Flat Universes

-24- Flat Universes

What is the shape of the universe? For a long time, people thought that it had to be the infinite three-dimensional space described by Euclid and therefore called it Euclidean 3-space. Then a few people realized that that wasn’t the only possibility. It might, for instance, be like the surface of a sphere in four-dimensional space, which nowa- days is called three-dimensional spherical space or just the 3-sphere. Then, it would be “finite but unbounded,” meaning that the total volume would be finite even though there would be no bounding walls. In that case, space would be curved, in a three-dimensional analog to the two-dimensional surface of the Earth, which has finite total area even though there is no edge for people to fall off. Still more recently, physicists have become aware that the universe might not only be finite and unbounded, but also exactly flat, in the sense that the local geometry is exactly Euclidean. It might, for instance, be the three-dimensional analog of a torus, which we shall call a torocosm. If you’ve played computer games such as Asteroids, you won’t find it too hard to understand. In that game, an asteroid or space ship that goes off one edge of the screen reappears from the opposite edge.

Flying around in an Asteroids universe: As the ship exits one side of the picture of its universe, it reappears on the other side. From the ship’s perspective, the universe is unbroken and continuous. This universe is ﬁnite, yet unbounded, and is topologically a torus.

(opposite page) The Hexacosm.

355

This chapter is about all the ways the universe might be ﬂat. First we’ll think about the two-dimensional analogs.

Compact Platycosms

If you lived in a small, ﬂat two-dimensional space like the Aster- oids universe, what would you see? Because light rays travel like everything else in that world, you’d see lots of images of any given thing.

Sight in the Asteroids universe: Light would travel like any other thing, wrapping around the universe. From the point of view of the ship, it would seem as if the light travelled across consecutive copies of the universe, and you would see lots of images of any given thing...includingyourself! Thesecopiesformtheuniversal cover of the torus, which is a Eu- clidean plane.

Downloaded by [University of Bergen Library] at 04:56 26 October 2016 The entire set of images you’d see would be related to each other by a lattice of translations. Geometrically, these translations form a copy of the plane group we call ◦, and the flat torus is just the orbifold for ◦. Thus, flat universes are described by space groups, in that they describe ways in which one might return to one’s starting place after “circumnavigating the universe.” We shall obtain the ten most in- teresting flat universes from ten of the space groups described in the next chapter. Technically these are the ten closed flat 3-manifolds.

Torocosms

The above figure represents the appearance of a two-dimensional Downloaded by [University of Bergen Library] at 04:56 26 October 2016 torus. A (flat) torocosm (also called a “3-torus”) is its three-dimensional analog. It’s like a room without walls with the strange prop- erty that if you leave from one side you come in at the opposite side. The figure on the next page suggests what the universe would look like if it were a sufficiently small torocosm. Geometrically, a torocosm is the orbifold of the space group generated by three inde- pendent translations. These need not be mutually perpendicular— there is a torocosm based on any parallelepiped.

The Klein Bottle as a Universe Games like Asteroids can also be played on other surfaces, for ex- ample, on a ﬂat Klein bottle. Downloaded by [University of Bergen Library] at 04:56 26 October 2016

Flying around in a Klein-bottle universe: Now, whatever leaves the screen on the top will re- enter it on the bottom, but reverse. However, what leaves at the right returns in the way it did in the original game.

Geometrically, this twisted Asteroids world is of course a Klein bottle, the orbifold of the plane group ××. The torus and Klein bottle are the only closed, ﬂat 2-manifolds. Why is this? Because the universal cover argument shows that any such space must be the

orbifold of one of the 17 plane groups, and each of the 15 other than ◦ and ×× has either a boundary (if its name involves ∗)oracone point, at which the local geometry will not be Euclidean.

The Other Platycosms The three-dimensional analogs of the torus and Klein bottle we call platycosms, meaning “ﬂat universes.” More precisely, they are the compact platycosms, meaning that they have ﬁnite volume. There are ten such spaces [8], whose individual names are

torocosm, dicosm, tricosm, tetracosm, hexacosm

(these are collectively called the helicosms)and

didicosm, positive and negative amphicosms, and positive and negative amphidicosms.

