Season 5 Episode 6
What does it mean to invert a point in a circle? What happens to lines and circles under inversion? Can we use this to solve geometry problems with lots of circles? Rebekah is here to answer all this and more in this episode.
Definition of inversion in a circle
Let’s restate the definition, just to have everything in one place. Given a circle $C$ with centre $O$ and radius $r$, we can invert the point $P$ in the circle $C$ by finding the point $Q$ such that $O$, $P$, and $Q$ are collinear with $P$ and $Q$ on the same side of $O$, and such that $|OP| \times |OQ|=r^2$.
Proof that inversion preserves angles
For a good explanation of why inversion preserves angles, see this free online textbook. Please note that the description here makes heavy use of complex numbers.
On the livestream we showed an animated GIF of Steiner’s Porism, and that’s at this page.
Wikipedia has some more gifs including a 3D version with spheres.
This website has some gifs, a proof, and interactives
And there’s a nice interactive Geogebra demo here.
For another look at Steiner’s Porism and a related modern result, see this excellent article. Kristian Kiradjiev considers the problem "given a list of $n$ radii, does there exist a pair of nonintersecting bounding circles such that we can construct a Steiner chain between the two bounding circles which consists of $n$ circles with the given radii?". Kiradjiev derives a set of criteria for when such a pair of circles exists, and shows how to construct the Steiner chain if it does exist. Kiradjiev was awarded the Institute of Mathematics and its Applications Early Career Mathematicians Catherine Richards Prize 2017 for the article and was an Oxford DPhil student at the time. You can see the brief news report we wrote about that at the time here.
For more tangent circle diagrams, see also the Apollonian Gasket.
Curves under inversion
Someone in chat asked for the image of other curves under inversion. They might like this StackExchange question (StackExchange is a website for mathematicians – or any other group of people – to ask each other questions).
We didn’t have time for Ptolemy’s Theorem. Here’s Numberphile to the rescue! A Miraculous Proof (Ptolemy's Theorem) - Numberphile
While we’re recommending Numberphile videos, you might also like Funny Fractions and Ford Circles - Numberphile
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk