U.S. patent number 3,722,153 [Application Number 05/034,163] was granted by the patent office on 1973-03-27 for structural system.
This patent grant is currently assigned to Zomeworks Corporation. Invention is credited to Stephen C. Baer.
United States Patent |
3,722,153 |
Baer |
March 27, 1973 |
STRUCTURAL SYSTEM
Abstract
A structural system in which the structural members are
interconnected so that they are parallel to the lines of the star
of the faces of an icosahedron. The system includes structural
members parallel to the lines of the star of the vertices of an
icosahedron and also includes structural members parallel to the
lines of the star of the midpoints of the edges of an
icosahedron.
Inventors: |
Baer; Stephen C. (Corrales,
NM) |
Assignee: |
Zomeworks Corporation
(Albuquerque, NM)
|
Family
ID: |
21874693 |
Appl.
No.: |
05/034,163 |
Filed: |
May 4, 1970 |
Current U.S.
Class: |
52/81.2; D25/13;
52/DIG.10; 52/653.1; 403/176 |
Current CPC
Class: |
E04B
1/32 (20130101); Y10T 403/347 (20150115); E04B
2001/327 (20130101); Y10S 52/10 (20130101); E04B
2001/3247 (20130101); E04B 2001/3294 (20130101) |
Current International
Class: |
E04B
1/32 (20060101); E04b 001/32 () |
Field of
Search: |
;52/81,80,79 ;135/1 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
|
|
|
|
|
|
|
1,116,877 |
|
1956 |
|
FR |
|
1,458,056 |
|
1966 |
|
FR |
|
1,196,348 |
|
1965 |
|
DT |
|
1,484,634 |
|
Jun 1967 |
|
FR |
|
Other References
Mathematical Models by Cundy & Rollett, Oxford Press 1961.
pages 76-83, 86-99, 120, 121 Table II, and 144-151 .
Space Arid Structures by John Borrego; MIT Press; 1968; pages 19,
88, 89 and 90. .
Dome Book 2, pages 10-12, 1971 distributed by Random House .
Useful Curves and Curved Surfaces, Architectural Record; April
1958; pages 245, 247, 249 .
Mathematical Models; 1962, Oxford University Press, page
111.
|
Primary Examiner: Abbott; Frank L.
Assistant Examiner: Raduazo; Henry E.
Claims
I claim:
1. A structural system comprising a plurality of interconnected
elongate structural members, said members joined to each other to
form a plurality of vertices, the angular relationship of the
joined members at the vertices being such that at least one member
is parallel to one of the lines of the star formed by diametral
lines through the midpoints of opposite faces of an icosahedron; at
least one member is parallel to one of the lines of the star formed
by diametral lines through each of the opposite vertices of an
icosahedron; and including additional elongate structural elements
each of which is parallel to at least one of the lines of the star
formed by diametral lines through the midpoints of opposite edges
of an icosahedron.
2. The structural system of claim 1 in which the primary structural
members comprise elongate structural members joined to each other
so that each is parallel to at least one of the lines of the star
formed by diametral lines through the midpoints of opposite faces
of an icosahedron.
3. The structural system of claim 1 in which the primary structural
members comprise elongate structural members joined to each other
so that each is parallel to at least one of the lines of the star
formed by diametral lines through each of the opposite vertices of
an icosahedron.
4. The structural system of claim 1 in which the primary structural
members comprise elongate structural members joined to each other
so that each is parallel to at least one of the lines of the star
formed by diametral lines through the midpoints of opposite edges
of an icosahedron.
5. The structural system of claim 1 in which the structural element
members comprise the edges of solid polyhedra.
6. The structural system of claim 2 in which the structural element
members comprise the edges of solid polyhedra.
7. The structural system of claim 3 in which the structural element
members comprise the edges of solid polyhedra.
8. The structural system of claim 4 in which the structural element
members comprise the edges of solid polyhedra.
