My name is Chris Lambie-Hanson. I am a mathematician, currently situated at the Department of Mathematics and Applied Mathematics of Virginia Commonwealth University, working primarily in the field of set theory.

Here, I approach infinity from all directions.

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In your post February 2017

Dante, Einstein, and the Shape of the World

describing a journey in 3-sphere you say ‘For a while, these successive 2-spheres have larger and larger radii, as is natural. Eventually, of course, they will start to shrink, contracting to a point before expanding and contracting as we return to our starting point at the Earth’s core’.

I don’t understand. Surely you keep going further from the earths core. At what point do you turn around and return? Sorry I’m clearly missing something and would be extremely pleased if you could explain.

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As I wrote in the post, it might help to think about things one dimension down, which is easier for us to visualize. Imagine that you’re walking on the surface of the earth (a 2-sphere, roughly), from the North Pole to the South Pole, and suppose that there are circles (i.e., 1-spheres) drawn on the surface of the earth to represent the lines of latitude. Initially, these circles of latitude have larger and larger radius. But when you reach the equator, even though you continue getting further away from the North Pole on the surface of the earth, the circles of latitude begin getting smaller until they contract to a single point at the South Pole. If you keep going, you start returning to the North Pole. This happens despite the fact that, if you were to try to “flatten” the surface of the earth with the North Pole at the center, these circles would appear to get larger and larger. This highlights a difference between the geometry of a 2-sphere and the geometry of flat 2-dimensional Euclidean space.

The situation in Dante’s depiction of the universe is similar, but one dimension higher and thus harder (perhaps impossible) for us to really intuitively visualize. The “equator” for him, where the concentric spheres begin to contract, is the edge of the Primum Mobile. If one were to keep going to the point at which these spheres contract to a point, one would reach the spot where Dante situated God. This corresponds to the South Pole in our analogy. And similarly, if one were to continue going in a straight line, one would start to return to one’s starting point on earth. One reason this is hard for us to visualize is that, in our everyday experience, the world appears to be flat 3-dimensional Euclidean space, in which this wouldn’t make sense. But Dante is not depicting the universe as flat 3-dimensional Euclidean space but as a 3-sphere, and, just as a 2-sphere is different from flat 2-dimensional space, a 3-sphere is different from flat 3-dimensional space.

I hope this is helpful! Thanks for reading!

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