Sunday, September 19, 2010

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.


Rob Bell

Wednesday, September 08, 2010


 Welcome to Zonotopia
 Zomes are zonohedral domes. Zomes are different, more beautiful and more functional than geodesic domes. In addition to being derived from higher dimensional geometry my Zomes embody an organic and spiritual nature which all people identify with. 
zome mani padme zome
Rob Bell

The forgoing text was the introductory page in a journal book placed inside of the Zonotopia Zome. During the entire week of Burningman 2010 the journal book provided a place for people to express themselves and write their thoughts about the zome and their playa experiences.

On the blustery, rainy Saturday afternoon of August 28th, Patricia and I were on the playa at the spot which was to become Zonotopia for the next eight days. Amazingly, everything for both structures fit on my Tacoma and a small flat-bed trailer.

Sunday morning we began. We received fantastic placement, just behind the temple in sight of the Man.

Chris K. Palmer, a fellow Zome enthusiast and geometer showed up to assist with the assembly. Here we are organizing parts and soaping the joints.

In short order we had the base ring and bench supports layed out.

Ominous storm clouds loomed as we proceeded past the equator. The custom interlocking wooden panel and connector system performed very well.

We received help and encouragement from friends and passersby.

It turned out to be a beautiful day to build on the playa - not too hot, it never rained and it wasn't too windy.

After about six hours or hard physical work fatigue and hunger began setting in. Help arrived in the form of the Mission Country Club, our friends and campmates who arrived like a calvary come to save the day.

Here we see two expert joint soapers, soaping the connectors with ivory soap. I do believe we had the freshest smelling structure on the playa!

While the soapers soaped other friends assembled last years zome, The Bodhisattva Zome. It was awesome to see it spring to life again on the playa.

It was a great feeling to see so many people happily engaged in my Zome building madness.

The Zonotopia Zome lent it's topmost rhombuses to help loft the Bodhisattva star.

Meanwhile, back at the big zome work proceeded every inwards and upwards. Here I am persuading one of the rhombs into it's proper position.

Like clockwork, around and around, twelve rhombs per revolution the Zonotopia Zome wound it's way toward it's central focus.

Two point two tons of thanks to the Mission Country Club for coming out and creating a wonderful zome building party on the playa.

The sun was setting and we still had two rows to go. We were also reaching the 'do not sit or stand' portion of our ladder. Alex, our campmate, friend and Black Rock City Ranger called in for support from Heavy Machinery. Help was on the way!

Here I am in the basket with all the remaining rhombs while Chris helps from the inside wearing his lucky zome hat. The machine operator was able to move me around the structure with surgical dexterity.

At last an orange moon rose above the horizon, after a long day of zome building, we had only the last row of rhombs to go. Time to call it quits for the evening, get some sleep and continue in the morning.

Early the next morning I finished the final row of rhombs and placed the summit flame. The Zome was finished!

As people emerged from the temple they saw this: Zonotopia.  For eight days, there were two complimentary Zomes adorning the playa with graceful architecture and craftsmanship. A manifestation of mathematics, design and simplicity in order to provide a space for experiences to happen.

 And Happen they did...

...there were unexpected meetings of beautiful friends at sunrise...

...there were weddings...

...there were thousands of smiles...

...there were good friends full of love and laughter...

...there was firedancing...

...there were ceremonies and prayers....

...there were men with zome hats spreading polyhedral cheer...

...there was yoga...

...there were portals which framed amazing art...

...there was dancing in dust storms...

...and most of all there was Love.

Welcome to my Spaceship, Beautiful Forever.

zome mani padme zome
Rob Bell

Tuesday, April 20, 2010

A Zome Pavilion by Rob Bell and Chris K. Palmer

Zome for G4G9
selected photos

A beautiful day for Zome Building


Our platform covers what was once an old mine shaft.

A view from within the nearest neighbor

Beautiful flowers abound nearby.

The preliminary gravel ground cover and pathway

The Zome is in good company

- zome mani padme zome -

Thursday, February 25, 2010

Zomadic Alchemy: The Triangle Question


The Question

Given a length of square stock and a standard compound miter chop-saw how would you make an equilateral triangle using only 45 degree cuts?

 This is a woodworking math puzzle I invented. For three years now no one I've asked
has been able to answer the question.

The Hints

You need only set up the saw once
You may fashion a simple length stop
You will make exactly four cuts



The Answer

Set the saw to a 45 degree x 45 degree compound cut as shown.
Cut off the tail of the stock.
Rotate the stock and off cut 180 degrees - this is critical.
partial rotation shown

Rotate stock 180 degrees
completed rotation shown

Use off cut as a length stop.
Slide stock to the stop.


Make the second cut.
remove first part
rotate stock 180 degrees and slide to stop


With exactly four cuts you can generate three parts.
Place the three parts flat on a surface


Rotate the two side parts by 90 degrees

Rotate one part around the fold axis.


Fold the second part over by 90 degrees.
Here is an equilateral triangle made from only 45 degree cuts.
That's It!


There is More!

It's interesting that the two angled sides
of the triangle are not the same.
Depending on the respective orientation from these triangles 
one can make edge truncated octahedrons and tetrahedrons.
Consequently it follows that one can generate an octet-truss.

Octahedron and Tetrahedron

An octet-truss generated from a single part.



These alder frames are glued and nailed.

Three triangles preparing to merge.

The start of a tetrahedron.

The start of an octahedron.

Truss components