U.S. patent application number 11/579307 was filed with the patent office on 2008-02-21 for three dimensional polyhedral array.
Invention is credited to Samuel J. Lanahan.
Application Number | 20080040984 11/579307 |
Document ID | / |
Family ID | 39082294 |
Filed Date | 2008-02-21 |
United States Patent
Application |
20080040984 |
Kind Code |
A1 |
Lanahan; Samuel J. |
February 21, 2008 |
Three Dimensional Polyhedral Array
Abstract
A polyhedral array comprises a plurality of discrete polyhedrons
and a connection network comprising connections that connect the
polyhedrons. The discrete polyhedrons are spaced apart from each
other at equilibrium in a predetermined generally regular pattern,
and each polyhedron is comprised of edges, faces and vertices. The
connection network at least partially constrains the discrete
polyhedrons relative to each polyhedron's six degrees of freedom.
In some embodiments, the connections extend along the bias of the
array. In other embodiments, the connections extend along radial
directions. All implementations can be constructed as regular
arrays or in lattice networks.
Inventors: |
Lanahan; Samuel J.;
(Portland, OR) |
Correspondence
Address: |
KLARQUIST SPARKMAN, LLP
121 SW SALMON STREET, SUITE 1600
PORTLAND
OR
97204
US
|
Family ID: |
39082294 |
Appl. No.: |
11/579307 |
Filed: |
August 15, 2006 |
PCT Filed: |
August 15, 2006 |
PCT NO: |
PCT/US06/31940 |
371 Date: |
October 31, 2006 |
Current U.S.
Class: |
52/79.9 ;
434/278; 446/85; 52/80.1 |
Current CPC
Class: |
E04B 2001/3294 20130101;
E02B 3/04 20130101; E02B 3/129 20130101; F28F 13/003 20130101; E04B
1/34384 20130101; E04B 1/344 20130101; F28F 13/06 20130101; E04B
2001/3223 20130101; E04B 1/3211 20130101 |
Class at
Publication: |
52/79.9 ;
434/278; 446/85; 52/80.1 |
International
Class: |
E04H 1/00 20060101
E04H001/00 |
Claims
1. A polyhedral array, comprising: a plurality of discrete
polyhedrons that are spaced apart from each other at equilibrium in
a predetermined generally regular pattern, each polyhedron being
comprised of edges, faces and vertices; and a connection network
comprising connections extending along bias directions to connect
the polyhedrons; wherein the connection network at least partially
constrains the discrete polyhedrons with respect to each
polyhedron's six degrees of freedom.
2. The array of claim 1, wherein each polyhedron's six degrees of
freedom are defined as the ability to translate in the X, Y and/or
Z directions of a coordinate reference frame, and the ability to
rotate about the X, Y and/or Z directions.
3. The array of claim 1, wherein the polyhedrons are arranged in
multiple, generally parallel layers.
4. The array of claim 1, wherein at least some of the plurality of
polyhedrons generally occupy a first plane, and wherein the bias
directions along which the connections extend intersect the first
plane.
5. The array of claim 3, wherein the bias directions are oriented
at angles of approximately 45 degrees to the first plane.
6. The array of claim 1, wherein the polyhedrons are arranged in
multiple layers, and wherein the bias directions along which the
connections extend are inclined at about 45 degrees relative to an
expected direction of a resolved load on the array.
7. The array of claim 1, wherein at least one of the connections
extends between an edge of a first of the polyhedrons and an edge
of a second of the polyhedrons.
8. The array of claim 1, wherein at least one of the connections
extends between a face of a first of the polyhedrons and a face of
a second of the polyhedrons.
9. The array of claim 1, wherein at least one of the connections
extends between a vertex of a first of the polyhedrons and a vertex
of a second of the polyhedrons.
10. The array of claim 1, wherein at least one of the connections
extends between one of a group consisting of a face, an edge and a
vertex of a first polyhedron, and a different one of the group
consisting of a face, an edge and a vertex of a second
polyhedron.
11. The array of claim 1, wherein the array is coherent.
12. The array of claim 1, wherein the array is omni-extensible.
13. The array of claim 1, wherein an existing array can be
increased in size by connecting additional discrete polyhedrons to
existing discrete polyhedrons with additional connections without
other modifications to the existing array.
14. The array of claim 1, wherein each discrete polyhedron is a
finitely closed structure having structural integrity independent
of the respective connections to which said discrete polyhedron is
connected and independent of other discrete polyhedrons in the
array.