9. The structural system of claim 1 and including a plurality of
connectors, the connectors adapted for joining the elongate
structural members along the star angles formed by the diametral
lines through the edge midpoints of an icosahedron, the diametral
lines through the face midpoints of an icosahedron, and the
diametral lines through the vertices of an icosahedron.
10. A structural system comprising a plurality of planar structural
members interconnected at the edges thereof to form a structural
surface, the edges of each of the joined planar members being such
that at least one edge of a first member is parallel to one of the
lines of the star formed by diametral lines through the midpoints
of opposite faces of an icosahedron; at least one edge of a second
member is parallel to one of the lines of the star formed by
diametral lines through each of the opposite vertices of an
icosahedron; and including additional planar structural members
each of which has one edge parallel to at least one of the lines of
the star formed by diametral lines through the midpoints of
opposite edges of an icosahedron.
Description
This invention relates to building structures and, more
particularly, to a structural system utilizing the five-fold
symmetries of the icosahedron and its dual, the dodecahedron.
The icosahedron and the dodecahedron are the two most complex of
the five regular Platonic polyhedra, the others being the
tetrahedron, the cube, and the octahedron. The regular Platonic
polyhedra are the only ones having faces which are equal regular
polygons and having equal vertices or corners. These polyhedra have
fascinated man since before the time of the ancient Greeks.
Plato believed that all objects are composed of four elements:
earth, air, fire, and water. The fundamental particles of fire were
supposed to have had the shape of the tetrahedron, those of air the
octahedron, those of earth the cube, and those of water the
icosahedron. The fifth shape, the dodecahedron, was reserved by God
for the shape of the Universe itself. These shapes are important
and fascinating because they are the distilled patterns of our
space; they are conjunctions of regularity and complexity. Our
world is filled with many variations and combinations of a
relatively few different patterns. There are only three dimensions
and five regular polyhedra. The icosahedron with 20 sides and the
dodecahedron with 12 sides have more corners and more faces than
the other three. They are the only regular polyhedra in which the
number "five" occurs: there are five triangles which meet to form
each of the vertices of the icosahedron and there are five sides of
each face in the dodecahedron.
The tetrahedron, cube, and octahedron and their symmetries often
appear in nature and in man's work. Crystals of sodium bromate are
in the form of tetrahedra; sodium chloride, common salt, has cubic
crystals; and alum crystals are in the form of octahedra. These
three simple shapes have been used extensively by man. The cube of
course is used as a container because of its convenient geometric
properties, that is, cubic containers will pack together and the
edges of a cube may be extended to form different shapes having the
same symmetries. The tetrahedron and the octahedron have been used
extensively in constructing space frames and rigid structures.
The dodecahedron and the icosahedron have not been so utilized.
Because each of them has five-fold symmetry, they cannot occur as
crystals. Crystals are built up of molecules which are located in
systems of regular points and it is impossible for such a system to
have five-fold symmetry. This may be demonstrated if one attempts
to cover a plane surface with regular pentagonal tiles. There will
always be gaps between some of the tiles in such a case. Other
regular tiles having three, four or six sides (triangular, square,
and hexagonal) will completely cover a plane surface. There are
crystals, for instance, MoAl.sub.12 which contain icosahedral
elements within their component structure but their basic
symmetries are not icosahedral. In these cases the icosahedral
elements merely "go along for the ride."
The icosahedron and the dodecahedron also appear in the structure
of the skeletons of certain tiny sea animals called radiolarians,
for example, Circogonia icosahedra and Circorrhegma dodecahedra.
Among man-made objects the icosahedron and dodecahedron are most
frequently used as decorations or ornaments. Christmas decorations
are sometimes made in the shape of stellated dodecahedra. Some desk
calendars are made in the shape of a dodecahedron with each month
occupying one of the 12 faces. During the last 15 years the
icosahedron and dodecahedron have appeared as surface patterns in
the arrangements of members of the now famous Geodesic Domes of R.
