The term ''zome'' is used in several related senses. A zome in the original sense is a building using unusual geometries (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
Abdullah Nooruddeen Durkee was a Muslim scholar, thinker, author, translator and the Khalifah (successor) for North America of the Shadhdhuli School for Tranquility of Being and the Illumination of Hearts, Green Mountain Branch. Nooruddeen Durke ...
, combining the words
dome
A dome () is an architectural element similar to the hollow upper half of a sphere. There is significant overlap with the term cupola, which may also refer to a dome or a structure on top of a dome. The precise definition of a dome has been a m ...
and
zonohedron
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
. One of the earliest models ended up as a large climbing structure at the
Lama Foundation
Lama Foundation is a spiritual community founded in 1967, located in the Sangre de Cristo Mountains of northern New Mexico, seventeen miles north of Taos. The original commune was co-founded by Barbara Durkee (now known as Asha Greer or Asha vo ...
. 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 brainchildren of inventor/designer
Steve Baer
Steve Baer (born 1938) is an American inventor and pioneer of passive solar technology. Baer helped popularize the use of zomes. He took a number of solar power patents, wrote a number of books and publicized his work. Baer served on the board ...
, his wife, Holly, and associates.
As building concept
Following his education at Amherst College and UCLA, Steve Baer studied mathematics at
Eidgenössische Technische Hochschule
(colloquially)
, former_name = eidgenössische polytechnische Schule
, image = ETHZ.JPG
, image_size =
, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ac ...
(Zurich, Switzerland). Here he became interested in the possibilities of building innovative structures using
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
. Baer and his wife, Holly, moved back to the U.S., settling in
Albuquerque
Albuquerque ( ; ), ; kee, Arawageeki; tow, Vakêêke; zun, Alo:ke:k'ya; apj, Gołgéeki'yé. abbreviated ABQ, is the most populous city in the U.S. state of New Mexico. Its nicknames, The Duke City and Burque, both reference its founding in ...
,
New Mexico
)
, population_demonym = New Mexican ( es, Neomexicano, Neomejicano, Nuevo Mexicano)
, seat = Santa Fe
, LargestCity = Albuquerque
, LargestMetro = Tiguex
, OfficialLang = None
, Languages = English, Spanish ( New Mexican), Navajo, Ker ...
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
Drop City was a counterculture artists' community that formed near the town of Trinidad in southern Colorado in 1960. Abandoned by 1979, Drop City became known as the first rural "hippie commune".
Establishment
In 1960, the four original foun ...
") — buildings intended to be appropriate to their environment, notably to utilize
solar energy
Solar energy is radiant light and heat from the Sun that is harnessed using a range of technologies such as solar power to generate electricity, solar thermal energy (including solar water heating), and solar architecture. It is an essenti ...
well. Baer was fascinated with the dome geometry popularized by architect
R. Buckminster Fuller
Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...
. Baer was an occasional guest at Drop City, an arts and experimental community near Trinidad, CO. He wanted to design and construct 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. ''Home Work'', a book published in 2004 and edited by
Lloyd Kahn
Lloyd Kahn (born April 28, 1935) is an American publisher, editor, author, photographer, carpenter, and self-taught architect. He is the founding editor-in-chief of Shelter Publications, Inc., and is the former Shelter editor of the ''Whole Earth ...
, 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.
Zomes have also been used in the artistic, sculptural, and furniture areas. Zomadic, based in San Francisco, CA and founded by Rob Bell, incorporates zome geometry into artistic structures constructed primarily from CNC machined plywood components. Bell is a frequent attendee at
Burning Man
Burning Man is an event focused on community, art, self-expression, and self-reliance held annually in the western United States. The name of the event comes from its culminating ceremony: the symbolic burning of a large wooden effigy, referred ...
, a yearly artistic showcase event located in the Black Rock Desert of Nevada.
Richie Duncan of Kodama Zomes, based in southern Oregon has invented a structural system based on a hanging zome geometry, suspended from an overhead anchor point. Constructed of metal compressive elements and webbing tensile elements, the structures are able to be assembled and disassembled. This suspended zome system has been used in furniture, performing arts, and treehouse applications.
Yann Lipnick of Zomadic Concepts in France has an extensive study of, and multiple project construction of zomes in many different materials. He highlights the universal appeal and healing atmosphere that zomes provide, and has training classes and reference books on zome construction.
Construction set
The ''Zometool'' plastic
construction set
A construction set is a set of standardized pieces that allow for the construction of a variety of different models. The pieces avoid the lead-time of manufacturing custom pieces, and of requiring special training or design time to constr ...
is produced by a privately owned company of the same name, based outside of
Longmont, Colorado
The City of Longmont is a home rule municipality located in Boulder and Weld counties, Colorado, United States. Longmont is located northeast of the county seat of Boulder and north-northwest of the Colorado State Capitol in Denver.
Longmont ...
, and 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 small
rhombicosidodecahedron
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
It has 20 regular triangular faces, 30 square (geometry), square face ...
, 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. The idea of shape-coding the three types of struts was developed by Marc Pelletier and Paul Hildebrandt. To create the "balls," or nodes, Pelletier and Hildebrandt invented a system of 62 hydraulic pins that came together to form a mold. The first connector node emerged from their mold perfectly on April 1, 1992.
In the years since 1992, Zometool has extended its product line, though the basic design of the connector node has not changed so all parts to date are compatible with each-other. From 1992 until 2000, Zometool produced kits with connector nodes and blue, yellow, and red struts. In 2000, Zometool introduced green struts, prompted by French architect Fabien Vienne, which can be used to construct the regular tetrahedron and octahedron. 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.