15. The array of claim 1, wherein the interconnecting network is
configured in a generally repeating pattern.
16. The array of claim 1, wherein the connections comprise at least
one of mechanical elements, bonds or guest molecules, ligands and
ligatures.
17. The array of claim 1, wherein the connections comprise
mechanical elements that have greater resiliency than the
polyhedrons.
18. The array of claim 1, wherein the connections are mechanical
elements comprising spring-shaped portions.
19. The array of claim 1, wherein each polyhedron occupies a unique
location at equilibrium that can be specified with Cartesian
coordinates.
20. The array of claim 1, wherein the array is anisotropic, with at
least one polyhedron having specific properties different from
another of the plurality of polyhedrons.
21. The array of claim 1, wherein at least one of the connections
can have predetermined properties different from another of the
connections.
22. The array of claim 1, wherein at least one of the connections
has a property that varies along its length.
23. The array of claim 1, wherein the discrete polyhedrons comprise
a first material and the connections comprise a second material
different from the first material.
24. The array of claim 1, wherein at least one of the plurality of
polyhedrons comprises a closed polyhedron having a majority of
closed faces.
25. The array of claim 1, wherein at least one of the plurality of
polyhedrons comprises an open polyhedron each having a majority of
open faces.
26. The array of claim 1, wherein the plurality of polyhedrons
comprises at least one interiorly hollow polyhedron.
27. The array of claim 1, wherein the plurality of polyhedrons
comprises at least one solid polyhedron.
28. The array of claim 1, wherein each polyhedron satisfies the
condition of having a face, an edge or a vertex approximately
coincident with one of the six sides of a cube circumscribing the
polyhedron.
29. The array of claim 1, wherein the polyhedrons are selected from
the list consisting of cubes, icosahedrons, truncated cubes,
truncated octahedrons, truncated icosahedrons, cubo-octahedrons,
dodecahedrons, truncated dodecahedrons, icosidodecahedrons,
rombicosidodecahedrons, snub dodecahedrons, truncated
cubo-octahedrons, stellated forms, deltahedrals and dual
tetrahedrals.
30. The array of claim 1, wherein at least one of the polyhedrons
is formed of two halves.
31. The array of claim 30, wherein the polyhedron is an icosahedron
and has a geodesic saw tooth-shaped equator defining the two
halves.
32. The array of claim 1, wherein at least one of the connections
extends continuously from a first polyhedron, to a second
polyhedron and to an nth polyhedron.
33. The array of claim 1, wherein at least some of the connections
are compression members configured primarily to resist compression
forces.
34. The array of claim 1, wherein the predetermined regular pattern
in which the polyhedrons are arranged can include spaces occurring
at generally regular uniform intervals.
35. The array of claim 1, wherein at least some of the polyhedrons
are icosahedrons.
36. The array of claim 1, wherein the plurality of polyhedrons
includes at least one interior polyhedron connected to twelve
adjacent polyhedrons.
37. The array of claim 1, wherein the discrete polyhedrons can move
relative to each other in response to applied loads, but remain
spaced apart from each other at equilibrium and within a selected
working load range.
38. The array of claim 1, wherein a force above a predetermined
working load range applied to the array can urge two adjacent
polyhedrons from an equilibrium position in which the adjacent
polyhedrons are separated from each other into an under load
position in which the adjacent polyhedrons are in contact with each
other.
39. A polyhedral array, comprising: a plurality of discrete
polyhedrons that are spaced apart from each other at equilibrium in
a predetermined generally regular pattern, each polyhedron being
comprised of edges, faces and vertices; and a connection network
interconnecting the discrete polyhedrons, the connection network
comprising radial connections, wherein a first end of each radial
connection and a corresponding first connection location on one of
the polyhedrons occupy a first plane, and wherein a second end of
each radial connection and a corresponding second connection
location on another of the polyhedrons occupy a second plane that
is not parallel to the first plane, wherein the interconnecting
network at least partially constrains the discrete polyhedrons
relative to each polyhedron's six degrees of freedom.
40. The array of claim 39, wherein the first and second planes of
the respective first and second connection locations are mutually
orthogonal.
41. A polyhedral array, comprising: a plurality of discrete
polyhedrons that are spaced apart from each other at equilibrium in
a generally regular pattern, each polyhedron being comprised of
edges, faces and vertices; and a connection network interconnecting
the discrete polyhedrons, the connection network comprising
connections having a first end connected in a first plane and a
second end connected in a second plane different from the first
plane, wherein the connection network at least partially constrains
the discrete polyhedrons relative to three Cartesian axes, and
adjacent polyhedrons are oriented in a face to face relationship
relative to each other.
