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Comment by jugg1es | Hacker News Reader

How can you have a 3d space with finite volume and no boundary?

In 2D the most intuitive way is the "Asteroids topology": when you exit from one edge you reappear from the opposite one. This space is flat, finite, and without a boundary. In mathematical terms it’s what you get when you take a rectangle and identify the left edge with the right and the top edge with the bottom one. It is also topologically equivalent to the (surface of) a torus, which is also flat and wraps around the same way. (Note that the embedding of a toroidal surface in three dimensions is curved, but the 2D space itself is flat: the angles of every triangle add up to exactly 180°.)

In 3D, just generalize from the 2D case.

> the 2D space itself is flat

It's not that simple. "Flat" is not a topological property, it's a metrical property. The correct statement is not that "the" 2D torus is flat, but that it is possible to put a flat metric on the 2D topological space that is called a "torus". That's what the "Asteroids" space does.

But it is also possible to put a curved metric on the same topological space--for example, just use the obvious metric derived from the embedding in 3D Euclidean space that we're all familiar with. The "torus" topological space in itself has no metric, and both the flat "Asteroids" metric and the curved "doughnut" metric are valid metrics on that topological space.

Yes, good point, I wanted to include the flatness property (which was the third condition mentioned in the article) but simplified a bit too much in the process.

> It is also topologically equivalent to the (surface of) a torus

Interesting, I thought it would be like the surface of a sphere. What's the difference?

Connectivity. When you go off the top, you come in on the bottom. When you go North on a globe, you come in elsewhere in the North.

Ah, this makes sense. I'm thinking of the Mercator projection of the globe.

When you wrap a rectangle around a sphere, all points at the top edge are identified – thecedge is compressed into a single point, the "north pole", and similarly with the bottom edge. When you go off the top/bottom edge of Mercator at longitude N, you emerge at another point at the same edge, namely at longitude N+180 (mod 360).

(Also, in Mercator it looks like you can approach the top or bottom edge diagonally, but this is an illusion, an artifact caused by the projection. You can only ever approach the north pole directly from the south, and once you cross it, you find yourself having "rotated" exactly 180° and are now facing south, in addition to having jumped to the opposite longitude. And vice versa for the south pole.)

The poles don't get identified with any point in the plane at all, really - the Mercator projection is infinitely tall, less a rectangle than an infinite strip.

The equirectangular projection does map the top and bottom edges to their respective poles, but Mercator just keeps going up (and down).

> the Mercator projection is infinitely tall

Well dang, I just realized that maybe I've based my homebrew D&D world on a Mercator projection.

Ah, indeed! Got my projections mixed up.

Amusingly this means a lot of old RPGs and such are arguably set on a toroidal world rather than a globe.

When you reach one of the boundaries you re-enter the space at a different boundary. It was one of the very few things I was able to understand.

There's "continuity" between two points that, in the 3D rectangle representation, don't appear to have continuity. This continuity is possible because spacetime may be all warpy like that... You can't exit the space at the boundaries. The door out of the room walks you into the same room from a different door.

eg. exit Face C at some point, and re-enter from Face A, because Faces C and A are aligned at that point.

I'm lost beyond that point though.

in 2d: take a square and glue the two pairs of opposite sides together. If you do this with one pair you get a cylinder, and then gluing the other pair gets you a torus. No boundary is left.

In 3d: take a solid cube and glue the three pairs of opposite sides together. Maybe a bit difficult to visualize, but the idea is that if you live in the cube and try to exit it on one side then you re-emerge into the cube from the opposing side.

Would this allow for Escher type spaces?

Yes, though not with a didicosm. (Unless your standards for Escher-esque spaces are low.)

Escher's most fantastical spaces would be classified as non-Euclidean, whereas the article here stresses that the spaces it describes are strictly Euclidean.

s/Euclidean/locally Euclidean/

I think they are rather forms of parallel projection (where farther objects aren't smaller, unlike perspective projection) combined with forced perspective, where things look locally connected from a specific angle:

https://en.wikipedia.org/wiki/3D_projection#Limitations_of_p...

In the same way a sphere, like the Earth's surface, is a 2d-space with finite area and no boundary.

Möbius strip:

https://en.m.wikipedia.org/wiki/M%C3%B6bius_strip

What does boundary mean in this context then? Because the mobius strip has a boundary on the sides even if it is continuous. I guess boundary means an edge in all directions?

A mobius strip is an example of a space that is finite but unbounded in a single direction. A two dimensional plane that is finite but unbounded in all directions is called a Kline bottle, but you can’t build them for real in 3D space, only distorted approximations.

You don’t have to do such geometric contortions though. The surface of a sphere is a two dimensional space that is finite but unbounded.

Not sure why you need a Mobius strip when an untwisted rectangle with two opposite edges joined (outside of a cylinder) has the same property. Unlike a sphere that would be a flat 2d space (but unlike a sphere it would be unbounded only in one direction)

Donuts are another option. That’s what you get when you join a square at each pair of opposite edges.

Indeed the gaussian curvature in a torus is 0.

(in 4d only)

The same way the surface of the earth is a 2D space with finite area and no boundary.

TFA tells you how.

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