# How to construct a genus 2 surface from 8-gon?

I am requesting some help or reference for visualization?

I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from Hatcher:

And here's my doodle.... I think I didn't get the identification of $b$ correctly at step (c), since I believe it $a$ shall go around the tube...

• One way is to "cheat", cutting the octagon diagonally, into two pieces, one with $c$ and $d$, the other with $a$ and $b$. Then each piece is a torus (minus a disk), and you can then glue them back together.
– user641
Aug 29, 2013 at 20:02
• Oh, that makes sense, @SteveD. Thank you. Aug 29, 2013 at 20:08
• I know this thread is really old, but I've recently found a reference that has a very nice picture explaining how to make a double torus from an octagon by identifying opposite sides. Have a look at pages 137-138 of dx.doi.org/10.1016/0370-1573(86)90159-6 Apr 3, 2017 at 10:02

Here are two visualizations. ${}{}{}{}{}{}{}{}{}{}$

Here's a third, from Steve D's comment:

• I like it! It is so cute!! Thanks Neal! Aug 30, 2013 at 1:07

You can find this too on p. 301 of Hilbert & Cohn-Vossen "Geometry and the Imagination."

On this page about classification of surfaces (unfortunately only in Frnech) you will find a collection of videos showing how to glue n-gons to get genus-g surfaces

http://analysis-situs.math.cnrs.fr/Classification-des-surfaces-triangulees-par-reduction-a-une-forme-normale.html

• Welcome to math SE. Could you edit your post to include the main idea in your link? Here, only a link is consider as a bad answer since if the link is broken, we lose that solution. Dec 10, 2019 at 13:47