42. The array of claim 41, wherein the connections are curved
mechanical elements, and wherein each curved mechanical element has
a first end connected to a first face of a first polyhedron and a
second end connected to a second face of a second polyhedron, the
first and second faces occupying different planes.
43. The array of claim 41, wherein when the array is subjected to a
load above a predetermined working load range, the array deforms
such that at least two of the polyhedrons make face to face contact
with each other.
44. The array of claim 41, wherein when the array is subjected to a
force or torque above a predetermined working range, the array
deforms such that at least one of the polyhedrons undergoes a
predetermined transformation in shape.
45. The array of claim 44, wherein the at least one of the
polyhedrons is reversibly transformed such that the at least one of
the polyhedrons returns to an original shape if the force or torque
is removed.
Description
CROSS REFERENCE TO RELATED APPLICATION
[0001] This application is a .sctn. 371 U.S. National Stage of
International Application No. PCT/US2006/031940, filed Aug. 15,
2006, which was published in English under PCT Article 21(2).
FIELD
[0002] This application relates to arrays, and in particular,
arrays of materials comprised of a plurality of geometric
polyhedrons and a connection network interconnecting the
polyhedrons.
BACKGROUND
[0003] Choices of materials, and how given materials are arranged
to achieve a resulting structure with desired characteristics, are
key aspects of many innovations. Decisions about materials govern
the built environment from large scale structures (e.g., buildings
and roads), to common articles of a familiar scale and to new
explorations in design of nanoscale, microscale and other small
scale technologies.
[0004] Designing objects by specifying materials arranged in known
arrays helps the designer predict the characteristics of the
finished object. Using known arrays helps specify the design to
those who assemble, build or configure it. Known arrays that follow
regular patterns allow for more predictability of interactions at
their boundaries, e.g., with other objects and/or the
environment.
[0005] Buckminster Fuller, among others, pioneered use of
polyhedron-based structures that provided superior
strength-to-weight ratios and other characteristics, particularly
for architecture applications. This prior work emphasized designs
in which a predetermined number of polyhedral-based modules were
assembled together, and in some cases, adjacent polyhedrons shared
components with each other. The issues of how to join the resulting
structures with other objects, including existing objects as well
as objects to be added at a later time, was not explored in any
detail.
[0006] Applicant's prior U.S. patent application Ser. No.
10/932,403, which is incorporated herein by reference, discloses an
array, referred to in some forms as a "structural fabric," which is
comprised of discrete icosahedral elements and interconnecting
elements in tension that interconnect the icosahedral elements. By
arranging the interconnecting elements primarily in tension, at
least some of the properties of a tensegrity ("tension integrity")
structure can be achieved. The prior application emphasized use of
icosahedrons and truncated icosahedrons, and arranging the
interconnecting elements orthogonally relative to the icosahedral
elements. The prior application recognized the importance of
providing a scalable design.
[0007] Applicant also recognized that characteristics of structural
materials in addition to their strength-to-weight ratio may also be
important to a particular application. Such characteristics may
include, for example, optical, acoustical, electrical and chemical
properties. While these properties may simply derive from the
substance of which a structural material is made, they may also
derive from a geometry, or a combination of substance and geometry.
For example, much attention has been given to the potential of the
carbon-60 (C.sub.60) molecule (commonly known as the "Buckyball")
due to its unusual geometry that may have unique useful
properties.
[0008] Others working at the nano scale in the fields of chemistry
and crystallography are investigating a variety of methods to
create what have been referred to as "supra-molecular arrays,"
"supramolecular architectures," "3D coordination polymers,"
"metal-organic frameworks (MOF's)," "structural topologies,"
"binary superlattices," and "porous open-framework solids," among
others. The mostly widely utilized methods can be described as
"bottom-up" and involve programmed construction of extended
high-dimensional metal-organic network solids using metal and
organic `building blocks` with known and desirable bonding or
functional properties. Some examples include: framework solids
constructed from divalent transition metals and citric acid;
formation of organic supramolecular structures via solid-state
self-assembly of triphenol adducts; synthesis of 1D and 2D (but not
3D) organic-inorganic infinitely extended structures through the
linkage of rare earth metal-organic cations; self-assembly of novel
metal-organic frameworks with aromatic polycarboxylates, bix and
metal salts; self-assembly of supramolecular porphyrin arrays by
hydrogen bonding; and co-crystallization of fullerenes and
porphyrins into tapes, sheets and prisms due to weak C-F
interactions.
