I threw together this little inversive geometry doodle last night to just make
something fun after a long while.

I might do some Steiner porisms and Appolonian gasket type stuff at some point.
Hoping some hyperbolic geometry on a Poisson disk.

I think a plane-filling Appolonian gasket pattern would be super cool, with an
inversion iteration level as a pattern argument.

Anyway, what's going on here is that things that are defined IN the gray circle,
get inverted to the outside of the circle.  Circles not passing through the
center of the inversion circle stay circles, and circles passing through the
center get inverted to lines (circles of infinite radius).  Tangency is
preserved.

Likewise, objects defined outside of the circle get inverted to the inside of
the gray circle.

The closer things are to the circle, the closer they are inverted, and the
closer they are to the center, the farther away they are, and the center maps to
"a point at infinity".

Rather than doing several pages of Euclidean geometry like Pappus, we use Felix
Klein's inversive geometry to leverage the special properties mentioned above.

We make 2 circles that pass through the center, and so get mapped to lines.
A circle tangent to both of those circles (blue) gets mapped to a circle outside
the inversion circle, and is tangent to the 2 lines from the inverted circles.
Then, since all of the subsequent circles are tangent to both of the circles and
the small circle next to it, they are easily made as circles outside the
inversion circle that are all just stacked circles of the same radius - they are
tangent to both lines and the circles next to them.
So when they get inverted to the inside of the circle, they preserve those
properties of tangency.

And out pops a Pappus chain.  Easy peasy.

- BE

Post a reply to this message

Attachments:
Download 'pappuschain.png' (75 KB)

Preview of image 'pappuschain.png'