Polar Zonohedron plugin for Sketchup

Polar Zonohedra

I finally sat down long enough to write a simple Sketchup plugin which generates a general case polar zonohedron.

The interface is quite simple. There are three parameters which define a primitive polar zonohedron: the frequency, the pitch and the edge length.

The Frequency is a positive integer value which must be 3 or greater. This defines the number of vector generators of which the entire polar zonohedron is formed. Shown above is an 8 frequency polar zonohedron - hence the eight petals which converge at the top.

The pitch defines the angle between each of the vectors and the horizontal plane. It may be surprising to note that the pitch of each and every edge throughout the polar zonohedron is identical. The pitch value is an angular value which is input as radians. There are certain special pitches which yield particularly beautiful and useful zonohedral forms.  

One particular pitch which equals about 35.264° is equal to the function atan ( sqrt(2) /2 ).There are many interesting things about this number. It is an angular result of a ratio called the silver mean. It is also the pitch of the edges of a cube when oriented in polar zonohedral form. More about this later. And lastly the third and final parameter defines the edges length of the polar zonohedron. Similar to the pitch it might be surprising that each and every edge throughout the polar zonohedron are exactly the same length.

That's basically it. A group is created and the raw face geometry for the polar zonohedron is added to it. Above is a 96 frequency polar zonohedron which took about 3 minutes to generate so have care when using larger numbers.

The plugin can be downloaded here...

Download the file polar_zonohedron.rb and drop it into your Sketchup plugins folder and launch Sketchup. An item called "Polar Zonohedron" will appear under the Plugins menu.

Enjoy!

Rob Bell