"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
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!
ReplyDeleteYep David Eppstein, that's exactly it!
ReplyDeleteI hope your wife is okay with you and these sexy models.
ReplyDeleteha!
ReplyDeleteWell, I think they are beautiful.
ReplyDeleteWould it be possible to show the spherical boundary of the ball model with transparent plastic?
ReplyDeleteSteve 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.
ReplyDeleteIf 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.
It reminds me a bit of a Georgia O'Keeffe painting, but in the form of a mathematical sculpture.
ReplyDelete