Building a 1.4m high polar zonohedron d

Tim Hutton - 2018-06-18 14:55:31+0000 - Updated: 2018-06-18 14:56:22+0000

Building a 1.4m high polar zonohedron dome ('zome') on Saturday. First strip cut out of cardboard.

Building a 1.4m high polar zonohedron dome ('zome') on Saturday. First strip cut out of cardboard.

Shared with: Public, Tim Hutton, Gerard Westendorp

+1'd by: Rubén Perblac, Doug Hackworth, Abdelaziz Nait Merzouk, Gerard Westendorp

Gerard Westendorp - 2018-06-18 21:18:53+0000 - Updated: 2018-06-18 21:21:38+0000

This pseudosphere looks a bit like a zonohedron. I am not sure the rhombi are planar, although they have equal sides.

Tim Hutton - 2018-06-19 09:18:51+0000

+Gerard Westendorp Oh, interesting, thanks. Yes, they can't be planar else the shape would be the same as a zome. After reading your link from last time I guess that means it's a Chebyshev net?

Gerard Westendorp - 2018-06-19 10:27:09+0000

+Tim Hutton Yes, it is a Chebyshev net.
I think you got a discretisation of the 'Sine Gordon' equation for the top angle (alfa) of the rhombi, if you demand that the angluar deficit per area is constant, ie constant Gausian curvature.
(area is proportional to the sin(alfa)). Thats cool, because the sine gordon equaiton has solitons, which then correspond to these shapes.

Tim Hutton - 2018-06-19 14:18:59+0000

+Gerard Westendorp I vaguely know of Sine Gordon as a PDE. Here's the damped version as a Ready file with a stationary soliton: https://drive.google.com/file/d/17z7190Hd-HigkwPpif6HI5KMWVD6Of5r/view?usp=sharing (I think, please check.)
But what's the connection between the solitons and the 3D shapes?

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