Comments on: The Dedekind tessellation

By: Dedekind or Klein ? | neverendingbooks

Dedekind or Klein ? | neverendingbooks — Tue, 07 Dec 2010 13:04:39 +0000

[...] left of a tessellation of the hyperbolic upper-halfplane, was called the Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The [...]

By: quivers versus quilts | neverendingbooks

quivers versus quilts | neverendingbooks — Sun, 20 Jan 2008 11:15:09 +0000

[...] the three red dots (all of them even points in the Dedekind tessellation) give (after the identification) the two points indicated by a whereas the blue dot (an odd point [...]

By: Quiver-superpotentials | neverendingbooks

Quiver-superpotentials | neverendingbooks — Mon, 14 Jan 2008 12:47:37 +0000

[...] in the Dedeking tessellation. (For more on these terms and constructions, please consult the series Modular subgroups and Dessins [...]

By: lieven

lieven — Wed, 27 Jun 2007 15:51:18 +0000

Doug,
the top picture is taken from John Stillwell so I cannot comment on his reasons but as for the other picture, there are only so many marks you can include and I had no intention to include full Fareys…
Yes there is a nice connection with the devils staircase and its on my list of ‘to do posts’. In short, the devil translates ONAG-numbers to Fareys.

By: Doug

Doug — Wed, 27 Jun 2007 15:28:27 +0000

1 – The Farey sequence in the top diagram of this thread appears to use ’2/7, 3/7, 4/7, 5/7′ but not ’1/7 or 6/7′.
This seems to be an incomplete order F7 sequence.
Why?
Likewise the bottom diagram appears to be similarly incomplete but with ’1/6 and 5/6′ absent as well.
Is this just a labelling convenience?

2 – Farey sequences of 3,6,9 order [F3,F6,F9] appear to contain Cantor sets?
Is there a relation to the ‘devil’s staircase’ through the Cantor Function?

By: Chris Atkinson

Chris Atkinson — Fri, 22 Jun 2007 21:22:53 +0000

Thanks for this. I noticed a couple typos:

\rho should be e^{(\pi/3)*i}

odd edges connect cusps to odd vertices.

-C