# Zometool Resources

**by Dr. Chris Hill**

Zometool is a mathematically-precise plastic construction set for building a myriad of geometric structures, from simple polygons to Platonic solids, from models of DNA molecules to geodesic domes, from “shadows” of four-dimensional figures to works of art. It is also a fantastic educational tool, as it facilitates mathematical discovery while unleashing creativity. This page provides a brief introduction and a growing collection of resources about Zometool. Comments, questions, and suggestions are welcome! Please contact Dr. Chris Hill.

*About me*: I'm a Zometool enthusiast (a.k.a. a *Zomer*). I love building with it and using it as a teaching tool. I have run several Zometool workshops and "barn raisings" with middle school and high school students and am looking for opportunities to do more.

### Introduction

This section provides an overview of how Zometool "works." However, it is not necessary to know any of this information before using Zometool. Zometool is exceedingly user-friendly.

Zometool has two types of parts: round connector balls and color-coded struts. Each connector ball has 62 holes: 30 rectangular holes, 20 triangular holes, and 12 pentagonal holes. The colors of the struts indicate into which holes in the balls they will fit. Blue struts fit into the rectangular holes, yellow struts fit into the triangular holes, and the red struts and green struts both fit into the pentagonal holes. However, a green strut features two bends which cause it to point in different directions than a red strut. A connector ball allows for construction in 92 different directions.

Each color of strut comes in short, medium, and long sizes. Red struts also come in an even shorter “hypershort” size. For each color of strut, the ratio between the lengths of one size of strut and the next smaller size (if there is one) is the golden ratio, which is about 1.618. (For computing such ratios, the “length of a strut” is equal to the distance between the middle of the node at one end of the strut to the middle of the node at the other end.) An elegant consequence of this fact is that for each color, the length of a long strut equals the length of a medium strut connected to a short strut. In brief, a long equals a medium plus a short. Similarly, in red, a medium equals a short plus a hypershort.

Different types of struts are required for the construction of different types of structures. For example, a regular dodecahedron can be built with blue struts of the same size, a rhombic dodecahedron with yellow struts of the same size, a rhombic triacontahedron with red struts of the same size, a regular tetrahedron with green struts of the same size, and more elaborate structures with combinations of strut types and sizes.

To make it easier to specify parts for a project, the strut types "blue," "yellow," "red," and "green" are abbreviated "B," "Y," "R," and "G," and the strut sizes "hypershort," "short," "medium," and "long" are denoted "00," "0," "1," and "2," respectively. For example, a long blue strut is a B2, a medium yellow strut is a Y1, a short green strut is a G0, and a hypershort red strut is an R00.

Prior to 2011, Zometool offered blue, yellow, and red struts in a size 3 that was longer than size 2 by a factor of the golden ratio. Per the comments about the golden ratio above, in a given color, a size 3 strut has the same length as a size 2 strut connected to a size 1 strut. Nevertheless, it's unfortunate that size 3 struts were discontinued. They allowed simple models to be built on a grander scale, and a few constructions (such as the compound of ten triangular prisms—see Projects, below) require them.

Since the ratio between the lengths of one size of strut and the next smaller size is the same for all types of struts, a Zometool model can be scaled up (respectively, down) by replacing each strut with the next larger size (respectively, smaller size), provided that size strut exists.

### Art

- George W. Hart is a mathematician/artist who creates the most splendid and varied mathematical sculptures. Hart works in a wide range of media, including Zometool. His name appears several times on this page.
- Pentidisc The world's largest Zometool construction is a pentagonal "disc" almost six meters in diameter, covered in quasiperiodic tilings, and containing over 90,000 parts. It was assembled during the Bridges 2013 conference in Enschede, the Netherlands.
- Kling Memorial Sculpture Assembled at the Bridges 2009 conference in Banff, Alberta, Canada by 150 mathematicians, artists, and some of their children, the mathematical sculpture has over 50,000 parts and is based on the "shadow" of a 6-dimensional cube. At the time, it was the world's largest Zometool construction. Additional photos are provided in Zometool's flickr stream.
- Zome-inspired Sculpture (pdf), by Paul Hildebrandt.
*Proceedings, Bridges London: Connections between Mathematics, Art, and Music*, Reza Sarhangi and John Sharp (editors). (2006) 335-342. - Zometool sculpture at Denver Art Museum

### For Educators

- Lesson Plans (pdf) Hands-on activities with Zometool for students of elementary grades through high school, with connections to mathematics, art, science, and architecture. A hard copy of the Lesson Plans may be purchased from the Zometool Corporation.
*Zome Geometry: Hands-on Learning with Zome Models*, George W. Hart and Henri Picciotto. Key Curriculum Press, 2001. This paperback book is a wonderful resource for middle school, high school, and college students and teachers. The book may be purchased from the Zometool Corporation. Note: the book was published and the addenda written before the Zometool Corporation began selling size 0 and size 00 struts and discontinued size 3 struts.- Zome Workshop (pdf), by Paul Hildebrandt. From the abstract: “This paper outlines how to conduct a Zome workshop for students, teachers or parents. I’ll discuss the discovery learning philosophy, preparing for the workshop, conducting the workshop, follow-up activities and additional resources that are available for educators.”

