Zomes are fun. The name comes from _zon

Tim Hutton - 2018-04-19 07:36:47+0000 - Updated: 2018-04-19 09:11:02+0000

Zomes are fun. The name comes from 'zonohedron' + 'dome'. I've made an interactive tool for playing with their parameters:
https://github.com/timhutton/zomes

The beauty is that the shape of the dome comes completely from the properties of rhombi - only the bottom row of triangles is chosen by the user.

https://en.wikipedia.org/wiki/Zonohedron
https://simplydifferently.org/Zome

timhutton/zomes

Shared with: Public, Tim Hutton, Félix de la Fuente, Chas Hofmeister, Abdelaziz Nait Merzouk

+1'd by: Félix de la Fuente, Gerard Westendorp, David Eppstein, Alison Grace Martin, Abdelaziz Nait Merzouk, Dean Calahan, Whitt Whitton, Michael Prescott, Rubén Perblac, José Lorenzo López, Luis Guzman

Abdelaziz Nait Merzouk - 2018-04-19 13:19:58+0000

Cool!
Maybe this is how the builders of Taj Mahal designed the domes. :D

Tim Hutton - 2018-04-19 13:38:05+0000 - Updated: 2018-04-19 13:55:38+0000

+Abdelaziz Nait Merzouk :) Looking at the Taj Mahal and other Mughal/Islamic domes I think the shape is different, which means they weren't using this approach. Which is a shame, because it would seem very appropriate given the importance of tiling patterns in Islamic art! And there's definitely a 'religious' feeling to the fact that the shape of the dome comes 'pre-ordained', without much input from the user.

I don't know if any classical real-world domes use this approach. It's an interesting question. None here seem to match, for example:
https://en.wikipedia.org/wiki/Dome

Russian onion domes often have rotating swirl patterns like zomes, but their shape is different again:

Onion dome - Wikipedia

Tim Hutton - 2018-04-19 16:07:31+0000

In the limit, the shape of zonohedra is the surface of revolution of a sine wave. You can see this in the app if you accidentally set the number of rings too high:

Gerard Westendorp - 2018-04-24 21:44:49+0000

I think this is related to Chebyshev (name has various spellings) nets. For example, there is a nice articel:

page.math.tu-berlin.de/~pinkall/forDownload/DiscreteK-surfaces.pdf

Félix de la Fuente - 2018-04-29 17:49:48+0000

+Tim Hutton I remembrer now a short article, I believe by Coxeter, where that surface of revolution was obtained. Watching this image of the sine wave makes me wonder, what if the vectors on the pole are not evenly distributed around a circle? Then their Minkowski sum would still close the zonohedron, whose upper half would be a zome, but it does not seem like an aproximation to a sine surface any more, does it?

Tim Hutton - 2018-06-18 14:53:50+0000

+Félix de la Fuente I found that Coxeter article you mentioned, thanks!

PolarZonohedra_1963_ChiltonCoxeter.pdf

Tim Hutton - 2018-06-18 17:28:18+0000

+Félix de la Fuente I tried your suggestion to change the vectors on the pole from being evenly distributed. I added uniform 3D noise to the first ring of vertices. https://timhutton.github.io/zomes/random.html (Reload to get new random samples.)

Tim Hutton - 2018-06-18 17:58:44+0000 - Updated: 2018-06-19 09:21:16+0000

The only zome I know of in architecture is London's Gherkin. And as Stephen Wolfram points out, it's only an approximation - it looks like they've elongated it and rounded the ends off. http://blog.wolfram.com/2008/10/10/russell-towle-1949-2008/

Tim Hutton - 2018-06-22 12:49:56+0000

Another almost-zome is Bruno Taut's 1914 Glass Pavilion. It was this that inspired me many years ago to explore rhombi domes (that I now know as polar zonohedra).

Looking at it now that I notice that his rhombi are actually kites of different proportions, as if he didn't appreciate the mathematical beauty of polar zonohedra.

Glass Pavilion - Wikipedia

Chas Hofmeister - 2018-07-18 01:35:03+0000

Howdy Tim,
For a while I was obsessed with polar zonahedra, to the point that I bought an inexpensive 3d printer so I could print out the various rotated sine waves I was creating in tinkercad.
Now I'd like to create a set of interlocking tiles to make zomes, but find myself lacking data. Once I create a shape I like with your github tool above, how do I calculate the angles or diagonals of the rhombi?
Thanks, Chuck H.

Tim Hutton - 2018-07-18 10:09:12+0000

+Chas Hofmeister I want to do something similar myself, and am planning to rework the app to spit out cutting plans etc. Maybe email me and I'll make sure that it does what you want too.

Chas Hofmeister - 2018-07-20 21:54:42+0000

+Tim Hutton Sounds great, Tim! For real world calculations, I think the length of the diagonals would be the most practical.

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