Mfcttrf

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.

Not with the usual definition of a circle as the set of points at a fixed distance $r$ from a center $C$. If circles are concentric that means they have the same center. If they intersect at one point then they have the same radius. That means they are the same circle.

That argument works in any geometry where distance is defined.

If the circles need not be concentric you can imagine a solution. Think of two towns. Consider "time to travel" as a measure of distance. Suppose the towns separated by a range of hills with several low passes. There can be exactly three isolated points each reachable in $10$ minutes from each town.

The situation is impossible if we make the following assumptions:

• Each circle has exactly one center, which is a point.


• Concentric circles have the same center.


• Each circle has exactly one radius, which is a number. (We make no assumptions, besides those listed, about the meaning of the word "number.")


• If two circles intersect each other at a point, then that point lies on both circles.


• Given any unordered pair of points, there is exactly one distance between those points, which is a number.


• If a point $p$ lies on a circle, then the distance between $p$ and the center of the circle is the radius of the circle.

From the above, suppose that $C$ and $D$ are two concentric circles that intersect at a point. The two circles have the same center, $e$, and call the intersection point $p$. Then the distance between $e$ and $p$ is the radius of $C$, but it is also the radius of $D$, so the two circles cannot have different radii.

We could make the situation possible by discarding some of the axioms, but for the most part, these axioms are so fundamental to the notion of geometry that if you discarded one, the result wouldn't be considered geometry any more (not even non-Euclidean geometry).

If I had to pick an axiom to discard, I would discard the statement that given two points, there is only one distance between those points. But even doing that, I'm not sure how I would go about creating a model where the given statement is true.

Consider the geometry in which the plane is identified with $(mathbb Z / 4 mathbb Z)^2$. Define the circle with center $C(a, b)$ and radius $r$ as the locus of all points $P(x, y)$ such that $(x - a)^2 + (y - b)^2 = r^2$.

Let $C = (0, 0)$ and consider the two circles centered at $C$ with radii $0$ and $2$. Since $2^2 = 0$ in $mathbb Z / 4 mathbb Z$, the two equations are clearly equivalent, and they both define the set $ (0, 0), (2, 0), (0, 2), (2, 2) $. Therefore we can say that there are two concentric circles with different radii that intersect at three different points (other than their center).