# Zome

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Zome

The term "zome" is used in several related senses. A zome in the originalsense is a building using unusual geometries [http://www.cyberarchi.com/actus&dossiers/logement-individuel/index.php?dossier=69&article=2896] (different from the standard house or other building which is essentially oneor a series of rectangular boxes).The word "zome" was coined in 1968 by Steve Durkee, now known as Nooruddeen Durkee, combining thewords "dome" and "zonahedron." One of the earliest models ended up as a large climbing structure at the Lama Foundation.In the second sense as a learning tool ortoy, "Zometool" refers to a model-construction toy manufactured by [http://www.zometool.com/ Zometool, Inc.] . It is sometimes thought of as theultimate form of the "ball and stick" construction toy, in form. It appeals toadults as well as children, and is educational on many levels (not the least,geometry). Finally, the term "Zome system" refers to the mathematics underlyingthe physical construction system. Both the building and the learning tool are the brain children ofinventor/designer Steve Baer, his wife, Holly, and others. Baer was educatedat Amherst College, UCLA, and Eidgenössische Technische Hochschule (Zurich,Switzerland), where he studied mathematics. Here he became interested in thepossibilities of building innovative structures using polyhedra(polyhedrons) other than rectangular ones. Baer and his wife, Holly, movedback to the U.S., settling in Albuquerque, New Mexico in the early 1960s. In New Mexico, heexperimented with constructing buildings of unusual geometries (calling them"zomes", see "Drop City"), intended to be appropriate to their environment,notably to utilize solar energy well. Baer was fascinated with the dome geometry introduced by architect R. Buckminster Fuller. Baer was an occasional guest at Drop City, an arts and experimental community near Trinidad, CO. He wanted to make buildings that didn't suffer from some of the limitations of the smaller, owner-built versions of geodesic domes (of the 'pure Fuller' design).

In recent years, the unconventional "zome" building-design approach with itsmulti-faceted geometric lines has been taken up by French builders in thePyrenees. A recent book, "Home Work", published in 2004and edited by Lloyd Kahn, has a section featuring these buildings. Whilemany zomes built in the last couple decades have been wood-framed and made useof wood sheathing, much of what Baer himself originally designed and constructedinvolved metal framing with a sheet-metal outer skin.

Construction Set

The plastic construction set is produced by Zometool, aprivately-owned corporation based in Denver, Colorado, which evolved outof Baer's company ZomeWorks. It is perhaps best described as a "space-frameconstruction set". Its elements consist of small connector nodesand struts of various colors. The overall shape of a connector node isthat of a non-uniform rhombicosidodecahedron, except that each faceis replaced by a small hole. The ends of the struts are designed tofit in the holes of the connector nodes, allowing for syntheses ofa variety of structures. These parts are made from state-of-the-art
ABS plastic injection-mold technology. In fact, the founders of Zometoolwere eventually forced to design a tool which would produce the smallconnector node using ABS plastics. The first connector node emergedfrom their mold perfectly on April 1, 1992. In the years since 1992, Zometool has widened and enrichedits line of products. Much of the development has focused on improvingthe style or the variety of struts available. Since 1992, the basic designof the connector node has not changed, and hence the various parts releasedhave remained universally compatible. From 1992 until 2000, Zometoolproduced many kits which included connector nodes and blue, yellow, andred struts. In 2000, Zometool introduced green lines, which were designedto allow the user to build, among other things, models of the regulartetrahedron and octahedon. In 2003, Zometoolchanged the style of the struts slightly. The struts "with clicks"have a different surface texture and they also have longer nibs whichallow for a more robust connection between connector node and strut.

