 (see Rmk. before), then we go from 0 to 1 in IR, like we see it in the picture. If we go 1/2 of the way of the loop g*y in S^1, then we have two antipodal points z and -z in S^1. Now we consider p: IR ---> S^1 given by p(t) = e^{2 pi i t}. Any two antipodal points z and -z in S^1 give integers x in p^{-1}(z) and x' in p^{-1}(-z) which differ just by a number of the form "n + 1/2" where n is an integer. Looking at the preimage p^{-1}(z) or p^{-1}(-z) means just looking at the bunch of points in IR above the points z and -z. But being something of the form "n + 1/2 = (2n + 1)/2 = odd integer/2" means just that the difference

d(t) = G*Y(t) - G*Y(t + 1/2) = q/2, where q is an odd integer.

But then

    G*Y(1) - 0 = G*Y(1) - G*Y(0) = G*Y(1) - G*Y(1/2) + G*Y(1/2) - G*Y(0)     = (G*Y(1) - G*Y(1/2)) + (G*Y(1/2) - G*Y(0)) = q/2 + q/2 = q.

So G*Y(1) = q not equal to 0 and this means that G*Y starts in 0 and ends in q. But this is a contradiction, because y is homotopic to the constant loop c that stays for its whole interval [0,1] at one point (i.e. c(t) = x_0), i.e. we can deform y in S^2 continuously to a point, like in this picture:

But then g*y is homotopic to a constant loop, hence G*Y, too. But we saw that G*Y doesn't stay at one point, but goes from 0 to q. So it can't be the constant loop. CONTRADICTION!

Q.E.D.

Next time we will look at an interesting consequence of the theorem, which has something to do with a division of the world. But this will not be written before next month. See you then!

Sources:

First picture: https://de.wikipedia.org/wiki/Karol_Borsuk#/media/File:Borsuk.jpeg, https://de.wikipedia.org/wiki/Stanis%C5%82aw_Marcin_Ulam#/media/File:Stanislaw_Ulam_ID_badge.png

Second picture: https://de.wikipedia.org/wiki/Erde#/media/File:The_Earth_seen_from_Apollo_17.jpg

Third and fifth picture: Modified pictures from