© Sigma Xi, The Scientific Research Societycontact perms@amsci.org

2001 September–October Front

We understand much about our planet,our galaxy and our universe. But a funda-mental mystery remains. What is the topo-logical shape of the universe?

Recent observations in astronomy arebeginning to limit the wide range of possi-bilities. One type of shape, called a Euclid-ean 3-manifold, has arisen as a prime can-didate. Amazingly, there are only 18Euclidean 3-manifolds and, of these, onlyten are probable candidates for the uni-verse. One of these, the 3-Torus, is the fo-cus of this insert.

Cut out the figure to the right and fold it intothe cube shown above. This is how a 3-Toruswould look if a green worm were sitting in themiddle of it; copies of the worm appear, contin-uing to infinity in every direction. This shape,on a much larger scale, may be the shape of ouruniverse.

One way to picture a 2-dimensionaltorus is to start with a square. Pretend thesquare is a piece of paper, and construct acylinder by gluing the left side of thesquare to the right side. The top and bot-tom sides of the paper have become thecircles on the top and bottom of the cylin-der. Gluing those two circles together cre-ates a torus (see below).

The 3-torus is the generalization of thetorus in a higher dimension. Instead ofgluing together the opposite edges of asquare, the opposite faces of a cube arejoined. In the 3-torus, every point on a faceof the cube is glued to the corresponding

The Shape of the Universe: One Possibility

Web insert for Colin Adams’s and Joey Lawrence’sThe Shape of the Universe: Ten Possibilities,American Scientist September–October 2001

cut

© Sigma Xi, The Scientific Research Societycontact perms@amsci.org

2001 September–October Back

point at exactly the same spot on the op-posite face. On the diagram to the rightthe blue face is glued to the other blueface, orange to orange, and front to back.For illustrative purposes, worms travelingthrough the glued faces demonstrate theabsence of rotation between the faces.

If you, or a green worm, were somehowin this 3-manifold and looked forward,you would see the back of your own head.You would see copies of yourself in eachface of the cube: forward, backward, left,right, above and below. Past these copies,other copies would be visible—copies asfar as the eye could see. Standing in a 3-torus and looking out is similar to stand-ing in a fun-house room of mirrors. But inthe 3-torus, the images are never reversed.Cut, fold and glue the figure on the tophalf of this page to better understand whatstanding in a 3-torus would feel like.

cut