Buckminster Fuller. Reference is made, for example, to the
illustrations in Fuller's U.S. Pat. No. 2,682,235 of June 29,
l954.
The present invention is radically different from the process one
would use in making icosahedral ornaments or Goedesic domes. The
present invention involves utilizing the five-fold symmetry of the
dodecahedron and icosahedron for a novel crystal-like structural
system. It has already been stated that five-fold symmetry is
impossible in natural crystal growth because there is no
intelligence to guide the growth of the structure. My invention is
a system and structure which allows one to exploit the simplicity
and elegance of the natural crystal in order to construct
structures far more complex and beautiful than possible with
inorganic crystalline shapes.
Definition of Terms
In order to more readily comprehend the explanation which follows,
certain of the terms which may be unfamiliar are defined below.
Zonohedron -- a polyhedron with every face a polygon composed of
pairs of equal parallel opposite sides.
Zone -- a set of edges in a zonohedron that are all parallel to one
another.
Zome -- a man made structure derived from zonohedra, where the
zones of the zonohedron may be stretched or shrunk or removed to
produce, if desired, an asymmetric dome shaped structure.
Stretching a Zone -- any zonohedron or zone may have its shape
altered without altering angles by stretching the set of lines and,
thus, a band of faces girdling the figure. The stretching may be
accomplished by lengthening the individual members or by adding
more components.
Six Zone Truss -- a truss system whose principal members are
substantially equal in length and parallel to the lines of the star
formed by the diameters through the vertices of an icosahedron (or
through the midpoints of the pentagonal faces of a
dodecahedron).
Ten Zone Truss -- a truss system whose principal members are
substantially equal in length and parallel with the lines in a star
created by the diameters through the midpoints of the triangular
faces of an icosahedron.
Thirty One Zone System -- a structural system utilizing members
which are parallel to the lines in the three stars of the vertices,
face midpoints, and edge midpoints of the icosahedron or its dual
the dodecahedron.
IN THE DRAWINGS
FIG. 1 is a perspective view of an icosahedron;
FIG. 2 is a perspective view of a dodecahedron;
FIG. 3A is a perspective view of an icosahedron showing the 10
diameters through the midpoints of the faces comprising the 10 zone
star of the icosahedron and some of the 15 diameters bisecting the
edges thereof;
FIG. 3B is a perspective of a dodecahedron showing the six
diameters through the midpoints of the faces comprising the six
zone star of the dodecahedron;
FIG. 4 is a perspective view of a triacontahedron, a simple 30
sided structure utilizing the principles of the present
invention;
FIG. 5 is a perspective view of an enneacontahedron, a 90 sided
structure showing an application of the 10 zone system of the
present invention;
FIGS. 6A and B show two basic cells of the six zone truss
systems;
FIGS. 7A - E show five basic cells of the 10 zone truss
systems;
FIG. 8 is a perspective view of an extended truss forming an arch
utilizing the basic cells of the six zone system; and
FIG. 9 is a perspective view of a structure illustrating the
application of panels of the 10 zone structural system to a domed
structure.
Description of the Preferred Embodiment
Referring now more particularly to the drawings in which the same
reference numerals refer to identical parts in each of the several
views, the present invention utilizes the relationships and angles
of the 20 faces, 30 edges, and 12 vertices of the icosahedron shown
in FIG. 1 and of the dual of the icosahedron, the dodecahedron,
illustrated in FIG. 2. The dodecahedron has 12 faces, 20 vertices,
and 30 edges. These two polyhedra are duals or reciprocals of each
other. Simply stated, the principle of duality is that each of the
vertices of a regular polyhedron may be replaced by a polar plane.
Thus, the polyhedra illustrated have many of the same symmetries.
Much of the following description will be equally applicable to
both the icosahedron and the dodecahedron because of their
relatiOnship as duals.