[0009] None of the prior efforts, however, has led to defining
overall configurations of discrete polyhedral arrays with many
advantages suitable for different applications in a wide range of
technologies.
SUMMARY
[0010] Described below are implementations of polyhedral arrays
configured to allow the array as a whole to have desired properties
and address some of the problems of prior approaches.
[0011] According to one implementation, a polyhedral array
comprises a plurality of discrete polyhedrons and a connection
network. The polyhedrons are spaced apart from each other at
equilibrium in a predetermined generally regular pattern, and each
polyhedron is comprised of edges, faces and vertices. The
connection network comprises connections extending along bias
directions or radial directions to connect the polyhedrons. The
connection network at least partially constrains the discrete
polyhedrons with respect to each polyhedron's six degrees of
freedom.
[0012] Each polyhedron's six degrees of freedom can be defined as
the ability to translate in the X, Y and/or Z directions of a
coordinate reference frame, and the ability to rotate about the X,
Y and/or Z directions.
[0013] In specific implementations, the polyhedrons in the array
can be arranged in multiple, generally parallel layers. The
predetermined regular pattern in which the polyhedrons are arranged
can include spaces occurring at generally regular uniform
intervals.
[0014] In specific implementations where at least some of the
plurality of polyhedrons generally occupy a first plane, the bias
directions along which the connections extend intersect the first
plane. The bias directions can be oriented at angles of
approximately 45 degrees to the first plane.
[0015] In specific implementations, the polyhedrons are arranged in
multiple layers, and the bias directions along which the
connections extend are inclined at about 45 degrees relative to an
expected direction of a resolved load on the array.
[0016] In specific implementations, at least one of the connections
in an array can extend between an edge of a first of the
polyhedrons and an edge of a second of the polyhedrons. In
addition, at least one of the connections in an array can extend
between a face of a first of the polyhedrons and a face of a second
of the polyhedrons. It is also possible that at least one of the
connections extends between a vertex of a first of the polyhedrons
and a vertex of a second of the polyhedrons. Further, at least one
of the connections can have a first end connected to one type of
connection feature (i.e., one of a face, an edge or a vertex), and
a second end connected to a different type of connection feature
(i.e., another of a face, an edge or a vertex).
[0017] In specific implementations, the array can be described as
being coherent. In specific implementations, the array is
omni-extensible. In specific implementations, an existing array can
be increased in size by connecting additional discrete polyhedrons
to existing discrete polyhedrons with additional connections
without other modifications to the existing array.
[0018] Each discrete polyhedron in the array can be a finitely
closed structure having structural integrity independent of the
respective connections to which it is connected and independent of
other discrete polyhedrons in the array. Each polyhedron in the
array occupies a unique location at equilibrium that can be
specified with Cartesian coordinates.
[0019] The interconnecting network can be configured in a generally
repeating pattern. In addition to discrete connections, at least
one of the connections can extend continuously, i.e., from a first
polyhedron, to a second polyhedron and to an nth polyhedron.
[0020] The connections can comprise at least one of mechanical
elements, bonds or guest molecules, ligands and ligatures. The
connections can comprise mechanical elements that have greater
resiliency than the polyhedrons. The connections can comprise
mechanical elements that include spring-shaped portions. At least
one of the connections can have predetermined properties different
from another of the connections. At least one of the connections
can have a property that varies along its length. At least some of
the connections can be compression members configured primarily to
resist compression forces.
[0021] The array can be anisotropic, with at least one polyhedron
having specific properties different from another of the plurality
of polyhedrons. The discrete polyhedrons can comprise a first
material and the connections can comprise a second material
different from the first material.
[0022] At least one of the plurality of polyhedrons can comprise a
closed polyhedron having a majority of closed faces. At least one
of the plurality of polyhedrons can comprise an open polyhedron
each having a majority of open faces. The plurality of polyhedrons
can comprise at least one interiorly hollow polyhedron. The
plurality of polyhedrons can comprise at least one solid
polyhedron.
[0023] Each polyhedron in the array can be described as satisfying
the condition of having a face, an edge or a vertex approximately
coincident with one of the six sides of a cube circumscribing the
polyhedron.
[0024] At least one of the polyhedrons can be formed of two halves.
The polyhedron can be an icosahedron and have a geodesic saw
tooth-shaped equator defining the two halves.
