a ··· the geometry of continued fractions · the geometry of continued fractions anton...

The geometry of

continued fractions

Anton Lukyanenko

It turns out that any real number can be written as a continued fraction:

x = a0 +1

a1 +1

a2+···.

We’ll see how working with continued fractions leads to thinking about hy-perbolic geometry, and then talk about generalizations to complex continuedfractions and Heisenberg continued fractions - and the related hyperbolicspaces. Lots of pictures, videos, and 3D prints will guide our intuition forthe geometries we encounter along the way.

Abstract for 08 September 2016

1

The geometry of

continued fractions

Anton Lukyanenko

It turns out that any real number can be written as a contin-ued fraction:

x = a0 +1

a1 +1

a2+···.

We’ll see how working with continued fractions leads to think-ing about hyperbolic geometry, and then talk about general-izations to complex continued fractions and Heisenberg con-tinued fractions - and the related hyperbolic spaces. Lots ofpictures, videos, and 3D prints will guide our intuition for thegeometries we encounter along the way.

1

The geometry of

continued fractions

Anton Lukyanenko

It turns out that any real number can be written as acontinued fraction:

x = a0 +1

a1 +1

a2+···.

We’ll see how working with continued fractions leadsto thinking about hyperbolic geometry, and then talkabout generalizations to complex continued fractions andHeisenberg continued fractions - and the related hyper-bolic spaces. Lots of pictures, videos, and 3D printswill guide our intuition for the geometries we encounteralong the way.

1

The geometry of

continued fractions

Anton Lukyanenko

It turns out that any real number can be written as a contin-ued fraction:

x = a0 +1

a1 +1

a2+···.

We’ll see how working with continued fractions leads to think-ing about hyperbolic geometry, and then talk about general-izations to complex continued fractions and Heisenberg con-tinued fractions - and the related hyperbolic spaces. Lots ofpictures, videos, and 3D prints will guide our intuition for thegeometries we encounter along the way.

Abstract for 08 September 2016

1