What shape is our universe

For me one of the central themes of The Library of Babel and Borges work in general is an attempt to understand the shape of the space that we live in. One of the sentences in The Library of Babel that stuck out for me: The Library is a sphere whose exact centre is any one of the hexagons and whose circumference is inaccessible. This, Borges declares, is a classic dictum about the library. It occurs at the end of the second paragraph. What I discovered yesterday whilst reading around for my History of Maths series is that this actually picks up on a quote from Pascal:Nature is an infinite sphere, whose centre is everywhere and whose circumference is nowhere. It turns out that Pascal's original version had the extra word FEARFUL sphere.

In the essays in my addition of Labyrinths there is an essay by Borges dedicated to The Fearful Sphere of Pascal. It traces back how the sphere has been an important symbol for many philosophers. Parmenides: The divine being is like the mass of a well-rounded sphere whose force is constant from the centre in any direction. The music of the spheres. Kepler's model of the universe as interconnected spheres.

But what is an infinite sphere? A sphere is a surface whose points are equidistant from one fixed point. If that distance is infinite then indeed one can take any internal point as a centre. But the interesting thing is that there is something rather unsatisfying about an infinite sphere because that surface - the circumference - is unattainble, unreachable. Even if you did reach it, the curvature of the sphere is such that it is an essentially flat surface. Think of a circle that gets bigger and bigger. The curvature of the circle gets flatter and flatter till at infinity it is really a straight line.

I think most people's perception of the shape of the universe is something like a huge infinite ball with us at the centre.

Interestingly Plato in his text Timaeus associates the dodecahedron rather than a sphere with the shape of the cosmos. The Platonic solids are the building blocks of his atomic theory:
the tetrahedron- fire
the cube - earth
the octahedron - air
the icosahedron (the most spherical of all the Platonic solids)- water.
and finally the dodecahedron is left as the shape which captures the universe.

These shapes are made from triangles, squares and pentagons. Once one gets to a hexagon, you find that the only way to put them together makes an infinite flat surface, like a honeycomb. This is if you are in a Euclidean universe. I'm intrigued to see whether you can get interesting shapes with hexagonal faces in non-Euclidean space. I'll try to chase this up.

But both the universe and the library are ultimately not believed to be infinite but rather finite but unbounded. For example the surface of a two dimensional sphere (like the surface of our earth) is finite but unbounded. Perhaps we live on the three dimensional surface of a four-dimensional sphere. Then we would actually be living on the circumference and the centre would be unattainable.

For me, the exploration of the library leads ultimately to a way to discover what sort of surface we are a living on - a 3-dimensional surface of a 4-dimensional shape. The building block of this surface is the hexagon (or rather a hexagonal box) and the way these are put together, the way they are connected gives us clues to the overall shape, a shape we can never see in its entirety. I'll post later my thoughts on what this shape might be.

This has been the fun part for me of reading the Library of Babel. It's been like a puzzle. Taking the clues that Borges gives us and then trying to find a shape that fits the clues. A lot of geometry recently has been about finding ways to describe a shape from its internal structure. How can I tell whether I'm on a sphere or a torus (a ball or a bagel) just from internal features of the surface.

By the way, I love the idea of the floor being a chalk surface that we can draw on. Also love our name: The Nineteenth Step.

Marcus.