[0025] In specific implementations, at least some of the
polyhedrons are icosahedrons.
[0026] In specific implementations, the plurality of polyhedrons
includes at least one interior polyhedron connected to twelve
adjacent polyhedrons.
[0027] In specific implementations, the discrete polyhedrons can
move relative to each other in response to applied loads, but
remain spaced apart from each other at equilibrium and within a
selected working load range. A force above a predetermined working
load range applied to the array can urge two adjacent polyhedrons
from an equilibrium position in which the adjacent polyhedrons are
separated from each other into an under load position in which the
adjacent polyhedrons are in contact with each other.
[0028] In a specific implementation, a polyhedral array comprises a
plurality of discrete polyhedrons that are spaced apart from each
other at equilibrium in a predetermined generally regular pattern,
each polyhedron being comprised of edges, faces and vertices, and a
connection network interconnecting the discrete polyhedrons, the
connection network comprising radial connections. A first end of
each radial connection and a corresponding first connection
location on one of the polyhedrons can occupy a first plane, and a
second end of each radial connection and a corresponding second
connection location on another of the polyhedrons can occupy a
second plane that is not parallel to the first plane. The
interconnecting network at least partially constrains the discrete
polyhedrons relative to each polyhedron's six degrees of freedom.
The first and second planes of the respective first and second
connection locations can be mutually orthogonal.
[0029] The connections can be curved mechanical elements, and each
curved mechanical element can have a first end connected to a first
face of a first polyhedron and a second end connected to a second
face of a second polyhedron, the first and second faces occupying
different planes. In specific implementations, when the array is
subjected to a load above a predetermined working load range, the
array can deform such that at least two of the polyhedrons make
face to face contact with each other.
[0030] In specific implementations, when the array is subjected to
a force or torque above a predetermined working range, the array
deforms such that at least one of the polyhedrons undergoes a
predetermined transformation in shape. At least one of the
polyhedrons can be reversibly transformed such that the at least
one of the polyhedrons returns to an original shape if the force or
torque is removed.
[0031] Disclosed below are representative embodiments of polyhedral
arrays. The described structures should not be construed as
limiting in any way. Instead, the present disclosure is directed
toward all novel and nonobvious features, aspects, and equivalents
of the various embodiments, alone and in various combinations and
sub-combinations with one another. The disclosed technology is not
limited to any specific aspect, feature, or combination thereof,
nor do the disclosed materials, structures, and methods require
that any one or more specific advantages be present or problems be
solved. For the sake of simplicity, the attached figures may not
show the various ways in which the disclosed structures can be used
in conjunction with other systems, methods, and apparatus.
BRIEF DESCRIPTION OF THE DRAWINGS
[0032] FIG. 1A is a perspective view of a regular icosahedron
circumscribed by a cube.
[0033] FIG. 1B is a perspective view of a regular dodecahedron
circumscribed by a cube.
[0034] FIG. 1C is a perspective view of a regular icosahedron
illustrating the mutually orthogonal planes that extending between
three pairs of opposite edges.
[0035] FIG. 1D is a perspective view of a cubooctahedron
circumscribed by a cube.
[0036] FIG. 1E is a perspective view of a portion of a polyhedral
array in which the connections are arranged on the bias.
[0037] FIG. 2 is a side elevation view of a portion of the bias
direction array shown in FIG. 1E.
[0038] FIGS. 3, 4, 5 and 6 are top, bottom, left and right side
views, respectively, of an interior polyhedron in the bias
direction array of FIG. 1E, showing the positions of the
connections to adjacent polyhedrons and their directions.
[0039] FIG. 7 is a perspective view of a portion of a radial
direction array showing a single interior polyhedron connected to
twelve surrounding polyhedrons by radial connections.
[0040] FIG. 8 is a perspective view of a portion of a radial
direction array showing a single interior polyhedron connected to
twelve surrounding polyhedrons by radial connections where the
polyhedrons can transform in shape from an ideal shape when
subjected to an influence.
[0041] FIGS. 9, 10, 11, 12 and 13 are side views of an interior
polyhedron of the array of FIG. 8.
[0042] FIG. 14A is an exploded perspective view of the interior
polyhedron of FIG. 9 and its connections.
[0043] FIG. 14B is a side view of an icosahedron element capable of
transformation.
[0044] FIG. 14C is a side view of an icosahedron element as
transformed by a tension force, according to a first example.
