"That's the sexiest model you've made"


"That's the sexiest model you've made"

Sarah Nemec-Nelson just told me that when I showed her these photos. The shape is a helicoid in hyperbolic 3-space. You can read a little more about this ruled surface in a previous post and the comments there.

https://plus.google.com/+RoiceNelson/posts/8heVohDiBsH

For the physical model, I also included the helicoid axis as a Dupin Cyclide, specifically a horn cyclide. That is the banana shape that a thickened hyperbolic line takes in the ball model. You can produce a horn cyclide from a cone by inverting it in a sphere!

https://en.wikipedia.org/wiki/Dupin_cyclide

Here is a fun and heavily visual paper about cyclides titled Ortho-Circles of Dupin Cyclides.

https://pdfs.semanticscholar.org/4e86/13b074369af1263efe97e299b90a7d81807f.pdf

To make the model more interesting, I chose an axis avoiding the origin and an asymmetric orientation of the helicoid twisting. You can purchase a copy for yourself (again, with no markup) here: http://shpws.me/Pny2









Comments

  1. David EppsteinJanuary 7, 2018 at 5:45 PM

    So it's just the same construction as a Euclidean helicoid, move a line perpendicularly along another line spinning it at constant speed as it moves, and see what surface it traces out? Neat!

    ReplyDelete
  2. Roice NelsonJanuary 7, 2018 at 6:01 PM

    Yep David Eppstein, that's exactly it!

    ReplyDelete
  3. John BaezJanuary 7, 2018 at 8:42 PM

    I hope your wife is okay with you and these sexy models.

    ReplyDelete
  4. Roice NelsonJanuary 7, 2018 at 8:54 PM

    ha!

    ReplyDelete
  5. Saul SchleimerJanuary 8, 2018 at 12:21 AM

    Well, I think they are beautiful.

    ReplyDelete
  6. Steve WennerJanuary 10, 2018 at 9:51 AM

    Would it be possible to show the spherical boundary of the ball model with transparent plastic?

    ReplyDelete
  7. Roice NelsonJanuary 10, 2018 at 10:00 AM

    Steve Wenner, I like that idea. I've searched for acrylic sphere's in the past, and even have a large 18" one in my shed - I bought it for an overly ambitious project I never finished.

    If you wanted to experiment with this idea, I'll be glad to send you the model file. It has a 12.5cm diameter, but could be scaled to match whatever sphere's were available. To get the model inside the sphere, you'd need to find a producer that makes sphere's in halves that can be assembled.

    ReplyDelete
  8. Richard GreenJanuary 10, 2018 at 12:19 PM

    It reminds me a bit of a Georgia O'Keeffe painting, but in the form of a mathematical sculpture.

    ReplyDelete

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