An icosahedron is illustrated generally at 10 in FIGS. 1 and 3A.
The 10 diameters which pass through opposite faces of the
icosahedron (and through the opposite vertices of its dual, the
dodecahedron) are numbered 12, 14, 16, 18, 20, 22, 24, 26, 28, and
30 and are shown in FIG. 3A. It can be demonstrated that these
lines which shall be here called the star of the faces of the
icosahedron or 10 zone star have a complex angular relationship to
one another. The lines from two sets of angles with each other, the
angles being: arc cos 1/3 (70.degree. 31' 44") and its supplement
109.degree. 28' 16" and arc sin 2/3 (41.degree. 48' 38") and its
supplement 138.degree. 11' 22" .
Also illustrated in FIG. 3A are some of the diametral lines which
bisect the edges of the icosahedron. In order to preserve clarity
only five of these lines are shown, 11, 13, 15, 17, and 19. Since
there are 30 edges in the icosahedron there are a total of 15 such
diametral lines. Because the faces of the icosahedron are all
regular, each of the 15 diametral lines through the midpoints of
the edges of the icosahedron will bisect four of the angles formed
by the lines of the 10 zone star. In the description which follows,
the 15 diametral lines will be referred to as the "edge star".
In FIG. 3B the six zone star is shown. The six zone star, or the
star of the vertices of the icosahedron (or of the faces of the
dodecahedron 11) as illustrated is made up of the six diameters
through the vertices of an icosahedron, or the six diameters
through the midpoints of the faces of a dodecahedron. These are the
lines numbered 32, 34, 36, 38, 40, and 42. These lines all make the
same angle with each other: arc tan 2 (63.degree. 26' 06") and its
supplement 116.degree. 33' 54". In the present invention the
structural members used to carry out the principles of the
invention are parallel to at least one of the lines of these stars.
It is an integral of the invention that structural members forming
a plane may be replaced by structural planar faces. Such structural
faces may be stressed skins, prestressed concrete slabs, rigid
plastic sheets, honeycomb sandwich panels, or any of the myriad
building material usable for making such sheets. In the use of
sheets for practicing the invention, the edges of the sheets are
parallel to the lines of the stars and are rigidly interconnected
to form a continuous skin for a structure.
The previously described edge star, 15 lines intersecting the edges
of the icosahedron, is also related to the lines of the six zone
star. It has already been noted that the six zone star and the 10
zone star are interrelated because of the duality of the
icosahedron and dodecahedron. Thus, the 15 diametral lines through
the midpoints of the edges of the icosahedron are identical to the
15 lines through the midpoints of the edges of the dodecahedron
although not separately illustrated as such. Each of the 15 edge
star lines will also bisect two of the angles formed by the lines
of the six zone star. As will be shown in the description below,
the lines of the 15 edge bisectors may be utilized to strengthen
structural units conforming to the six zone and 10 zone alignments.
In addition, because of the interrelationship between the 15 edge
bisectors and both the six zone and 10 zone stars the 15 edge
bisector lines form a link between 10 zone and six zone structures
and permit utilization of combinations of the two systems for even
more intricate and beautiful structural designs.
A triacontahedron, a 30 sides zonohedron associated with the star
of the faces of the dodecahedron is shown in FIG. 4. This figure
may be used to illustrate the beginnings of the structural system
of the present invention. The triacontahedron shown generally at 44
in FIG. 4 is made up of a plurality of diamond shaped faces 46, the
edges of which are aligned with the star lines of the dodecahedron
so that the acute angles of this figure are 63.degree. 26' 06" and
the obtuse angles are 116.degree. 33' 54". For purposes of clarity
in the illustrations the angle 63.degree. 26' 06" will hereinafter
be referred to as angle .alpha. and 116.degree. 33' 54" as angle
.beta.. In FIG. 4 the diamond shaped faces are bisected by a
diagonal 48 which in the case of an open framework such as is
illustrated would considerably stiffen the structure. It is
contemplated as an integral part of the present invention that each
of the faces 46 of a structure such as is illustrated in FIG. 4
could be made of an integral planar sheet as previously described,
each of the edges of which is parallel to at least one of the lines
of the six zone star, 10 zone star, or of the 15 lines of the edge
star.
