Pecha Kucha : The Geometry of Desire

Coinciding with the ending of our Zonotopia fundraiser was a Pecha Kucha event which emphasized Burningman projects so I took the opportunity to tell the story of Zonotopia.  The theme was Desire.  It was a great time and everyone laughed in all the right places.

The Geometry of Desire

In 2005, I was working as a software engineer. It was then that I began a journey into the world of zonotopal architecture. Here are a few of the paper models I built to begin educating myself in the structural systems of space.

To manifest a desire one should state an intention. In 2006 I started a fabrication studio and this model was one of the first things I cut on my cnc and it was a declaration of intention as you will soon see.

In my studies of different coordinate systems and explorations of non-rectilinear geometries I learned that computer architecture is a dangerous place were magic is easy. It's quite difficult to build something completely new in the real world and that works.

 In 2009 we built the first Zome of Zonotopia, the Bodhisattva Zome. It defines a space where people can enter and not feel enclosed, to see and to be seen. It creates a form which is compelling and beautiful or in a word... Desirable.

 And in 2010 at Burningman Zonotopia consisted of two Zomes. The original eight frequency Bodhisattva Zome and seen here a newer 20' tall twelve frequency Zome. You can determine the frequency by counting the number of petals which emanate from the center.

I call this collection of zonohedral pavilions Zonotopia: The Once and Future Utopia of Zomes,
because now that I understand how simple they are I see no reason that they could not have existed long, long ago.

As this acrobat can attest, when properly constructed Zomes are incredibly strong. All of my Zomes are built entirely with interlocking wooden fasteners and we don't burn them. They will be heading to the playa this year for the third and hopefully not the last time.

To me Zonotopia represents not only possibility of what might be but also of what once might have been. Imagine if this were only the top of an ancient larger structure. What would that structure be?

What would a would an ancient people, knowledgable and skilled in this architectural space build had they the means? And were I allowed to continue digging down to unearth it what sort of undreamt of wonders might I find?

This is how a cube may propagate a helix in 3-space. Polar Zonohedra are hyper dimensional cubes which when transformed in similar fashion create polytopes in their higher dimensions which cast volumious, helical shadows into our 3 space.

This is the design for Zonotopia 2011. A 35' tall tower I call the Miracula Mirabilis. And the fact that it can be constructed using the same panel and connectors from last years Zome even blew my mind.

Now that the design was nailed down our next task was straightforward. Raise the money. So on a rare and beautiful hot day in San Francisco we set up our Zome in Dolores Park to kick off our Indiegogo fundraiser. We set up in a little over an hour.

Unfortunately police came and shut us down almost immediately. And then, to add insult to injury we were labeled Renegade Hippies by the local Mission Blog. We may be renegades but we're no hippies.

Furthering the fundraising efforts I made this 1/8th scale model of the Miracula Mirabilis. This model and models like it are some of our donation rewards. This one is currently on dislay at Coffee Bar, just a block from my fabrication studio. 

Production is well underway. This is one unit of 1" ACX Pine. One of the great challenges on a project like this is how to minimize waste, maximize efficiency and create systems where friends and supporters can come into the shop and help.

 Our joinery systems have evolved and become more sophisticated. To the lower right is one of the new panels fresh off the cnc. The curvy connectors I call frogs and are a new design created to easily and accurately join parts together.

This is my frog editor. Not only do these forms create a positively locking joint but both positive and negative can be efficently cut in a single continuous pass. Generally something of this complexity would take too much time to be cost effective.

Based on the joinery need at hand Frogs come in many shapes and proportions. Here you see the first Legion of my Wooden Frog Army coming off the cnc and I'm making more every day...soon I will be invincible.

Zonotopia would not exist and none of this would be possible without the help and love of my friends and supporters. I'm doubly blessed to be able to manifest this madness and to have such wonderful people encouraging and helping me to do it.

If you like what you see and you'd like to see this and more of what still lies underground, I encourage you to help us with our fundraising efforts on idiegogo. I look forward to seeing you under a Zome on the playa and beyond.

Thank you one and all.

Pecha Kucha Inspire Japan

For the second time I got the privilege to speak at a Pecha Kucha event. This time it was for a fundraiser benefit for Japan and the theme was Inspiration. The event was held at Yoshi's in San Francisco. It was a great time and everyone's presentation was awesome. The format is 20 slides - 20 seconds each, so each presentation lasts just under 7 minutes...so here we go!

