I am trying to find a story I read as a child. I remember little about it except for the following:

  • I suspect I read it before 1985, but it was a school library book which means it could be God knows how much older than that.

  • This may mean it was a book meant for younger readers.

  • In memory it "feels" longer than a short story, but I was a bit young to be reading novels. If I had to guess, I would guess novella length or shorter.

  • English language, available in the United States.

  • Contains several descriptions and ideas that stuck with me:

    • The overall idea was that there were "parallel" worlds, parallel like the pages of a book, except for one that was (for whatever reason) skew to the rest, or warped somehow, and therefore intersecting the rest

    • Geometry between that world and the rest seemed fundamentally incompatible as a result

    • The most evocative example of this (by definition, since it is the only one I recall with clarity) is a description of... something... as looking like "Two concentric circles, intersecting in three places." This is obviously gibberish (concentric circles intersect nowhere unless they also have the same radius in which case they overlap perfectly; non-concentric circles that do intersect do so only at two points) but I was just old enough to realize it was gibberish being used to a purpose. It stuck with me.

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    Technically speaking, in 3 dimensions you could have two concentric circles of equal radius, at 90 degrees to one another, which cross at two points. – Fifth_H0r5eman Aug 19 '19 at 11:05
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    @Clockwork Non-Euclidean Geometry – popctrl Aug 19 '19 at 14:52
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    Two concentric circles, intersecting at three points: this is possible if the two circles are identical in dimension, position, and, assuming 3D space, orientation. Of course, they actually intersect at ∞ points. If you're looking for circles that intersect at exactly three points, that's a different story.... – JakeRobb Aug 19 '19 at 18:04
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    @Clockwork As popctrl said, you're probably thinking of Non-Euclidean Geometry, but that is not a branch of geometry where "everything is possible". Euclidean geometry states that if you have distinct lines A, B, and C such that neither A nor B intersect C, then A and B never intersect each other. (i.e. If A is parallel to C and B is parallel to C, then A is parallel to B). Non-Euclidean Geometries (e.g. elliptic or hyperbolic geometry) are any geometries that do not have that requirement. The rest of Euclid's axioms still apply. (On large scales, the universe may be non-Euclidean.) – Ray Aug 19 '19 at 22:01
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    There's now a question on math.stackexchange about the specific geometry of this story. – Ray Aug 20 '19 at 9:53

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. At first his efforts met only with failure and madness. But now The Thresholders have risen in fury. Somehow Bob Benedict must make one more trip into that land of peril and pacify them before they succeed in hurling his planet—piece by piece into the oblivion of infinity.

It was first published in 1951 (the edition I've linked above is from 1965) so it matches your time frame, and it's only 220 pages long (in hardback form) so it's only a short novel.

The paragraph you remember is:

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 blackness was around him, the cold of the place dug into him and the fear hung around him like a cloak. He had the hallucinatory sense of being torn apart, roughly, in a jagged line from top to bottom and of seeing, unclearly, the parts of his body in mutilated distortion. Did he appear to them as some grotesque geometric distortion, a crooked, twisted, impossible mass of lines and shadows and forms? He must; they couldn't possibly see him as he saw himself in his own place. Yet something had happened, now he could sense that the Thresholders were all around him, with a pervading sense of excitement.

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    Yes, this is definitely it. I remembered the description of a human body decomposed into fragments, too, but not well enough to include it. That is the clincher. Thanks! – Novak Aug 19 '19 at 19:30

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