Dr. Lorxus @lorxus@vulpine.club

Take the resulting circle and push it outside the sphere – it is a topological object, and thus more stretchable than rubber and less substantial than mist. Swing it directly through the sphere if you want, or let it grow past infinity like we’re living in S^3 instead of R^3, or simply shrink it to a point and grow it back out again. (Our secret’s safe. We’re not undergrads anymore; we can do as we like, as long as we know what to avoid, and why.) Invoke van Kampen’s theorem. Shade the circle and a little patch around it (within the space) in green and shade the sphere and a little patch around it (in the space) in red. The overlap is a pure yellow – these mix like light, not like paint. Now what flavors of loops can live in this space stably is more clear: some combination of loops through the circle and loops in the sphere. Pull a blackboard Z out of the circle and a 1 from the sphere and an amalgamated free product machine from the overlap, and combine; we get Z.

What are the loops in this space? Draw a circle in electric blue on the surface of the sphere and watch it contract itself down to a point. Draw a larger one and watch it do the same. Let the electricity flicker instead to the center of the sphere and let it loop around the sphere and through the rest of the diameter. Watch the trail of light pause as it joins back to itself at the center, flowing subtly clockwise. Watch it try to shrink back down to a point with a tugging sensation, and fail. Good. Watch it keep looping, as many times as there are natural numbers, then quickly rewind and shoot past zero, turning a reddish-pink as it flows counterclockwise. This is a good start, but we’ll need more than just guesses. (All of this has taken maybe a few seconds, easier done than said.)

Back to this object’s irritating geometricity, its proud rigidity. This won’t do. Give yourself more power over it, change it into a topological object. There are still constraints here, but they are weaker. Now we can slide the ends of the diameter together, both to the north pole, with a faint scraping noise. No need to touch them to do this; simply will it to be so.

A survey on mathematical reasoning prompted me to describe a visualization of a mathematical object:

Let's calculate the fundamental group of a 2-sphere together with one of its diameters. See the sphere, a deep blue, shiny like enamel and just as opaque. See its opacity fade to translucence, except for the part of the equator facing us; the back becomes dashed instead, or perhaps only dimly visible through the sphere. The diameter attached to its innards is revealed now; observe its two points of connection, thickened in black where they meet the equator, each maybe the size of a peppercorn to the sphere's basketball and the diameter's guitar string.

While this is still a purely geometric object, it is rigid: the sphere has a texture like marble. If you tapped on it, it would make a dull, hollow sound. Pluck the diameter, with great effort, and watch it rapidly revert to straightness.

@Kyresti forgot to ask you - what dopamine precursors are you taking? do they seem to be doing anything for you?