Characteristics of Zometool

The color of a Zometool strut is associated with its crosssection and also with the shape of the hole of the connector nodein which it fits.Thus, each blue strut has a rectangular cross section, each yellow struthas a triangular cross section, and each red strut has a pentagonal crosssection. The cross section of a green strut is a rhombus, where the ratio ofthe diagonals is √2. The green struts, fittingin the "red" pentagonal holes, are not a partof the 1992 release of Zometool, and, consequently, using themis not as straightforward as the other colors. One mayfind a variety of colors of connector nodes, but these all have the same purposeand design. At their midpoints, each of the yellow and red struts has an apparenttwist. At these points, the cross-sectional shape reverses.This design feature forces the connector nodes on the ends of the strut to havethe same orientation.Similarly, the cross section of the blue strut is a non-square rectangle,again ensuring that the two nodes on the ends have the same orientation.Instead of a twist, the green struts have two bends which allow them to fitinto the pentagonal holes of the connector node. Among other places, the word "Zome" comes from the term "zone". A zone is a partitionof Euclidean space into mutually-parallel lines. The Zome system allows no more than 61 zones. The cross-sectional shapescorrespond to colors, and in turn these correspond to "zone colors". Hence theZome system has 15 blue zones, 10 yellow zones, 6 red zones, and 30 green zones. Two shapes are associated withblue-green. The blue-green struts with a rectangular cross section are designedto lie in the same zones as the blue struts, but they are half the lengthof a blue strut; hence these struts are often called "half-blue".The blue-green struts with a rhombic cross section lie in the same zonesas the green struts, but they are designed so that the ratio of a blue strutto a rhombic blue-green strut is the square root of two. It is importantto understand that the blue-green struts having a rhombic cross sectiondo not belong to the Zome system.

A definition of the Zome system

Here is a mathematical definition of the Zome system, on which the physicalZometool construction set is based. It is defined in terms of thevector space $R^3$, equipped with the standard inner product,also known as 3-dimensional Euclidean space. Let $varphi$ denote the Golden ratio andlet $H_3$ denote the symmetry group of the configuration ofvectors $\left(0,pmvarphi,pm 1\right)$, $\left(pmvarphi,pm 1,0\right)$, and $\left(pm 1,0,pmvarphi\right)$.The group $H_3$, an example of a Coxeter group, is known as the icosahedral group because it is the symmetrygroup of a regular icosahedron having these vectors as its vertices. Define the "standard blue vectors" as the $H_3$-orbit of the vector $\left(2,0,0\right)$.Define the "standard yellow vectors" as the $H_3$-orbit of the vector $\left(1,1,1\right)$.Define the "standard red vectors" as the $H_3$-orbit of the vector $\left(0,varphi,1\right)$.A "strut" of the Zome System is any vector which can be obtained from the standardvectors described above by scaling by any power $varphi^n$, where $n$ isan integer. A "node" of the Zome System is any element of the subgroup of $R^3$generated by the struts. Finally, the "Zome system" is the set of all pairs $\left(N,S\right)$,where $N$ is a set of nodes and $S$ is a set of pairs$\left(v,w\right)$ such that $v$ and $w$ are in $N$and the difference $v-w$ is a strut. One may check that there are 30, 20, and 12 standard vectors having thecolors blue, yellow, and red, respectively. Correspondingly,the stabilizer subgroup of a blue, yellow, or red strut is isomorphic to the dihedral group of order4, 6, or 10, respectively. Hence, one may also desrcibe the blue, yellow, and redstruts as "rectangular", "trianglular", and "pentagonal", respectively. One may extend the Zome system by adjoining green vectors.The "standard green vectors" comprise the $H_3$-orbit of the vector $\left(1,1,0\right)$.and a "green strut" as any vector which can be obtained by scaling a standardgreen vector by any integral power $varphi^n$. As above, one may check that thereare 60 standard green vectors and that the stabilizer subgroup of such a vectoris a two-element group generated by a reflection symmetry of the regularicosahedron. One may then enhance the Zome system by including these green struts.Doing this does not affect the set of nodes. The abstract Zome system defined above is significant because of the following fact: Every connected Zome modelhas a faithful image in the Zome system. The converse of this fact is only partially true,but this is due only to the laws of physics. For example, the radius of a Zometool node is positive,so one cannot make a Zometool model where two nodes are separated by an arbitraryprescribed distance. Similarly, only a finite number of lengths of struts will ever be manufactured.