Each of the diagonals 48 which, as has been noted, serve to form a
series of triangles to stiffen the structure already illustrated,
is parallel to one of the 15 lines of the previously described edge
star. The diagonals 48 triangulate (divide into triangles) the
diamond shaped faces of the triacontahedron 44 and as will be shown
are parallel to diagonals also of the enneacontahedron illustrated
in FIG. 5. The enneacontahedron is a 90 sided polyhedron having a
plurality of diamond shaped faces of two kinds: a slender diamond
52 and a larger diamond shape 54. Each of the edges of the diamond
shaped faces comprising the enneacontahedron shown in FIG. 5 is
parallel to one of the 10 lines making up the 10 zone star of the
faces of the icosahedron (and the vertices of the
dodecahedron).
Each of the diamond shaped faces 46 of the triacontahedron 44
comprises four edges 56 of equal length. If the length is expressed
as L, the length of the diagonal 48 is approximately 1,0514L. The
polyhedron structure 44 may be made up of a plurality of
interconnected structural elements 56 and 48 to result in an open,
airy form. The structural elements may be pipes, rods, beams, or
any other elongate form. In FIG. 4, the elements are shown
interconnected by means of ball-shaped connectors 58. The
connectors may be any device which permits the elements to be
joined together in the manner and at the angles illustrated. The
connectors in this application may be in the shape of dodecahedra
with the structural members welded or otherwise attached to the
faces thereof.
The lines of the edge star are arranged in such a way that each
bisects one 41.degree. 48' 38" , one 138.degree. 11' 22", and one
109.degree. 28' 16" angle formed by the lines of the 10 zone star,
and each is perpendicular to two of the lines of the 10 zone star.
Also, each lines of the edge star bisects one 63.degree. 26' 06"
angle and one 116.degree. 33' 54" angles of the six zone star and
each is perpendicular to two lines of the six zone star. The lines
of the edge star are related to each other is such a fashion that
all angles between lines of the edge star are multiples of
36.degree., 60.degree., or 90.degree..
With the structure of the triacontahedron 44 now noted, the growth
of the system from such a starting point, as with a crystalline
array may be described. In FIG. 6A, one of the basic cells of the
six zone truss is shown, the other basic cell of the six zone
system being illustrated in FIG. 6B. The six zone A cell comprises
a plurality of structural elements 56 of the same length L as the
elements of the triacontahedron 44 of FIG. 4. The diagonal members
48 used previously are used in the A cell for strengthening the
structural unit. The connectors 58 are again identical to those
utilized for the triacontahedron. As might be expected from the use
of the identical elements, there are six faces in the A cell, each
being the same as face 46 of FIG. 4. The angles in the faces are
the angles of the six zone star the angles .alpha. and .beta..
The six zone B cell in FIG. 6B is in the obtuse configuration, that
is, the obtuse angles .beta. of three faces meet at two diagonally
opposite vertices. As a result, these vertices tend to be closer
together than in the more conventional A cell. The short diagonal
47 between the obtuse vertices is approximately .562 times the
length of one of the edge members 56. The short diagonal member 47
while strengthening a six zone cell is actually a member of the 10
zone system, being parallel to one of the lines of the 10 zone
truss system.
Since the faces of the A and B cells of the six zone system are
congruent to the faces of the triacontahedron, a series of suitably
arranged A and B cells may be superimposed on the faces of that
polyhedron to cause the structure to grow in any of the face
directions.
As previously noted there are two sets of faces making up the
surface of the enneacontahedron, the slender diamond 52 and a
larger diamond shape 54 which is also known as a Maraldi diamond.
