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Square torus, again

Joppy

Well-known member
MHB Math Helper
Mar 17, 2016
256
I've been having a look at Richard Schwartz' book - Mostly Surfaces. He introduces the square torus on page 2 and describes what a 'bug' might see if it were walking around on this surface.

He (and I've seen this description elsewhere also) says that when the bug sits on one of the horizontal edges (technically it's sitting on both) and looks up or down, it sees 'a little half disk'. Similarly, the bug sitting on one of the corners (all the corners) sees 'four little quarter disks'.

Now maybe I'm overthinking things but isn't the point of pretending to be a bug supposed to help us put on our 'flatlander glasses'? We wouldn't be able to see any such disk if that were the case, so I assume the bug is a small, 3D being. But the question remains, what are the little half and quarter disks? I can't seem to visualise them at all.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,643
Leiden
Here's my 2 cents.

Suppose it's foggy and the bug can only see up to a certain fixed distance.
And suppose someone drew lines (walls in flatland) where the edges are.
Then wouldn't even a flatlander bug see the disks you described?
 

Joppy

Well-known member
MHB Math Helper
Mar 17, 2016
256
Here's my 2 cents.

Suppose it's foggy and the bug can only see up to a certain fixed distance.
And suppose someone drew lines (walls in flatland) where the edges are.
Then wouldn't even a flatlander bug see the disks you described?
Hmm..(Sweating).

I'm not sure I understand how there is any difference between what the bug sees when it sits on an edge, and when it sits the surface of the square.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,643
Leiden
Hmm..(Sweating).

I'm not sure I understand how there is any difference between what the bug sees when it sits on an edge, and when it sits the surface of the square.
Without markers on the square there is indeed no way that the bug can tell that it's on an edge.

So suppose we draw a horizontal line and a vertical line through the center of our square (instead of on the edges).
And we put the bug in the center instead of on a corner.
Then we have an equivalent situation where the bug sees 4 quadrants doesn't he? (Thinking)
 

Joppy

Well-known member
MHB Math Helper
Mar 17, 2016
256
Without markers on the square there is indeed no way that the bug can tell that it's on an edge.

So suppose we draw a horizontal line and a vertical line through the center of our square (instead of on the edges).
And we put the bug in the center instead of on a corner.
Then we have an equivalent situation where the bug sees 4 quadrants doesn't he? (Thinking)
Ah... I think I see it now. Am I correct in saying that for the case you describe here, the bug might also think that it is sitting on top of a sphere (with limited viewing distance as you previously suggested)?

This is starting to bug me! (not really :))
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,643
Leiden
Yep. It won't be able to tell the difference with a sphere.
That is, until it traces a loop that would be a square respectively a digon. (Mmm)