In the secret space of dreams
Where I dreaming lay amazed
— Attics of My Life, Grateful Dead

How many spaces are possible! Outer space, inner space, head space, physical space, metaphysical space, green space, dead space, open space, living space, crawl space, breathing space. We could go on, but let us not waste space.

Why does space get so much space? Maybe space, which begs to be filled, can be filled in so many different ways. And when there’s plenty of space to go around, as there often is, there’s room for tons of interesting variety and structure.

Mathematics has a field day with space. In this section we’ll look at a few well known two-dimensional spaces called two-manifolds or, in more common parlance, surfaces. We’ll also look at a few three-manifolds, analogue of surface but one dimension higher.¹

I highly recommend two books: The Shape of Space by Jeffrey Weeks, and The Symmetries of Things by John Conway, Heidi Burgiel, and Chaim Goodman-Strauss. These books go into the themes of this section, and well beyond, with far greater depth and clarity than I ever could.