I first saw Vladimir give a talk in 2010 and his work and images were totally inspiring to me.

I first saw Vladimir give a talk in 2010 and his work and images were totally inspiring to me. That has continued ever since!

Originally shared by Vladimir Bulatov

I've posted slides from my talk on hyperbolic tilings
http://bulatov.org/math/180110

Comments

  1. Melinda GreenJanuary 18, 2018 at 3:01 PM

    Hm, he says it's non repeating but the left half looks identical to the right to me.

    ReplyDelete
  2. Roice NelsonJanuary 18, 2018 at 7:03 PM

    Melinda Green, it's aperiodic in the same way as a Penrose tiling. That is, a shifted copy of the tiling will never match the original (shift meaning Euclidean translations). Aperiodic Penrose tilings can have reflection symmetry as well, even as much as 5-fold dihedral symmetry.

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  3. Melinda GreenJanuary 19, 2018 at 12:00 AM

    Roice Nelson But that's exactly how it looks to me. Use your stereo vision to fuse both halves. They look pixel-for-pixel identical to me.

    ReplyDelete
  4. Roice NelsonJanuary 19, 2018 at 11:36 AM

    Melinda Green, I see, you're right! Looking through the slides again, I think the title image is not a horosphere section. Not only is it repeating under a translation, the tiles shrink in size instead of remaining constant. I'm guessing Vladimir Bulatov used a section of a hyperbolic plane for the title image.

    ReplyDelete
  5. Vladimir BulatovJanuary 19, 2018 at 2:51 PM

    Melinda Green Roice Nelson is 100% correct. The title image is just pretty looking image. More exactly it is unwrap of cylindrical projection from a surface of a sphere intersecting the tiling inside of Poincare sphere. The region where that sphere intersect the surface of Poincare sphere has tilings of infinitely small size. Therefore we have that illusion of horizon.
    https://lh3.googleusercontent.com/SeSoJxk1Vqlrx55KY3o50Y_84ukXcxG4_mgJyj9dkd5PbLReivWHlF1mNeZ58cckUcRi85FnfFfMhQisAe4i_n4Txm2PNWbNzec=s0

    ReplyDelete
  6. Melinda GreenJanuary 19, 2018 at 2:57 PM

    Vladimir Bulatov OK but are the two halves identical or not?

    ReplyDelete
  7. Vladimir BulatovJanuary 19, 2018 at 3:12 PM

    Melinda Green yes, they are identical as tiling has symmetry plane via center of the sphere.

    ReplyDelete

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