Henry Segerman's recent shapeways model of the {3,3,6} H3 honeycomb inspired me to make a similar model using...

Henry Segerman's recent shapeways model of the {3,3,6} H3 honeycomb inspired me to make a similar model using different media... laser engraved crystal.  I took the tessellation a few levels deeper and rendered edges with proper thicknesses in the Poincare ball model.

The print has a very ghostly feel to it, which is quite nice for including lots of tetrahedra and still being able to see the the cells living towards the interior of the ball.  I think it would work well with even more cells added.  Lighting has a big effect on how things look!

The {3,3,6} is one of 15 regular honeycombs in hyperbolic 3-space.  It is one of 11 that have infinite cells and/or vertex figures.  For more info, see:

Henry's video about his shapeways model:
Regular triangulation of H³

Coxeter's paper, "Regular Honeycombs in Hyperbolic Space":
http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf









Comments

  1. Henry SegermanMay 7, 2013 at 9:26 PM

    Wow, that looks amazing! Can I buy one?

    ReplyDelete
  2. Roice NelsonMay 7, 2013 at 9:38 PM

    Yes, but as luck would have it, you don't need to :)  They accidentally sent me two of these, so I have an extra with your name on it.  Shoot me an email with a mailing address, and I'll drop it in the mail this week.  (I don't have an extra lighted base, however - those are available at http://www.lasercrystalawards.com/catalog/presentation-bases.html)

    ReplyDelete
  3. Roice NelsonMay 7, 2013 at 9:38 PM

    For anyone else interested, these can be purchased from www.lasercrystalawards.com for a little under $40.  You can upload a dummy model file during checkout and leave a note pointing them to the (large) stl file here:  http://www.gravitation3d.com/lasercrystal/H3_336.stl

    Trying to upload this large file to their servers fails, which is the reason for the dummy model file.

    ReplyDelete
  4. Henry SegermanMay 7, 2013 at 9:44 PM

    Roice Nelson Thanks so much! I'll email you my address. Someone needs to do a Shapeways for crystal engraving...

    ReplyDelete
  5. Vladimir BulatovMay 7, 2013 at 10:25 PM

    Very nice! How the STL file is rendered? As the vertices of triangles or do they generate some new dots?

    ReplyDelete
  6. Allen KnutsonMay 8, 2013 at 5:19 AM

    Fantastic. If it's not too much trouble, I'd love to see a video where the camera moves around the cube!

    ReplyDelete
  7. Roice NelsonMay 8, 2013 at 8:09 AM

    Vladimir Bulatov, the rendering is the result of many tiny dot microfractures in the glass.  The stl I made is just triangles though (about 1.3 million of them).  Laser crystal awards does the rest.  I know Bathsheba has made her own special algorithms for her engraved crystals controlling exactly where dots go, but I haven't found this necessary.

    http://www.bathsheba.com/crystal/process/

    I will say that on the {3,3,6} model, only the surface gets turned into dots, and this does have a little bit of a downside near the really crowded vertices (leading to small voids), so that is potentially improvable if I tried to control the dots myself.

    By the way, this is the third model I've had engraved, and all have been very pleasing.  Here's some pics of another:

    https://lh3.googleusercontent.com/hdkOy3am_6Ao8sOgilX1Fl98c_QIuK42FEs999VRVg4ANPZ5Zb18YRd995T3Erd8rSnsVPHrRSOjv4fg_nobvGwDgLUwiHn_v00=s0

    The last was a portion of a 120-cell, but it was a gift and unfortunately I didn't take pics before giving it.

    ReplyDelete
  8. Roice NelsonMay 8, 2013 at 8:35 AM

    Allen Knutson, I just made a video for you.  All I have to record is my phone, and things are a little blurry, but hopefully it gives a view like you were hoping :)

    {3,3,6} H3 Honeycomb

    ReplyDelete
  9. Roice NelsonMay 8, 2013 at 11:39 AM

    Vladimir Bulatov, sorry, I misunderstood your question earlier, so edited my previous response.  In short, they generate new dots (it is not just triangle vertices that are rendered).

    ReplyDelete
  10. Kevin CliftMay 8, 2013 at 6:11 PM

    This may be of interest to you Mark Bruce, perhaps you could have your bust in optical crystal by this means.

    ReplyDelete
  11. Robbye RichardsMay 10, 2013 at 1:14 PM

    DON,T UNDERSTAND A WORD BUT LOVE THE PICTURE OF YOU AND SARAH  SEND ME ONE AND THE CHRYSTAL IS BEAUTIFUL  LOVE GRANDMA

    ReplyDelete
  12. Paul NelsonMay 12, 2013 at 8:59 PM

    Looks cool. Maybe I will try to 3d print this stereolithography file on the rapid prototype machine at work. :)

    ReplyDelete
  13. Henry SegermanMay 15, 2013 at 8:32 PM

    Roice Nelson I'm told that the package has arrived in Oklahoma, I'm looking forward to seeing it in person!

    ReplyDelete

Post a Comment

Popular posts from this blog

Hyperbolic Hopf Fibrations

Image
Hyperbolic Hopf Fibrations The Hopf Fibration of S^3 is amazing and beautiful.  Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson. nilesjohnson.net/hopf.html To understand it better (and fibrations in general), I recommend this talk by Niles too. www.youtube.com/watch?v=QXDQsmL-8Us It turns out there is an analogue of the Hopf fibration for H^3.  In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case.  There is a 2-parameter family of them, plus one additional fibration that does not fit the family.  As with S^3, fibers in the H^3 cases are geodesics.  They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration.  I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this. www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf I'm posting three early pictures of H^3 fibrations belo...
Read more

76 Unique Honeycombs

76 Unique Honeycombs Last weekend, Tom Ruen and I hit the milestone of uploading to wikipedia at least one image for 9 families of compact, Wythoffian, uniform H3 honeycombs, a total of 76 unique honeycombs. http://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space You can easily browse all the images on my wiki user page: http://commons.wikimedia.org/wiki/User:Roice3 - Compact means the cells are finite in extent.  - Wythoffian means we can generate them using a kaleidoscopic construction, that is by reflecting in mirrors.    - Uniform means they are vertex transitive and have uniform polyhedral cells. There may even be more honeycombs that meet all these criteria, I don't know. ( update: see Tom's comment below! )  I do know there are hundreds more which don't meet one or more of these criteria, many undiscovered.  In fact, there are infinitely more because there are some infinite families of honeycombs.  wendy krieger continues to discover and enumerate more...
Read more