# Closed-space sphere "displacement"?

This is strictly a geometric question from Lemaitre's writing.

He is presenting an example of a closed space represented as a sphere (I'm assuming the surface only?), and makes a "displacement" of the sphere to demonstrate a point. I am not following his operation on the sphere based on any of the ways I can think of by which he means "displacement".

To read his words (translated from the French original), see page 340 at this site where the section heading is "Closed Space". I've read this page slowly (I have this same in a book) about a hundred times and still have no idea what he is doing to this sphere representing a closed space; Any help is welcome.

It is the second paragraph where I lose the path. I imagine three ways to interpret "displacement" of the sphere:

- a rotation
both B and B' are on the original sphere

- a change in radius
where B is outside the original sphere
where B is inside the original sphere
where B is on the original sphere (not a rotation)

- translation of three forms
where B is outside the sphere
where B is inside the sphere
where B is on the sphere (same as rotation if B' is also on the sphere)

I guess some combinations of these are possible, but what he is demonstrating is supposed to be simple and elementary... but I can't isolate what kind of displacement he is indicating.

His mention that the segments A-B and A'-B' remain representations of similar segments in closed space seems to suggest that he is only considering rotation, which keeps the whole of both segments in the surface of the rotated sphere.

His mention (again with regard to the A-B and A'-B' segments) that the representation that was previously interior to the sphere is now exterior to the sphere seems to suggest a change in radius or translation.

He also seems to be including the interior of the sphere as part of the closed space, "... a portion of space which has already been represented in the interior of the initial sphere..."? But he is also using an antipodal definition where A and antipodal A' represent the same point, so maybe something about that is why?

The third paragraph begins with mentioning the exterior sphere... suggesting a change in radius.

Thanks, if anyone can clear this up for me.

## Answers and Replies

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I must have been working with this problem while I slept; the first time I thought about it today I understood it immediately.

It is simple increase in radius with A and A' on the original surface, B and B' on the new surface, with the radial segments A-B and A'-B' both on the same diameter line through both opposite surfaces of both the spheres. Therefore A, A', B, B', A-A', and B-B' are all on the same line which contains a common diameter of each sphere. Since the point pair A & A' represents the same point, so does B & B', and since both of the segments A-B and A'-B' and the inner diagonal A-A' are located on the same diagional B-B', the two segments are also representations of the same point (B and B').

The same point is represented by A, A', B, B', A-A', B-B', A-B, A'-B', A-B', and A'-B ... all points and segments on the common diagonal represent the same point because these segments are closed lines..