[Rogers & Hildebrandt (2002)]
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. 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 of √2 aspect ratio, but as the connector nodes do not include holes at the required positions, the green struts instead fit into any of the 12 pentagonal holes with 5 possible orientations per hole, 60 possible orientations in all; using them is not as straightforward as the other struts.
At their midpoints, each of the yellow and red struts has a twist where 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 which are at a slight offset from the strut’s axis.
Among other places, the word zome comes from the term zone. 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. The blue 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.
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
, equipped with the standard inner product, also known as 3-dimensional Euclidean space.
Let
denote the
Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
and let
denote the symmetry group of the configuration of vectors
,
, and
. The group
, an example of a
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
, is known as the icosahedral group because it is the symmetry group of a regular
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
having these vectors as its vertices. The subgroup of
consisting of the elements with determinant 1 (i.e. the rotations) is isomorphic to
.
Define the "standard blue vectors" as the
-orbit of the vector
. Define the "standard yellow vectors" as the
-orbit of the vector
. Define the "standard red vectors" as the
-orbit of the vector
. A "strut" of the zome system is any vector which can be obtained from the standard vectors described above by scaling by any power
, where
is an integer. A "node" of the zome system is any element of the subgroup of
generated by the struts. Finally, the "zome system" is the set of all pairs
, where
is a set of nodes and
is a set of pairs
such that
and
are in
and the difference
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 cyclic group of order 2, 3, or 5, respectively. Hence, one may also describe the blue, yellow, and red struts as "rectangular", "triangular", and "pentagonal", respectively.
One may extend the zome system by adjoining green vectors. The "standard green vectors" comprise the
-orbit of the vector
and a "green strut" as any vector which can be obtained by scaling a standard green vector by any integral power
. As above, one may check that there are
=60 standard green vectors. 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).
Zome as a modeling system
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
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
, and the 4-dimensional polytopes related to the
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
and the
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from " ...
. However, the list of mathematical objects which are amenable to zome is long, and an exhaustive list is not forthcoming. Besides those already mentioned, one may use zome to model the following mathematical objects:
* Three of the four
Kepler-Poinsot polyhedra
* Regular
polyhedral compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
The outer vertices of a compound can be connected ...
s
*
Regular 4-dimensional polytopes and some compounds
* Many stellations of the
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
* Many stellations of the regular
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
*
Zonohedra
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
, especially the
rhombic enneacontahedron
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 Rhombus, rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedr ...
and
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
*
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
s in dimensions 61 or fewer
* Most
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...
(a major exception being those formed using the
snub
A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub.
In Awards and Lists
For awards, the term "snub" ...
operation)
* Many
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
s
* Thorold Gosset's
exceptional semiregular polytopes in 6, 7, and 8 dimensions
* A few of the
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s
*
Desargues configuration
In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues.
The Desargues configuration can be constructed in two dimensions f ...
* Two of the
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
The Catalan sol ...
s
* Classical and exceptional
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
s
*
Triality
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8) ...
(from Lie theory)
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
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
,
nanotube
A nanotube is a nanometer-scale hollow tube-like structure.
Kinds of nanotubes
* BCN nanotube, composed of comparable amounts of boron, carbon, and nitrogen atoms
* Boron nitride nanotube, a polymorph of boron nitride
* Carbon nanotube, includes g ...
, and
viral structures, and to make
soap film surfaces.
Notes
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
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', 3rd edition, Dover, 1973. .
* Brian C. Hall. ''Lie Groups, Lie Algebras, and Representations: An Elementary Introduction'', Springer, 2003. .
*
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. .
* Paul R. Hildebrandt & Marc G. Pelletier (1985)
"Geometric Modeling Kit and Method of Making Same" United States Patent No. 4,701,131.
* Paul Hildebrandt. Zome-inspired Sculpture. ''Proceedings, Bridges London: Connections between Mathematics, Art, and Music'', Reza Sarhangi and John Sharp (editors). (2006) 335–342.
* Paul Hildebrandt & Clark Richert (2012)
"Domes, Zomes, and Drop City" ''Bridges 2012''.
* Cass Wester (1973)
"Steve Baer and Holly Baer: Dome Home Enthusiasts" ''Mother Earth News'' 22.
* 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.
* Steven F. Rogers & Paul R. Hildebrandt (2002
"Connections for Geometric Modeling Kit" United States Patent No. 6,840,699 B2.
* Sadler, Simon (2006)
"Drop City Revisited" Journal of Architectural Education 59 (3), pp. 5-14.
* Michael Stranahan, Carlos Neumann, & Paul Hildebrandt (2011)
"How Not to Bring Your Product to Market" Bridges 2011.
External links
Zome buildings:
Examples of European zome buildingsExamples of zome usage in North American prefabricated housing construction
Zomes as art:
Kodama ZomesZomadicHeliss
Zome modelling system:
Zome modelisation- Zome modelisation - Open-Source Sketchup Plugin
Zome Creator- Source code for free Zome modelisation software
* {{MathWorld , urlname=Zome , title=Zome
ZometoolThe manufacturer's site.
by David Richter
by
George W. Hart
George William Hart (born 1955) is an American sculptor and geometer. Before retiring, he was an associate professor of Electrical Engineering at Columbia University in New York City and then an interdepartmental research professor at Stony Bro ...
an
Henri PicciottovZomefor building virtual Zome models
at the London Knowledge Lab
Japan Zome Cluba user's club in Japan (Japanese)
Metazomea project making Zome models with Zome
''Energy Management Company:
ZOME Energy Networks, a smart energy company
Construction toys
Educational toys
Building
Building engineering