The relationship among the diamond faces is more clearly seen in
FIG. 5. The angles of the two diamonds 52 and 54 are as
follows:
In the slender diamond 52, the acute angle is 41.degree. 48' 38"
which will be referred to as .gamma. and its supplement is
138.degree. 11'22" which will be referred to as .DELTA. and the
angles of the Maraldi diamond 54 are 70.degree. 31' 44" which will
be referred to as angle .phi. and its supplement is 109.degree. 28'
16" which will be referred to as .theta.. Each of the diamond faces
52 and 54 of the enneacontahedron 50 comprises four edges 62 of
equal length. If the length is expressed as L the length of the
short diagonal 64 is approximately 0.714L and the length of the
long diagonal 66 of the Maraldi diamond 54 is 1.155L. The
polyhedral structure 50 may be made up of a plurality of
interconnected structural elements 62 with additional reinforcing
members 64 and 66. As with the polyhedral structure 44, the
structural elements of the enneacontahedron 50 may be pipes, rods,
beams or any other elongate forms. They may be interconnected by
means of ball-shaped connectors 68 or any other device which
permits the elements to be joined together at the specific angles
of the 10 zone system.
The five basic cells of the 10 zone structural system of the
present invention are illustrated in FIGS. 7A through 7E. These
cells are the basic building blocks of the 10 zone system and
illustrate the variety of angular relationships which the
structural elements of the 10 zone system may have when
interconnected. It is possible to join one cell to another at one
of the edges thereof in order to expand the structural units in any
direction. Thus, in the construction of a truss utilizing the basic
cells of the present invention one need merely add additional rows
of similar truss cells end to end to gain length, side by side to
gain lateral strength, and several rows deep for greater
stiffness.
Each of the cells of the 10 zone system illustrated in FIGS. 7A
through 7E comprises a plurality of primary structural elements 62,
the primary elements being joined together in the form of four
sided rhomboidal planar figures in the shape of the slender
diamonds 52 and the Maraldi diamonds 54 making up the faces of the
enneacontahedron and each of which utilize the basic angles of the
10 zone system. Included within the cell members illustrated are
additional structural elements 64 and 66 for strengthening the
structural units.
The rhomboidal faces of the five cells of the 10 zone system are
made up of combinations of the slender and Maraldi diamonds 52 and
54 previously described as making up the face of the
enneacontahedron 50. In FIG. 7A the A cell of the 10 zone system is
illustrated, each of the faces of the A cell is a Maraldi diamond
54. In the case of the A cell the faces are joined together in the
acute configuration. For convenience of reference, because of the
perspective of the illustration the angles of the diamond faces are
shown. Thus, in the A cell FIG. 7A the acute configuration means
that in two corners thereof the three acute angles .phi. meet
together as illustrated. In the B cell illustrated in FIG. 7B, each
of the faces of the cell are Maraldi diamonds 54 in the obtuse
configuration. This means that at two of the corners of the A cell
the three obtuse angles .theta. meet to yield the configuration of
the B cell. One of the interesting features of the configuration of
the B cell is that the central diagonal 62 reinforcing the cell
between the two corners having the three obtuse angles meeting is
of the same length as all of the edge members. Not only is the
central diagonal 62 of the B cell of the same length as each of the
edge members 62 of the B cell, but it is parallel to one of the
lines of the 10 zone star. Thus, it is a standard member of the 10
zone system.
The C cell illustrated in FIG. 7C comprises two parallel slender
diamonds 52 and four Maraldi diamonds 54 arranged into parallel
sets. The arrangement of these rhomboidal faces is in the acute
configuration, that is, in which two of the corners have the acute
angles of the diamond faces meeting each other. Thus, in FIG. 7C
the rear left corner of the C cell has two vertices with .phi.
angles and one with the .gamma. angle meeting, and this combination
of angles is repeated in the front right hand corner of the cell
illustrated.