[0045] FIG. 14D is a side view of an icosahedron element as
transformed by a tension force, according to a second example.
[0046] FIG. 15 is a perspective view of a bias direction array of a
specific implementation where the connections are mechanical
elements and the polyhedrons are formed with corresponding
connection receiving features.
[0047] FIG. 16 is a perspective view of one of the polyhedrons of
FIG. 15.
[0048] FIG. 17 is a perspective view of one of the connections of
FIG. 15.
[0049] FIG. 18 is a perspective view of a specific polyhedron
construction, showing the polyhedron formed of two geodesic
equatorial "saw tooth cut" with a hollow interior.
[0050] FIG. 19A is a front view of another array, also referred to
as a trigonal lattice.
[0051] FIG. 19B is a left side view of the trigonal lattice of FIG.
19A.
[0052] FIG. 19C is a top plan view of the trigonal lattice of FIG.
19A.
[0053] FIG. 19D is a side elevation view of a repeating element of
the trigonal lattice configured with the mechanical elements of
FIG. 15.
[0054] FIG. 20A is a front view of another array, also referred to
as a cubo-octahedral lattice.
[0055] FIG. 20B is a left side view of the cubo-octahedral lattice
of FIG. 20A.
[0056] FIG. 20C is a top plan view of the cubo-octahedral lattice
of FIG. 20A.
[0057] FIG. 21 is a perspective view of a slice of a polyhedral
array absorbing a load and showing some of the polyhedrons moved
from their equilibrium positions, with the connections between the
polyhedrons omitted for clarity.
[0058] FIG. 22 is a perspective view of a slice of a polyhedral
array showing one of the polyhedrons rotated from its equilibrium
position in response to an applied torque.
[0059] FIG. 23 is a perspective view of a polyhedral array
implemented as the core of a catalytic converter.
[0060] FIG. 24 is a perspective view of a polyhedral array
implemented as an artificial reef in a marine setting.
[0061] FIG. 25 is a perspective view of a polyhedral array
implemented as heat exchange element in a heat exchanger
system.
[0062] FIG. 26 is a perspective view of a polyhedral array
implemented as an artificial biological material.
[0063] FIG. 27 is a perspective view of a polyhedral array
implemented as a multi-function road and/sidewalk surface.
[0064] FIG. 28 is a perspective view of polyhedral arrays
implemented as stormwater and erosion control structures.
[0065] FIG. 29 is a perspective view of polyhedral arrays
implemented as various construction materials.
DETAILED DESCRIPTION
[0066] Described below are various embodiments of polyhedral
arrays. Each array is comprised of a plurality of discrete
polyhedrons, and a connection network comprising individual
connections that interconnect the polyhedrons. The connection
network serves to constrain each polyhedron with respect to its six
degrees of freedom within three-dimensional space, such as is
defined by a three-axis Cartesian coordinate reference frame.
[0067] Each polyhedron is discrete relative to adjacent polyhedrons
in the array. Thus, each polyhedron is separate from the other
polyhedrons, and adjacent polyhedrons do not share common vertices,
edges or faces. The connections serve to interconnect the
polyhedrons, and when the array is in equilibrium, i.e., at rest
and not under an applied load, to maintain them in a spaced apart
configuration. When the array is subject to an external influence,
e.g. a load such as a force or torque applied to the array, one or
more of the polyhedrons may be moved in response to the influence.
In some implementations, at least one polyhedron, although
constrained by its connections, may move from its equilibrium
position into contact with an adjacent polyhedron when the array is
subjected to a load. Depending upon the specific implementation,
the polyhedron's movement may be a translation, a rotation, or a
combination of a translation and a rotation. The ability of a
polyhedron to move within the array but remain connected to its
neighboring polyhedrons allows the array to have "flexibility"
under compression and resist the forces being applied.
[0068] In general, the array is three-dimensional and may comprise
multiple layers depending on the application and the
characteristics desired for the application. Each layer is
generally planar, and the layers are generally parallel to each
other. The connections extend to adjacent polyhedrons in the same
layer and in other layers. The connections can be provided in many
different configurations and materials to allow the array to have
selected properties.
[0069] According to one specific implementation, the connections
between the polyhedrons extend along the bias (also described as
"in bias directions"). In other words, the connection or
connections between a first polyhedron and its nearest neighbor are
arranged to extend at acute angles relative to the respective plane
or plane(s) of these polyhedrons. In one specific implementation,
and as described in greater detail below, the direction of the bias
or "bias directions" are oriented at about 45 degrees with respect
to the plane or planes. At equilibrium, these bias directions are
mutually orthogonal, so any connections in the array that do not
extend in parallel directions are also orthogonal with respect to
each other. Examples of bias direction arrays, which are discussed
below in greater detail, are shown in FIG. 2 and FIG. 15.
