# Around and around and back and somewhere else

I dreamed that I was navigating through an enormous house, with lots of twists and turns. I eventually became convinced that the house was only locally Euclidean.

Yes, I sometimes have topology dreams.

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1. Reminds me of this: http://www.youtube.com/watch

2. You are in a maze of twisty little passages, all alike

3. Joshua says:

Join the party.

I often have oddball topography in my dreams. Basically space can be overlayed when accessed through doors from different directions.

4. Me says:

What does it mean that the space was only locally euclidean?

Would a Torus space (like in pacman) be euclidean?

[A torus space is not Euclidean because it loops back on itself. However, it is locally Euclidean, because if you look at a small piece at a time (the space immediately surrounding the pacman's current location), that small piece looks Euclidean. It is only when you look at large pieces that you notice that the space is warped funny. Another example of a locally Euclidean space is a spiral ramp. To an ant walking on the ramp, it looks like a regular flat space, but if the ant has a good memory, he may realize, "Wait a second, I walked a complete circle but I did not come back to my starting point." That was the sort of behavior I was experiencing in my dream. -Raymond]
5. Joshua says:

@Me: I suspect it means in this case that building walls are also domain walls.

6. Although a torus space is not Euclidean in the sense that it is finite, it does obey the parallel postulate.

7. Jake says:

Hey! That happened to me last night too!

Only, I was playing Antichamber. http://www.antichamber-game.com

Can you be sure you didn't dream a video game?

8. That sounds more like a nightmare to me :-S .

By the way, this reminds me of the classical "And he built a crooked house" (http://www.math.union.edu/…/crooked-house.pdf); it's a particularly good read if you try to follow the movements of the characters in the tesseract.

9. j b says:

If Euclid hadn't overruled the old Phoneicians, space would have had two large circular dimensions and one small linear one.

10. Dave says:

Other people's dreams might be the most boring thing in the world.

11. Henning Makholm says:

@Maurits: The torus fails not the fifth but the first postulate (there are more than one straight line joining any two points).

12. Reminds me of the Heinlein short story AND HE BUILT A CROOKED HOUSE.

en.wikipedia.org/…/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22

13. PavelS says:

Now I wonder if there is a space which is not locally Euclidean. eg. no matter how small piece you take, it's still not flat.

[Curved spaces are nowhere locally Euclidean. An example of a space that is locally Euclidean in some places but not others would be a line segment. The body of the line segment is locally Euclidean, but the endpoints are not: In Euclidean space, you can always turn 180 degrees and walk the other way, but at the endpoint, you can walk in one direction but not the other. -Raymond]
14. Charles says:

"Curved spaces are nowhere locally Euclidean."

The sphere is definitely locally Euclidean: puncture a sphere and you can flatten it to form a subset of the plane.

An example of a nowhere locally Euclidean space would be (I believe) the Cantor set. So is any countable product of the discrete space.

[I stand corrected. I've been away from this stuff for too long. -Raymond]
15. @Charles

Depending on what you mean by "locally Euclidean", a sphere may or may not be locally Euclidean.

Specifically, if you include "zero curvature" in your definition of Euclidean, a sphere's surface is not locally Euclidean. But a cylinder's surface is.

@Henning Makholm

Depends on how you define a line. For example, you could define a "line" on a torus as "the intersection of the torus with a plane that contains the point C at the torus's center of gravity." Clearly, C and any two points on the surface of the torus define such a plane.

16. Charles says:

@Maurits

I was using what I thought was a fairly definition for a topological manifold. If you're working with Riemann manifolds (which impose more structure) then I guess the notion is different; that's not an area I'm familiar with. Is that where you were headed?

(And I was wrong when I said the Cantor set is nowhere locally Euclidean (topologically); E^0 is Euclidean, just not very interesting)

17. @Charles

Yup. A topologist would say that a there is a homomorphism between (T^2 = (S1)^2) {0, 0} and R^2, and that there was a homomorphism between any sufficiently small neighborhood in T^2 and the open disk in R^2.

A differential geometer (or a physicist) would counter that there was no diffeomorphism between T^2 and any subset of R^2, so sufficiently accurate measurements of curvature in even a fairly large T^2 would reveal the non-Euclidean nature of the space.

18. Silly says:

@PavelS. Mabye a region of infinite (aka 1#J) curvature could never appear to be flat? Or any surface who's 'x' component goes like A/(x-B) where 'x' goes to 'B'? I've also heard wild tales of what physics might be lurking near the Planck scale, so maybe curved spaces "in reality" that seem flat at small distances go all kooky again at super high energy. Well I obviously don't know, but I do know that I like the way the word 'Euclid' rolls off the toungue.

At any rate, I've had a few real-life strange experiences in houses as a guest… groping around at night in pitch black looking for a light switch or a door handle to get me into the hallway, and then doing much the same in the hallway looking for the toilet. And the rooms were spinning too.

19. Er, I mean that there's a homomorphism between a punctured *sphere* and R2, not a punctured torus.

20. kinokijuf says:

Raymond is lucky that his dreams are at least locally Euclidean. I dreamed once that I was a bird trying to fly in an elliptic universe that seemed to be contracting.