In the 10 zone D cell illustrated in FIG. 7D the cell comprises a
combination of four of the slender diamonds 52 and two of the
Maraldi diamonds 54. The Maraldi diamonds are the top and the
bottom faces of the cell as illustrated and all the other faces are
the slender diamonds having the .gamma. and .DELTA. angles. As has
already been noted, each of the edge members of the D cell, in the
same manner as the A, B, and C cells, comprises the primary
structural element 62 and in addition utilizes the additional
structural elements 64 and 66 as diagonals for reinforcement
purposes.
In the ten zone E cell illustrated in FIG. 7E the cell comprises a
combination of four Maraldi diamonds 54 and two of the slender
diamonds 52 in the obtuse configuration. This configuration is
similar in many respects except for the angles to the obtuse
configuration of the six zone B cell illustrated in FIG. 6B. In the
10 zone E cell the slender diamonds are at the extreme left and
right of the cell. The obtuse angles at which the structural
elements making up the cell meet in two of the corners are
illustrated by the angles .theta. and .DELTA.. Because of the
obtuse configuration of the E cell the structure is not as rigid as
would be desired. Therefore, an extra structural element 66 may be
connected between the connection points of the obtuse angles. This
extra structural member 76 is approximately 0.419 of the length of
the primary structural elements 62 and is disposed at an angle
which makes it part of the six zone system. Thus, it may be seen in
this particular structural unit of the present system that the six
zone system and the 10 zone system are complementary and, in
certain cases structural elements parallel to the six zone start
may be used in structures whose primary elements are parallel to
the 10 zone star.
It should be borne in mind in viewing the cells of the six and 10
zone systems that a diagonal bisector through the obtuse angles of
the cells will subdivide the cell into a non-regular octahedron and
two non-regular tetrahedra. Each of the acute parallelepiped cell
structures forms a strong truss cell when thus subdivided. The
cells in the obtuse configuration are suitable as shells only. They
do not subdivide in the same manner as the acute cells and thus
require the "special" diagonals which have previously been
described. The effect of the "special" diagonals is to form six
non-regular tetrahedra having a common edge which is the "special"
diagonal.
In FIG. 8 a truss utilizing the angles of the six zone system is
illustrated. Each of the primary structural members of the truss is
equivalent to the primary structural elements 56 of the six zone
cells shown in FIG. 6A and 6B. The ball connectors 58 are used in
the same manner to connect the primary structural elements 56 and,
in addition the reinforcing structural members, the diagonals 48
are used to give rigidity to the overall structure. It should be
noted that because of the arrangement of the primary structural
members in one of the six angular directions of the six zone system
it is possible for the structural elements to join together in
cellular arrays with smooth transitions from one direction to
another. The same, of course, applies to the use of structural
elements of the 10 zone system, or of a combination of elements of
the six and 10 zone systems with elements of the edge star
relationship in what I have called the 31 zone system. Cellular
arrays of incredible complexity and beauty are available by simply
adding units one to another in the directions dictated by the
angular relationships of the zonal systems of the present
invention.
An even more striking and beautiful structure utilizing the ten
zone system is shown in FIG. 9. FIG. 9 is a building utilizing as
its basic components a plurality of diamond shaped structural
panels the edges of which conform to the angles of the 10 zone
system. Thus, each of the faces of the building structure 80 is
either a slender diamond 52 or a Maraldi diamond 54 as found in the
enneacontahedron previously described. By joining the structural
panels along the lines of the 10 zone system an extraordinary
structure results. It should be noted that the structure 80
comprises a generally spherical dome 82 resting on a partial
spherical dome 84 and fused with an elongate partially cylindrical
structure 86.