[0070] According to another implementation, the connections between
the polyhedrons are described as radial connections. As used
herein, "radial connections" for any given polyhedron extend from a
face, edge or vertex of the polyhedron in a direction perpendicular
to a vector originating its center. Over its length extending
towards an adjacent polyhedron, the radial connection changes
direction (or "bends"), as is described and illustrated below, such
that its opposite ends are oriented in different planes that are
mutually orthogonal to each other. Among other benefits, certain
implementations of the radial connection array (hereinafter,
"radial array") allow it to deform such that two or adjacent
polyhedrons enter into face to face contact when subjected to a
load above a predetermined working load range. Examples of radial
direction arrays, which are described below in greater detail, are
shown in FIGS. 7 and 8.
Polyhedrons and Connections
[0071] According to well known geometry principles, one way
polyhedrons are described is in terms of the number of faces,
vertices and edges that each type of polyhedron has. For example,
an icosahedron has twenty faces, twelve vertices and thirty edges.
In a regular icosahedron, each of the twenty faces is an
equilateral triangle, so all of the angles are equal and the edges
are of equal length. In an irregular polyhedron, however, at least
some of the angles and edges are unequal. Certain regular
polyhedrons and irregular polyhedrons can be used in arrays, as
described below and depending upon the specific implementation.
[0072] As described above, the array comprises discrete
polyhedrons. In specific implementations, a suitable polyhedron is
one that, if circumscribed by a cube, has an edge, a face or a
vertex approximately coincident with each of the six cube faces.
Stated differently, suitable polyhedrons have at least three pair
of connection locations, where each pair of connection locations is
mutually orthogonal to the others, and each pair of connection
features comprises two opposite faces, two opposite edges, two
opposite vertices, or two opposite features of different types
(e.g., one face and one opposite edge, or one edge and one opposite
vertex).
[0073] For example, as shown in FIG. 1A, a regular icosahedron 50
can be circumscribed by a cube 52 such that three pairs of the
icosahedron's edges are coincident with the faces of the cube 52.
Thus, there are (1) a first pair of opposite edges, including a
left edge 54 and an opposite right edge (obscured, but along line
56), (2) a second pair of opposite edges, including a top edge 58
and an opposite bottom edge (obscured, but along line 60) and (3) a
third pair of opposite edges, including a front edge 62 and an
opposite back edge (obscured, but along line 64), each of which is
coincident with a respective one of the six faces of the cube
52.
[0074] Similarly, as shown in FIG. 1B, a regular dodecahedron 70
can be circumscribed by a cube 72. Thus, three pairs of opposite
edges of the dodecahedron (74 and 76, 78 and 80, and 82 and 84) are
coincident with the faces of the cube 72. Referring to FIG. 1D, a
cubooctahedron 86 can be circumscribed by a cube 88. Thus, the
three pairs of opposite square faces of the cubooctahedron 86 (one
of each pair being shown at 90, 92 and 94) are coplanar with the
faces of the cube 88.
[0075] FIG. 1C is another illustration of a regular icosahedron,
such as is shown in FIG. 1A, except in FIG. 1C the icosahedron is
depicted in a slightly different orientation for better perspective
and with transparent faces. As shown in FIG. 1C, each pair of
opposite edges of the icosahedron lies in the same plane, and the
three planes defined by the three pair of opposite edges are
mutually orthogonal to one another.
[0076] Referring again to the circumscribing cube, each polyhedron
in the array, like an object in space, can be described as having
as many as six degrees of freedom, namely the ability to translate
in the X, Y and/or Z directions as defined by a fixed coordinate
axis (see FIG. 1C), and/or to rotate about one or more of these
axes. These rotations are sometimes described as "pitch," "yaw" and
"roll." The connections serve to constrain some or all of each
polyhedron's six degrees of freedom in at least a limited way. In
some specific implementations, the array can be configured such
that the connections among the polyhedrons eliminate one or more
degrees of freedom for at least one polyhedron (i.e., constrain the
polyhedron to substantially no movement within that degree of
freedom).
