Andrew Mihal, Matt Moskewicz, Yujia Jin, Will Plishker, Niraj Shah, Scott Vorthmann, Scott Weber

{mihal, moskewcz, yujia, plishker, niraj, sjweber}@eecs.berkeley.edu

University of California, Berkeley

February 2003


Revised November 2003:


1. Overview

Metazome is a cool project you can do with the Zome Construction System. To make Metazome, you use regular Zome parts to make models of Zome parts. Then you use your Metazome to make Metamodels. Metazome parts fit together perfectly, just like regular Zome. Anything you can make with Zome can be made with Metazome.

Node

Metanode

Struts

Metastruts

Model

Metamodel

 


2. Metazome Instructions

Here are the instructions for making individual Metazome components.

  White Blue Red Yellow Green
Standard Parts Node Short
Medium
Long
Short
Medium
Long
Short
Medium
Long
Extra-Short
Short
Medium
Long
Advanced Parts   Extra-Extra-Short
Extra-Short
Half-Short
Half-Medium
Half-Long
Extra-Extra-Short
Extra-Short

Short Purple
Medium Purple
Long Purple

Extra-Extra-Short
Extra-Short

Short Orange
Medium Orange
Long Orange

Half-Short
Half-Medium
Half-Long

Short Blue-Green
Medium Blue-Green
Long Blue-Green

Connections   Connection Connection Connection Connection

 


3. Metazome Geometry

Metazome illustrates the scientific principle of self-similarity. An object is said to be self-similar if it has a feature that repeats itself on several different scales. An example from mathematics is the fractal. When you zoom in on a fractal, the same shapes keep popping up again and again. Self-similarity is common in nature as well. A floret of broccoli has the same shape as an entire head of broccoli. A frond of a fern is composed of hundreds of leaflets, each of which follows the same shape as the frond itself. A Metazome model is composed of Metanodes and Metastruts, which are made of regular nodes and regular struts that have the same shapes.

Metazome is possible because of the inherent self-similarity in the Zome System itself. The feature that keeps appearing on many different scales in Zome is the golden ratio, tau (). The golden ratio appears not only in the lengths of the struts, but also in the cross-sections of the struts and the size and shape of the nodes.

Look closely at the cross section of the blue strut and the rectangular hole in the Zome node. The ratio between the long edge and the short edge is . This is an easy shape to make with Zome. You simply build a rectangle using two medium blue struts and two short blue struts. In fact, all of the shapes on the Zome node are easy to make with blue struts: pentagons, rectangles, and triangles. With this in mind it is straightforward to build a Zome model of the Zome node. The result is a Metanode.

Furthermore, the ratio between the diameter of the Metanode and the diameter of the regular node is also based on . We empirically found the value to be equal to . This relation holds even though the diameter of the node is slightly different when measured between faces with different shapes. The following figures demonstrate this fact.

Diameter measured between rectangular faces
Diameter measured between pentagonal faces
Diameter measured between triangular faces

The remaining challenge in Metazome is to figure out how to build Zome struts out of Zome. There are four points to consider:

  1. Metastruts must be the right length relative to each other. That is, they scale up by the golden ratio .
  2. Metastruts must be the right length relative to the diameter of a Metanode. The ratio between the length of a Metastrut and the diameter of a Metanode must be the same as the ratio between the length of a regular strut and the diameter of a regular node. Otherwise, the Metamodels won't look right. They'll be out of proportion.
  3. You have to be able to connect Metastruts to Metanodes. The connections have to hold together, and you have to be able to make all the same connections that you can make with regular Zome.
  4. Red and yellow struts have twists in the middle. The red and yellow Metastruts must have similar twists.

There are many ways to solve these issues. The instructions on this website are for one possible solution that we have found works well.

How long should a Metastrut be? We'll answer this question by looking at the fundamental Zome component, the short blue strut. The lengths of all other struts are based on this strut.

We define to be the fundamental Zome length, measured along a short blue strut from the center of a node to the center of a node. To get the length of the strut itself, we subtract two node radii, or one node diameter. The result is . In order to satisfy consideration number 2, the ratio between the length of the short blue Metastrut and the regular short blue strut must be equal to the ratio between the diameter of the Metanode and the diameter of the regular node. That ratio is . Therefore the length of the short blue Metastrut must be:

From the figure above, we see that is equal to two long blue struts. The length is an extra-extra-extra-long blue strut. We can restate this quantity in regular Zome lengths using the property that .

(1 extra-extra-long blue strut, 1 extra-long blue strut)

(2 extra-long blue struts, 1 long blue strut)

(3 long blue struts, 2 medium blue struts)

Therefore, the final length for the short blue Metastrut is 4 long blue struts and 4 medium blue struts. Here are the instructions for building it. It is straightforward to calculate the length of the medium blue strut:

This length is equivalent to 8 long blue struts and 6 medium blue struts. The derivation for the long blue Metastrut is the same.

The red and yellow Metastruts are slightly more complicated. While the basic length relations still hold,

(short red Metastrut)
(short yellow Metastrut)

and are not simply two long struts. They are slightly longer than a long red strut and a long yellow strut, respectively. There are no regular Zome struts that make up the difference exactly. Furthermore, the red and yellow struts have twists in the middle. The twist in the red strut rotates the pentagonal cross sections at each end by 36 degrees. The twist in the yellow strut rotates the triangular cross sections at each end by 60 degrees. The Metastruts must have equivalent twists, without effecting the overall length of the strut.

The solution is to build each twist in such a way that the length of the twist structure and the extra diameter length add up to a regular Zome length. Then, when you subtract these lengths from the term, you are left with a length that can be made out of regular struts.

Here is one possible solution for the red Metastrut, and one possibility for the yellow Metastrut. The short red and short yellow struts that just hang off in these pictures are not part of the twist structure. They are only there to illustrate how the twist structures fit into a Metastrut.

Red Twist

Yellow Twist

The length of the red twist is and the length of the yellow twist is . These lengths are defined as follows:

(2 long red struts, 2 extra-short red struts)
(2 long yellow struts, 2 extra-short yellow struts)

After some simple algebra we arrive at the number of struts you need to make the main sections of the short red and short yellow Metastruts.

(6 long red struts)
(6 long yellow struts)

Three long struts go on each side of the twist structure. When scaling up to medium Metastruts, the length of the twist structure does not also scale.

(10 long red struts, 2 medium red struts)
(10 long yellow struts, 2 medium yellow struts)

 


4. Acknowledgements

The parts we used to make Metazome were kindly donated by Zometool. The original Metazome model appeared in a display window at The Construction Site in Waltham, Massachusetts between November 2002 and January 2003. Many thanks to Paul Hildebrandt and Crispin Richey.

These Metazome models were made with custom-colored struts provided by Paul. The Metanodes normally require short and medium blue struts - we got these made in white plastic. The red Metastruts require some short blue struts which were manufactured in red. The yellow Metastruts require short red, short blue, and medium blue struts which were manufactured in yellow. The difference is purely cosmetic, of course. All of the strut lengths are the same.

 


5. Metazome Gallery

Here are some pictures of Metazome projects.

Lots of Metazome!
Niraj and Andrew build some parts.
Takin' a break!
You need lots of hands to build Metazome.
Another view of the original Metamodel.
Rocket Ship
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Back to Andrew's Zome Projects page