Zome

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The term zome is used in several related senses. A zome in the original sense is a building using unusual geometries [1] (different from the standard house or other building which is essentially one or a series of rectangular boxes). The word "zome" was coined in 1968 by Steve Durkee, now known as Nooruddeen Durkee, combining the words "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 or toy, "Zometool" refers to a model-construction toy manufactured by Zometool, Inc.. It is sometimes thought of as the ultimate form of the "ball and stick" construction toy, in form. It appeals to adults as well as children, and is educational on many levels (not the least, geometry). Finally, the term "Zome system" refers to the mathematics underlying the physical construction system.

Both the building and the learning tool are the brain children of inventor/designer Steve Baer, his wife, Holly, and associates.


Contents

[edit] The Zome as Building Concept

Steve Baer was educated at Amherst College, UCLA, and Eidgenössische Technische Hochschule (Zurich, Switzerland), where he studied mathematics. Here he became interested in the possibilities of building innovative structures using polyhedra (polyhedrons) other than rectangular ones. Baer and his wife, Holly, moved back to the U.S., settling in Albuquerque, New Mexico in the early 1960s. In New Mexico, he experimented with constructing buildings of unusual geometries (calling them by his friend Steve Durkee's term: "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 its multi-faceted geometric lines has been taken up by French builders in the Pyrenees. A recent book, Home Work, published in 2004 and edited by Lloyd Kahn, has a section featuring these buildings. While many zomes built in the last couple decades have been wood-framed and made use of wood sheathing, much of what Baer himself originally designed and constructed involved metal framing with a sheet-metal outer skin.

[edit] Construction Set

Zometool logo
Zome Model of the Great Grand Stellated 120-Cell

The plastic construction set is produced by Zometool, a privately-owned corporation based in Denver, Colorado, which evolved out of Baer's company ZomeWorks. It is perhaps best described as a "space-frame construction set". Its elements consist of small connector nodes and struts of various colors. The overall shape of a connector node is that of a non-uniform rhombicosidodecahedron, except that each face is replaced by a small hole. The ends of the struts are designed to fit in the holes of the connector nodes, allowing for syntheses of a variety of structures. These parts are made from state-of-the-art ABS plastic injection-mold technology. In fact, the founders of Zometool were eventually forced to design a tool which would produce the small connector node using ABS plastics. The first connector node emerged from their mold perfectly on April 1, 1992.

In the years since 1992, Zometool has widened and enriched its line of products. Much of the development has focused on improving the style or the variety of struts available. Since 1992, the basic design of the connector node has not changed, and hence the various parts released have remained universally compatible. From 1992 until 2000, Zometool produced many kits which included connector nodes and blue, yellow, and red struts. In 2000, Zometool introduced green lines, which were designed to allow the user to build, among other things, models of the regular tetrahedron and octahedon. In 2003, Zometool changed the style of the struts slightly. The struts "with clicks" have a different surface texture and they also have longer nibs which allow for a more robust connection between connector node and strut.

[edit] Characteristics of Zometool

The color of a Zometool strut is associated with its cross section and also with the shape of the hole of the connector node in which it fits. Thus, each blue strut has a rectangular cross section, each yellow strut has a triangular cross section, and each red strut has a pentagonal cross section. The cross section of a green strut is a rhombus, where the ratio of the diagonals is √2. The green struts, fitting in the "red" pentagonal holes, are not a part of the 1992 release of Zometool, and, consequently, using them is not as straightforward as the other colors. One may find a variety of colors of connector nodes, but these all have the same purpose and design.

At their midpoints, each of the yellow and red struts has an apparent twist. At these points, the cross-sectional shape reverses. This design feature forces the connector nodes on the ends of the strut to have the 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 fit into the pentagonal holes of the connector node.

Among other places, the word "Zome" comes from the term "zone". A zone is a partition of Euclidean space into mutually-parallel lines. The Zome system allows no more than 61 zones. The cross-sectional shapes correspond to colors, and in turn these correspond to "zone colors". Hence the Zome system has 15 blue zones, 10 yellow zones, 6 red zones, and 30 green zones.

Two shapes are associated with blue-green. The blue-green struts with a rectangular cross section are designed to lie in the same zones as the blue struts, but they are half the length of a blue strut; hence these struts are often called half-blue (and were originally made in a light blue color). The blue-green struts with a rhombic cross section lie in the same zones as the green struts, but they are designed so that the ratio of a rhombic blue-green strut to a blue strut is 1:1 (as opposed to the green strut's √2:1). It is important to understand that, due to this length ratio, the blue-green struts having a rhombic cross section do not mathematically belong to the Zome system.

[edit] A definition of the Zome system

Here is a mathematical definition of the Zome system, on which the physical Zometool construction set is based. It is defined in terms of the vector space R3, equipped with the standard inner product, also known as 3-dimensional Euclidean space.

Let \varphi denote the Golden ratio and let H3 denote the symmetry group of the configuration of vectors (0,\pm\varphi,\pm 1), (\pm\varphi,\pm 1,0), and (\pm 1,0,\pm\varphi). The group H3, an example of a Coxeter group, is known as the icosahedral group because it is the symmetry group of a regular icosahedron having these vectors as its vertices.

Define the "standard blue vectors" as the H3-orbit of the vector (2,0,0). Define the "standard yellow vectors" as the H3-orbit of the vector (1,1,1). Define the "standard red vectors" as the H3-orbit of the vector (0,\varphi,1). A "strut" of the Zome System is any vector which can be obtained from the standard vectors described above by scaling by any power \varphi^n, where n is an integer. A "node" of the Zome System is any element of the subgroup of R3 generated by the struts. Finally, the "Zome system" is the set of all pairs (N,S), where N is a set of nodes and S is a set of pairs (v,w) 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 the colors blue, yellow, and red, respectively. Correspondingly, the stabilizer subgroup of a blue, yellow, or red strut is isomorphic to the dihedral group of order 4, 6, or 10, respectively. Hence, one may also describe the blue, yellow, and red struts as "rectangular", "trianglular", and "pentagonal", respectively.

One may extend the Zome system by adjoining green vectors. The "standard green vectors" comprise the H3-orbit of the vector (1,1,0). and a "green strut" as any vector which can be obtained by scaling a standard green vector by any integral power \varphi^n. As above, one may check that there are 60 standard green vectors and that the stabilizer subgroup of such a vector is a two-element group generated by a reflection symmetry of the regular icosahedron. 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 model has 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 (as opposed to a node being a single point mathematically), so one cannot make a Zometool model where two nodes are separated by an arbitrarily small prescribed distance. Similarly, only a finite number of lengths of struts will ever be manufactured, and a green strut cannot be placed directly adjacent to a red strut or another green strut with which it shares the same hole (even though they are mathematically distinct).

[edit] Zome as a modeling system

Compound of Five Cubes, rendered in ZomeCAD
Compound of Five Tetrahedra, rendered in vZome

The Zome system is especially good at modeling 1-dimensional skeleta of highly symmetric objects in 3- and 4-dimensional Euclidean space. The most prominent among these are the five Platonic solids, and the 4-dimensional polytopes related to the 120-cell and the 600-cell. However, the list of mathematical objects which are amenable to Zome is long, and currently an exhaustive list is not forthcoming. Besides those already mentioned, one may use Zome to model the following mathematical objects:

[edit] Other uses of Zome

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

[edit] 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.

[edit] External links

Zome buildings:

Zome modelling system:

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