Zome Geometry and the Silver Ratio

Hello Everyone. For about 5 years now I've been exploring an architectural space called Zomes.  My zomes are polar zonohedral domes and I believe that the space they create is very inspirational. I designed and built this pavilion two years ago and from my observations, everyone who experiences it feels the same.  And beyond the simple beauty of the structure I've discovered to my delight that the form is very functional.

Nature has provided me with an important source of Inspiration. Natural architecture, living and inanimate, provides many examples of economy and function packaged in countless beautiful and varied designs.  I believe that were all designers to thoughtfully study what Nature has to teach we would all be better off.

And people have also provided me with inspiration.  Buckminster Fuller, set an example of forward and free thinking that most of us can  hardly conceive. He provided me with invaluable ideas I would have never come up with on my own.  Not the least of which is that the  geometry we are taught in school is not exactly the way Nature works.

More recently I had the honor of corresponding with and being inspired by another interesting character, Russell Towle.  An autodidact, he surprisingly discovered a new and wonderful class of polyhedron called Spirallohedra which were secretly hiding inside of every polar zonohedron.  But then what is a polar zonohedron?

Polar Zonohedra are all around us and one of them is quite ubiquitous.  At the far top left that's a three frequency polar zonohedron also called a cube.  One can say that the frequency defines the number of dimensions of the polyhedron. So here we are looking at projections  3,4,5,6,7 and 8 dimensonal cubes.

Here's the first Zome I built in 2007.  This is an eight frequency polar zonohedron.  Note that the door is a perfect square and therefore the edges of that square are at 45 degrees to the floor.  In fact, all the edges are 45  degrees to the floor.  This angle is called the pitch.

One can change the pitch to make the polar zonohedron tall and skinny or short and squat. Whatever you do each and every edge is always the same pitch as every other edge. That middle one with the Silver Rhombus label...that one is special and pretty soon I'll tell you why.

Sometimes something is so big that it needs to be framed to be put into context .  I think the Universe is like that.  So when people tell you to think outside the box, say no, I'd rather just increase the number of dimensions of my box - and in this case it is eight.

Many people know about the Golden Ratio. Here's a geometric sketch of how it is derived.  The silver ratio is lesser know but derived in a similar fashion - but using a square instead of a pentagon.  It's only at the pitch of the silver rhombus that polar zonohedra are perfect three dimensional projections of higher dimensional cubes.

Here's another source of my inspiration. Japanese joinery!  I've been designing some of my own systems and techniques which will allow me  to build a strong and completely habitable structure using only interlocking fasteners. Here's one of my early working joint designs.

This is my workshop on a somewhat typical day.  There in the back you can see a computer controlled router. I use this tool for most of  my build work - that's the tool that makes most of this - if not sane - at least possible.

Good ideas in the computer sometimes just don't work in real life and conversely good techniques in real life can be very difficult to model.  So there's a lot of trial and error before success.  But I've found that persistence and tenacity will take you a long way.


With the proper choices of pitch and frequency one can recombine rhombuses from two or more zomes to make a bigger one or conversely  take a big one, and rebuild it as two or more smaller ones. For example you can see those green squares on the far left cube get used again in the six and again in the twelve frequency zonohedron.

This last summer I took on quite the challenge. To build a twelve frequency Zome over 20'.  I had invaluable help from my friends. We fabricated all the parts and drove them to the desert without having had time to see if it would actually work.  Here I am applying a little persuasion to a tight joint.

I have had some wonderful building successes. This was beautiful zome greenhouse my partner Patricia and I built together. An elongated  twelve frequency structure in which each panel was tongue and grooved into it's neighbor. But this new project was another matter altogether.  And the new joints had only been tested in my shop.

 
Here's the digital model of Zonotopia. I've written some parametric design software which automatically generates all of the parts there's but still plenty to pay close attention to and lots of places to go wrong.  It certainly looked cool here - but would it look cool there - and more importantly would it work?

It did!  It took countless times up and down a ladder but we did it and we did it with mostly our bare hands. I love how the Rising Sun creates cresents of shadow light from across the structure. Everyone else loved it too.

And at night we illuminated it from within with white black-light paint and LED black lights. The effect was astounding and ghost-like. With three doors, nine benches and 84 portals Zonotopia was almost always occupied.

And here it is again - all of it - along with my ever dependable Toyota truck. The truck and I have been together for 187,000 miles - that's the same distance light travels in one second and I don't think I could have done much of this without it. - thank you Japan.

And lastly here we see my two silver pitch zomes - an eight frequency and a twelve frequency.  They have provided me and thousands of others with the inspiration that we can do things we never thought we could do and that we will get to see wonderful things we never thought we would  see.

Thank you.

-Rob Bell

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

Zonotopia

 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. 

Enjoy!

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