An additional feature of the structural system of the present
invention resides in the ability to form a plurality of blocks or
hollow, solid forms in the shape of the previously described cells
of the six, 10, or 31 zone systems. It is thus possible to utilize
such blocks as, for example, a child's game in which the child is
able to combine the blocks into myriad possible shapes by utilizing
the 15 planes found in the six zone system or the 45 planes of the
10 zone system. In addition to use as toys, the blocks of the
present structural system could be made in extremely large sizes to
permit their use as building blocks for the construction of
buildings of normal size. The possible applications of the resent
invention are limited only by the ingenuity of the mind which sets
about its utilization.
The concept of using blocks in combinations related to the six and
10 zone solid-cell blocks was mentioned, but not pursued, in a book
published in Germany in 1938: Gerhard Kowalewski, Der Keplerische
Koerper und Andere Bauspiele. Kowalewski was concerned with two of
the regular polyhedra unknown to the ancient world and discovered
by Kepler: the two stellated dodecahedra and particularly with
coloring of the various facets of the constructs. As far as I have
been able to determine no one previously has discovered the
interrelationship of the six and 10 zone cells heretofore described
and their utilization for both practical and esthetic
applications.
Because of the previously described interrelationship of the six,
10, and 15 zone system--either separately or when combined into the
31 zone system which utilizes all of the interrelated stars of the
dodecahedron--building blocks having equal length edges, and in
many cases, congruent faces, permit combining the building units
into regular shapes, e.g., dodecahedra or icosahedra or into
structures as complex as the domed structure in FIG. 9.
As is well known, the golden section is one of the regularly
occurring relationships in polyhedra having five-fold symmetry. The
golden section which is sometimes also called the divine proportion
is a relationship between the portions of a line in which the ratio
of the whole to the larger part is equal to the ratio of the larger
to the smaller part. This is expressed mathematically as .tau. =
(.sqroot.5 + 1)/(2) = 1.6180. I have noted that there is a striking
relationship among the rhomboidal faces of the cells previously
described in which the golden section figures prominently.
Thus, if the equal edges of the planar faces 46 of a typical six
zone cell are of a length a, and the short diagonal 48 of the face
is length c, the long diagonal of the face is c.tau., where .tau.
is the golden section. If the sides of the slender diamond 52 and
of the Maraldi diamond 54 of the 10 zone system are of a length b,
the short diagonal 66 of the Maraldi diamond is equal to length c,
the long diagonal of the slender diamond 52 is c.tau. and its short
diagonal 64 is c/.tau.. The specific relationship of the lengths a,
b, and c is a = c cos 18.degree. or 0.9510565c, and b = c cos
30.degree. or 0.8660254c.
Other typical relationships among the edge members and the zonal
cells are, e.g., the long diagonal from acute vertex to acute
vertex in the six zone A cell is b.tau..sup.2 where the edges 56
are of length a. The short diagonal 47 in the six zone B cell is
b/.tau. where the edges 56 are of length a. The short diagonal 76
of the ten zone E cell is a/.tau..sup.2 where the edge members 62
are of length b.
In the previous description, each of the cells, and the rhomboidal
faces thereof have been regular, that is, all the edges have been
of the same length. It is an integral part of the present invention
to utilize the angular relationships of the previously described
rhomboidal faces but altering the shape thereof by elongating sets
of edges. I have called this process "stretching a zone." Thus, a
zonohedron may be elongated by elongating parallel edge members
thus stretching a plurality of faces making up a band along or
around the structure. The angular relationships among the
structural members or the edges of the planar faces are identical
to those in which equal length members or edges are utilized.
I call combinations of stretched (or shrunk) zones (rhomboidal
faces) zomes. These are the resultant structures, non-regular
polyhedral structures in which planar faces and other structural
members are combined utilizing the angular relationships heretofore
outlined. The zomes may be formed into combinations of dome shaped
clusters in which symmetries of the system permit unlimited
expansion and soaring, fantastic constructs that are both beautiful
and livable at the same time.
While certain embodiments of the invention has been shown and
described, it will be obvious that other adaptations and
modifications can be made without departing from the true spirit
and scope of the invention.
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