Toroid’s surface has peculiar properties compared to a much simpler shape such as a sphere.
On a spherical surface you may start travelling in any direction and end up where you started. This property is valid for any point on the surface.
A spherical surface is a finite two-dimensional surface with no boundaries. It is a finite surface hence you end up reaching the same point. It has no borders or sharp edges that you may fall off.
As a result a spherical surface is a useful model to study a universe with no boundaries but a finite shape like ours.
A spherical surface has also constant curvature. Nowhere on it there is a point that sits on an area that is more or less curved than any other point on the sphere.
On a toroid on the other hand you may find points across its surface sitting on different curvatures. A point on its outer rim is on a surface with positive curvature, where a point on the inner rim is on a surface with negative curvature.
A simplified modelling can explain this “curvature business” that is the essence of Einstein’s general relativity.
You may construct a toroid by bending a cylinder along axis and joining its ends.
Imagine you make a toroid from a rubber cylinder. You will see that the outer surface of the toroid will stretch, and the inner surface will squash forming wrinkles.
Take a rubble rectangular slab, and on its surface paint little circles with identical diameter placed in equal distance from each other. Form a cylinder from the slab by rolling it so that the circles we drew can be viewed from outside. We should observe that despite introducing a positive curvature, the sizes of circles and the distance between them remain the same. There is no stretching or squashing; the curvature is positive but constant.
In contrast when we form a toroid using a rubber cylinder by bending it along its axis and joining both ends, we should observe that the circles on toroid’s outer surface grow in size and the distance between them increases too. The circles on toroid’s inner surface should shrink in size and the distance between them should become shorter. This is because we have introduced variable curvature changing from positive (stretching) to negative (squashing) from outer to inner rim respectively.
Imagine the toroid represents a certain space-time configuration.
It is possible a light beam to travel from circle A to its neighbour B on the inner surface in shorter time than between neighbouring circles on the outer surface. Space-time is clearly stretched on the outer rim, shrunk on the inner one.
A black hole is like a toroid that its inner hole is infinitely small hence the circles you drew on its inner surface collapse onto each other. There is no way to identify or differentiate those circles from one another, the information about them are sucked by black hole’s event horizon.
It is fascinating to think that there may be oddly shaped universes with multitude of curvatures. A universe that its space-time properties are shaped like a toroid is possible. But other weirder shapes too. Some of them may even have collapsed regions where curvature is transformed in odd directions that we may not easily imagine their shape in our Euclidian minds.
The reason we could visualise and construct a toroid is because we may construct it from Euclidian shapes that we are familiar with such as a cylinder. However we should note that we might not do so if we could not stretch and squash the outer and inner surfaces respectively. You cannot construct a non-Euclidian shape without introducing variable curvature.
A toroid is a non-Euclidian shape whereas a cylinder is a Euclidian one. Our education system has given emphasis on shapes with Euclidian geometry perhaps because there are economical benefits of realising them. Euclidian geometry has given us ability to make useful approximations. We can build pipes by modelling them as cylinders for instance.
However studying non-Euclidian geometry such as toroids, can be crucial in modelling and understanding the cosmos.
