I have written a program for Rhino which performs the mathematical transformation of ‘inversion with respect to a sphere’. This transformation has many beautiful properties and reveals some surprising symmetries. It also relates in a deep and beautiful way to Einstein’s theory of Relativity.
Q. How does a mathematician capture a lion in the desert ?
A. He builds a circular cage, enters it and locks it. He then performs an inversion with respect to the cage. Now he is outside and the lion is locked in the cage.
Inversion can be intuitively understood as a way of turning the entire world inside out. That is, it moves everything outside a given sphere (the inversion sphere) to the inside and everything inside it to the outside. It does this in a very particular way, and the following are some of the important properties of this transformation:
1. When any sphere is inverted, its position and size may change, but it remains spherical
2. All spheres which touch the centre of the inversion sphere (the origin) map to flat planes
At first impression, 1 may seem to contradict 2 – surfaces cannot be both flat and spherical at the same time can they ? To make sense of this we need to understand one of the more bizarre results of inversion.
Points at the origin are mapped to the single point at infinity (and vice-versa)
This means that if a point on a sphere is at the origin, the inversion of that sphere has one point which is infinitely distant. It therefore has infinite radius and as curvature is defined as 1/r its curvature is 0, therefore it is a flat plane. This is because
1/∞ = 0 and 1/0 = ∞
I know that division by zero is often described as ‘undefined’ or ‘illegal’, and it does tend to confuse calculators, but bear with me, in this context it actually makes some sense.
It is important to note that we are talking about a single point at infinity. Unlike the Real number line which extends to infinity in two different directions:
-∞…-9, -8, -7, -6, -5, -4, -3, -2, -1, 0,1 ,2, 3,4 ,5, 6,7 ,8, 9….∞
or the Euclidean plane which extends infinitely far in any direction you choose, the point at infinity which inverts to the origin is a strangely a single unique point.
This might all sound like abstract nonsense, but when we apply it we will see that it achieves some remarkable results.
One of the most useful properties of inversion is that it preserves angles (it is conformal). This means that if we take two surfaces that meet at a particular angle and invert them, the surfaces may change shape but the angle between them stays the same.
Therefore, although the overall shape of large arrangements of objects is distorted, if we look closely at any small enough part, its shape remains the same*. This can be thought of as a higher dimensional version of the way the earth is globally round, but appears locally flat.
*apart from a change of chirality
For example, in the following video, I have inverted a 3D model of a room with respect to an inversion sphere inside the room (notice that the walls still meet at perpendiculars) :
This is one way of making an architectural model of an enclosed interior space which can be looked at without taking it apart! I’d love to make an stl model of one of these objects set inside a transparent sphere; I think it would be very strange to turn an object in your hands yet be able to feel you are inside it.
If you’re thinking these objects look similar to the ‘Wee planets’ images you may have seen on Flickr, thats no coincidence – those images are produced using stereographic projection, which is closely related to inversion and we’ll come to soon. (Some more great photos like this here). (by the way, photography like this can never quite turn an interior space fully inside out, the only ones which look like ‘objects floating in space’ are taken outside – can you see why?)
P.S. I know Googlevideo have had a few server problems recently so here are the same videos on Youtube