### Mathematics

- The Mathematics of Zome (pdf), by Tom Davis. One of the remarkable aspects of Zometool is that, as you're building with it, when you want to connect two nearby nodes with a strut, very often the two nodes have holes of the same type and orientation lined up and there is a strut of the right length to connect them. In brief, things tend to work out. Davis provides the beautiful mathematics that underlies this phenomenon.
- Metazome, by Andrew Mihal, Matt Moskewicz, Yujia Jin, Will Plishker, Niraj Shah, Scott Vorthmann, and Scott Weber. The Metazome website proves mathematically that by using Zometool to build large meta-versions of nodes and struts, one can, in principle, construct an enormous meta-version of any Zometool model. Also see Projects, below.
*Regular Polytopes*, 3rd ed., by H.S.M. Coxeter. Dover, 1973. Although written before Zometool was invented, this classic provides mathematical background for Zometool-constructible regular polytopes.

### Projects

- Advanced contructions An online addendum, by George Hart, for the book
*Zome Geometry*(see above).- Compound of Ten Triangular Prisms This may be the most challenging of Hart's advanced constructions. It's certainly the most challenging and satisfying Zometool model I've ever made. (See the photo to the right.) The link goes to a page in Hart's website where he outlines its construction. Andrew Mihal provides an extremely helpful step-by-step procedure for building the compound, which I used to build it. Note that this model requires Y3 struts.

- Metazome, by Andrew Mihal, Matt Moskewicz, Yujia Jin, Will Plishker, Niraj Shah, Scott Vorthmann, and Scott Weber. The Zometool node has the shape of a certain polyhedron (an elongated rhombicosidodecahedron) and this polyhedron can be built using Zometool. Think of the resulting Zometool model as a significantly scaled-up node—a
*meta-node*. Scaled-up versions of Zometool struts can also be built using Zometool, giving us*meta-struts*. The Metazome website shows how meta-nodes and meta-struts can, in principle, be used to construct an enormous meta-version of any Zometool model. A Metazome project was the focus of the St. Bonaventure Zometool Workshop & Geometric Barn Raising 2011. Note that the Metazome website was created before the Zometool Company discontinued size 3 struts. However, as mentioned in the Introduction, any size 3 strut may be replaced by a size 2 strut connected to a size 1 strut. - Zometool polyhedra, by George Hart. A list of some of the types of polyhedra that can be built using Zometool. These polyhedra are discussed in Hart and Picciotto's book
*Zome Geometry*(see Educators, above). - Zome Projects, by David A. Richter. A list of advanced projects, with varying amounts of detail for their constructions. Most of the projects are three-dimensional projections of four-dimensional figures.

### Software

- vZome, by Scott Vorthmann. A program for building virtual Zome models. A version without the ability to save files is available immediately, and a fully capable version is available upon request, both at no cost.
- Zometool Shape Approximation A team from RWTH Aachen University in Germany has developed a program that takes a digital 3-d model and produces a digital Zometool representation of the model. The digital Zometool representation, called a
*Zometool mesh*, may be used to build a physical Zometool approximation of the model.

### Website and Social Media

- www.zometool.com
- Parts Chart A lists of all parts currently offered by the company. Parts may be purchased in system kits, in project kits, and in bulk.

- Zometool on facebook
- Zometool on flickr

### Workshops and Barn Raisings

- Zome Workshop Zome Workshop (pdf), by Paul Hildebrandt. From the abstract: “This paper outlines how to conduct a Zome workshop for students, teachers or parents. I’ll discuss the discovery learning philosophy, preparing for the workshop, conducting the workshop, follow-up activities and additional resources that are available for educators.”
- Organized by Chris Hill
- Geometry Barn Raising at ALCS – Spring 2016 Project: runcitruncated 600-cell. People: over 60 of Linda Dodd-Nagel's 8th-grade math students at Allegany-Limestone Central School.
- The Connect Four Project 2014–2015 Project: Stage-6 Sierpinski tetrahedron. People: 87 middle school students and 9 teachers. Time: spread over a few months. Locations: Four middle/high schools in western New York and St. Bonaventure University.
- Zometool Workshop & Geometric Barn Raising – Fall 2011 Project: mega-icosahedron. People: 13. Time: 2 hours. Location: St. Bonaventure University.

- Mathcamp 2010 Project: expanded 120-cell. Time: 4 hours. People: unspecified.
- Mathcamp 2008 Project: truncated-ambo 120-cell (also called the cantitruncated 120-cell). Time: 8.5 hours. People: initially about 100; a small subset worked throughout.
- Knotting Mathematics and Art 2007 Conference at USF Project: truncated 120-cell. Time: 4.5 hours. People: 10.
- SUNY Stony Brook 2006 Sangaku Zome Construction Project: truncated-ambo 600-cell (also called the cantitruncated 600-cell). Time: 4 hours. People: unspecifiied.
- Bridges 2006 Conference in London Project: omnitruncated 120/600-cell. Time: 8 hours. People: about 40.