Is there a known non-euclidean geometry where two concentric circles of different radii can intersect? (as in the novel “The Universe Between”)

From the 1951 novel The Universe Between by Alan E. Nourse.

Bob Benedict is one of the few scientists able to make contact with the invisible, dangerous world of The Thresholders and return—sane! For years he has tried to transport—and receive—matter by transmitting it through the mysterious, parallel Threshold.

[…]

Incredibly, something changed. A pause, a sag, as though some terrible pressure had
suddenly been released. Their fear was still there, biting into him, but there was something else. He was aware of his body around him in its curious configuration of orderly disorder, its fragments whirling about him like sections of a crazy quilt. Two concentric circles of different radii intersecting each other at three different points. Twisting cubic masses interlacing themselves into the jumbled incredibility of a geometric nightmare.

The author might be just throwing some terms together to give the reader a sense of awe, but maybe there’s some non-euclidean geometry where this is possible.

Answer

Yes, with the appropriate definition of “circle”. Namely, define a circle of radius R centered at x on manifold M to be the set of points which can be reached by a geodesic of length R starting at x. This seems pretty reasonable, and reproduces the usual definition in Euclidean space.

It’s not hard to see that concentric circles on a torus or cylinder can have four intersection points.

enter image description here

(Here’s how to interpret this picture: The larger circle has been wrapped in the y-direction, reflecting a torus or cylinder topology. Coming soon: A picture of this embedded in 3D.)

By flattening one side of the torus a bit, you can make one side of the larger circle intersect the smaller circle at two points, while the other side just grazes at a single point*. Thus you get three intersections.


*As a technical point, this can definitely be accomplished in Finsler geometry, though I’m not sure if it can be done in Riemannian geometry.

Attribution
Source : Link , Question Author : Mindwin , Answer Author : Yly

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