We describe them by pictures. We have already seen the hexacosm (on page 354) and the torocosm (previous page), which is the helicosm with N = 1. In another helicosm, say the N-cosm, the images of an object fall into parallel layers and one moves up a layer by rotating it through 360◦/N . The images of any one layer are related to each other by a lattice of translations that has this rotation as a symmetry—for instance, it must be a square lattice for the tetracosm. So, in an N-cosm, one’s images face in just N diﬀerent directions, which are related by a cyclic group of N rotations. They all have Downloaded by [University of Bergen Library] at 04:56 26 October 2016 the same handedness: if you were to wave your right hand, each of your images would wave its right hand, in contrast with what would happen in a mirror, where your image would wave its left hand. This is described technically by saying that the helicosms are all orientable manifolds. The only other orientable platycosm is the didicosm,inwhich one’s images are in four diﬀerent orientations. They can be head up or feet up and face either left or right, but are all of the same handedness.

The dicosm. The tricosm. Downloaded by [University of Bergen Library] at 04:56 26 October 2016

The tetracosm.

We shall show the remaining platycosms with ships as before, but for clarity we will also use blocks that have the letters b, d, p,or q on them; each kind of block stacks top and bottom, right and left, to form layers, but the letters are always on the tops of the blocks.

The didicosm. Downloaded by [University of Bergen Library] at 04:56 26 October 2016

The amphicosms and amphidicosms are nonorientable, meaning that half the images of an object have one handedness and half the other. So, if you were to wave your right hand in one of these four platycosms, half of your images would wave their right hands, the other half their left hands. It is hard to illustrate these correctly (the four pictures in [31] represent only two manifolds). The cockpits of the ships in our pictures are always on top.

The amphicosms and amphidicosms: from left to right, +a1, −a1, +a2, and −a2.

The positive amphicosm +a1. The negative amphicosm -a1. Downloaded by [University of Bergen Library] at 04:56 26 October 2016

The positive amphidicosm +a2. The negative amphidicosm -a2.

The reason that these ten platycosms are the only compact ones is much the same as the reason that the torus and Klein bottle are the only compact two-dimensional ﬂat surfaces, namely that the corresponding space groups are the only ones in which there is no ﬁxed point for any nonidentical element. This is easily checked by hand from the list in the next chapter. The 35 prime space groups have lines that are analogous to order-3 cone points in a plane group and so need not be considered. The ten space groups that arise are those called

c1 (◦), c2 (21212121)=(×¯ ×¯ ), c3 (313131), c4 (414121), c6 (613121), c22 (2121×¯ ), +a1 (¯◦0)=(∗:∗:) = (××0), -a1 (¯◦1)=(∗:×)=(××1), +a2 (2121∗ :) = (*¯ : *¯ :) = (××¯ 0), -a2 (2121×:) = (∗ : ×¯ )=(××¯ 1),

in the next chapter. Some of these groups have more than one name as a consequence of the alias problem discussed there. We summarize the names and notations for the ten compact platycosms below.

helicosms didicosm orientable c1 c2 c3 c4 c6 c22 +a1 -a1 +a2 -a2 nonorientable amphicosms amphidicosms Downloaded by [University of Bergen Library] at 04:56 26 October 2016

Infinite Platycosms The compact platycosms are the only ones of finite volume, but there are eight more infinite, or noncompact, ones. The infinite two- dimensional flat surfaces are the Euclidean plane, the cylinder, and the Möbius cylinder. The following figures suggest what it would look like to live in these spaces. The infinite platycosms are named in the captions.

∼ Circular Prospace = circle x Euclidean plane. Circular Mospace.

∼ To ro id al P ro space = torus x Euclidean line. Toroidal Mospace.

∼ Kleinian Prospace = Klein bottle x Euclidean Orientable Kleinian Mospace. line. Downloaded by [University of Bergen Library] at 04:56 26 October 2016

Nonorientable Kleinian Mospace. Euclidean three-space.

Here, “Prospace” abbreviates “product space” because those three spaces are the direct products indicated. “Mospace” means “M¨obius space”: the four Mospaces are related to the Prospaces in the way that the M¨obius cylinder relates to the ordinary one.

Where Are We? This chapter has described the three-dimensional ﬂat universes (platycosms). The enumeration of the ten compact ones was deduced from that of the 184 composite space groups in the next chapter. Downloaded by [University of Bergen Library] at 04:56 26 October 2016