The Poincaré Conjecture (1904)

What is the shape of the universe?

Let’s start off easy. We know the shape of the earth. It is more or less spherical. Next, our galaxy, the Milky Way. Somewhat surprisingly to Edwin Hubble (1889-1953) who discovered it, it is barred spiral shaped, i.e. the shape of a disk with spiral arms. What about the observable universe? Is it spherical? It certainly seems to be, as it is expanding outwards and light is more redshifted the further we look. What about the global universe, beyond the horizon of what is observable to us?

The answer is, we do not know. We can conjecture. It may be finite or infinite, with or without boundary, with or without curvature, simply connected like a sphere, or multiply connected like a donut. What we do know, is that it appears to be expanding. To what? We do not know. But we can conjecture.

Introduction

The shape of the universe as it was, is now and may become in the future is very hard for us to discern empirically. Einstein helped us somewhat by showing us that matter and energy (three-dim…