Zome as a modeling system

The Zome system is especially good at modeling 1-dimensionalskeleta of highly symmetric objects in 3- and4-dimensional Euclidean space.The most prominent among these are the five Platonic solids,and the 4-dimensional polytopes related to the 120-cell andthe 600-cell.However, the list of mathematical objects which are amenable toZome is long, and currently an exhaustive list is not forthcoming.Besides those already mentioned, one may use Zome to model the followingmathematical objects:
* Kepler-Poinsot polyhedra
* Regular Polyhedral compounds
* Regular 4-dimensional polytopes and some compounds
* Many stellations of the rhombic triacontahedron
* Many stellations of the regular icosahedron
* Zonohedra, especially the enneacontahedron
* Hypercubes in dimensions 61 or fewer
* Most uniform polyhedra
* Many 4-dimensional uniform polytopes
* Thorold Gosset's exceptional semiregular polytopes in 6, 7, and 8 dimensions
* A few of the Johnson solids
* Desargues' configuration
* Two of the Catalan solids
* Classical and exceptional root systems
* Triality (from Lie theory)

Other uses of Zome

The uses of Zome are not restricted to pure mathematics.Other uses includethe study of engineering problems, especiallysteel-truss structures,the study of some molecular, nanotube, and virus structures,to make soap film surfaces, and as an artistic medium.

References

* Steve Baer. "Zome Primer." Zomeworks Corporation, 1970.
* David Booth. The New Zome Primer, in "Fivefold Symmetry," István Hargittai (editor). World Scientific Publishing Company, 1992.
* Coxeter, H. S. M. "Regular Polytopes", 3rd edition, Dover, 1973. ISBN 0-486-61480-8.
* Brian C. Hall. "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction", Springer, 2003. ISBN 0-387-40122-9.
* George Hart, Four-Dimensional Polytope Projection Barn Raisings. "Proceedings, Sixth International Conference of the Society of Art, Math, and Architecture, Texas A&M University." May 2007.
* George Hart and Henri Picciotto. "Zome Geometry: Hands-on Learning with Zome Models." Key Curriculum Press, 2001. ISBN 1-55953-385-4.
* Paul Hildebrandt. Zome-inspired Sculpture. "Proceedings, Bridges London: Connections between Mathematics, Art, and Music", Reza Sarhangi and John Sharp (editors). (2006) 335-342.
* David A. Richter. Two results concerning the Zome model of the 600-cell. "Proceedings, Renaissance Banff: Mathematical Connections between Mathematics, Art, and Music", Robert Moody and Reza Sarhangi (editors). (2005) 419-426.
* David A. Richter and Scott Vorthmann. Green Quaternions, Tenacious Symmetry, and Octahedral Zome. "Proceedings, Bridges London: Connections between Mathematics, Art, and Music", Reza Sarhangi and John Sharp (editors). (2006) 429-436.

Zome buildings:
* [http://www.zomes-concept.com/EJaccueil.htm The zome building concept explained]
* [http://www.cyberarchi.com/actus&dossiers/logement-individuel/index.php?dossier=69&article=2896 Examples of European zome buildings]
* [http://archinstitute.blogspot.com Examples of zome usage in North American prefabricated housing construction]

Zome modelling system:
*
* [http://www.zometool.com/faq.html The Zome FAQ] at the manufacturer's site.
* [http://zometool.com/zomeforum/index.php The official Zometool forum]
* [http://homepages.wmich.edu/~drichter/zomeindex.htm Advanced Zome Projects] by David Richter
* [http://www.georgehart.com/zomebook/zomebook.html Zome Geometry] by George W. Hart and [http://www.keypress.com/x2527.xml Henri Picciotto]
* [http://www.vorthmann.org/zome/ vZome] for building virtual Zome models
* [http://www.lkl.ac.uk/bridges/zome.html Zome at Bridges London] at the London Knowledge Lab
* [http://www.zome.jp/ Japan Zome Club] a user's club in Japan (Japanese)
* [http://www.zome.jp/metazome/ Metazome] a project making Zome models with Zome

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