Some details from Vladimir; It is the same 3D bending of 2D hyperbolic symmetry group generated by reflections in the sides of quadrilateral I've described in http://bulatov.org/math/1107 All the tiles are the same and fit perfectly and can be mapped to each other using Moebius transformation. Angles and devils belongs to 2 different classes though. All the circular holes in the holed tilings are equivalent (including the largest hole with the center at infinity - outside of the disk) On Tue, Jan 24, 2012 at 5:26 AM, <math-fun-request@mailman.xmission.com>wrote:

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2. bending M.C.Escher "Circle Limit IV". (Stuart Anderson)

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Message: 4 Date: Mon, 23 Jan 2012 16:44:36 +0000 From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] bending M.C.Escher "Circle Limit IV". Message-ID: <CAN57YquTA=LVrv+gASRnEp+S2mgP4dQcLhexgGf2tLU3mOtTeg@mail.gmail.com

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I'm intrigued by the blank circles in images on right-hand side: it looks as if he has some way of mapping multiple hyperbolic infinities to a lattice in the plane, suggesting a (2-D) space peppered with black holes of varying masses.

If the tiles were now blown up back to congruence again, what shape does this space have: eg. is it describable by some simple metric?

Fred Lunnon

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Message: 6 Date: Mon, 23 Jan 2012 18:26:09 +0000 From: Veit Elser <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] bending M.C.Escher "Circle Limit IV". Message-ID: <37BDD28D-9961-4189-8965-B00B67227F18@cornell.edu> Content-Type: text/plain; charset="us-ascii"

The cover art for this book by Hermann Weyl

http://www.amazon.com/Open-World-Hermann-Weyl/dp/0918024706

shows the surface of the globe conformally mapped to a flat torus -- the singularities are cleverly hidden in the oceans.

Veit

On Jan 23, 2012, at 11:44 AM, Fred lunnon wrote:

...

I'm intrigued by the blank circles in images on right-hand side: it looks as if he has some way of mapping multiple hyperbolic infinities to a lattice in the plane, suggesting a (2-D) space peppered with black holes of varying masses.

If the tiles were now blown up back to congruence again, what shape does this space have: eg. is it describable by some simple metric?

Fred Lunnon

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