[0077] Of the five regular Platonic polyhedrons, namely the
tetrahedron (4 equilateral triangle faces, 6 edges and 4 vertices),
the cube (6 square faces, 12 edges and 8 vertices), the octahedron
(8 equilateral triangle faces, 12 edges and 6 vertices), the
dodecahedron (12 regular pentagon faces, 30 edges and 20 vertices),
and the icosahedron (20 equilateral triangle faces, 30 edges and 12
vertices), only the regular tetrahedron and the regular octahedron
do not provide at least three pair of connection locations that are
mutually orthogonal to one another (although a special case of the
octahedron does meet this criterion). Many other polyhedrons other
than the Platonic polyhedrons are also suitable, such as truncated
versions of the cube, octahedron and icosahedron (also referred to
as a "fullerene" or a "bucky ball"), cubo-octahedrons,
dodecahedrons, icosidodecahedrons, rombicosidodecahedrons, snub
dodecahedrons, etc. Stellated forms, deltahedrals, dual
tetrahedrals and other similar variations of Platonic and
Archimedian polyhedrons are also suitable. The specific polyhedrons
enumerated here are not to be construed as an exhaustive list, as
other polyhedrons having the appropriate number and arrangement of
connection features are also suitable
[0078] Two important polyhedrons are the icosahedron and the
truncated icosahedron. As described above, the icosahedron is a
practical choice because it has three pairs of parallel faces. The
truncated icosahedron is derived by "slicing off" the twelve
vertices of the icosahedron, thereby forming twelve regular
pentagons. The truncated icosahedron has twelve regular pentagonal
faces, twenty regular hexagonal faces (for a total of 32 faces), 60
vertices and ninety edges. The pentagons and hexagons provide a
plurality of pairs of parallel surfaces.
[0079] Polyhedrons with fewer rectangular faces (including square
faces) are generally more desirable because the triangular faces
are more stable than rectangular faces. Although the cube is an
acceptable polyhedron with which an array can be formed, in
practice, other polyhedrons offer greater advantages.
[0080] In some implementations, it is possible to form a mixed
array that comprises more than one type of polyhedron. For example,
it is possible to form a mixed array having icosahedrons and
truncated icosahedrons. Other such mixed arrays are also possible.
It is also possible to use polyhedrons of different sizes in the
same array, particularly if the differently sized polyhedrons are
geometrically scaled relative to each other.
[0081] The polyhedrons in the array may be solid, or they may have
solid faces that define a hollow interior. One or more faces of a
polyhedron may have openings defined therein. It is also possible
in some implementations to use polyhedrons having a "wireform"
configuration with material defining only the edges and the faces
being open. In some implementations, the polyhedrons include
enhancements departing from ideal regular forms to facilitate
connecting the polyhedrons to each other, such as is described in
connection with one implementation shown in FIGS. 15 and 16.
[0082] Connections as used herein refers to any matter, including
any arrangement of matter (such as a structure), or material that
serves as part of the connection network in connecting the
polyhedrons together and maintaining their spaced apart
configuration in the array at equilibrium. Connections can include
mechanical elements, molecules or bonds, such as guest molecules,
ligands, ligatures, etc.
[0083] Each connection can be a separate element, or multiple
connections can be formed together, depending upon the particular
geometry, materials and requirements of the array. In arrays with
one than one type of connection locations, e.g., vertices and
faces, there may be connections extending between a vertex at one
end and a face on the opposite end.
Response to Loading
[0084] As described above, the polyhedral arrays can be designed to
withstand loadings within a predetermined working range. For
example, the polyhedral arrays can be designed to withstand forces
and torques, including compression forces, tensile forces and
torsional forces applied to the array, as well as other forces,
such as shear forces, that may be developed internally within the
array. Bending moments and axial compression forces can also
comprise part of the loadings. As described, depending upon the
magnitude of the loading, one or more of the polyhedrons in the
array may be caused to move from its position at equilibrium. FIG.
21 is a perspective view of a slice of a representative polyhedral
array P withstanding a substantial force F applied to the array,
and most directly, to the polyhedron G. In FIG. 21, the connections
are omitted for clarity and to emphasize that the polyhedron G has
been moved by the force F. More specifically, the force F has urged
the polyhedron G to translate (in a generally downward direction).
Although the connections are not shown, they serve to assist in
absorbing and spreading the force F to other polyhedrons and
connections in the array. If the force F is within the expected
working range, and if the array P is so designed, the array P (and
specifically, the polyhedron G) will return to its original
equilibrium state if the force F is removed.
[0085] In addition to withstanding forces, the polyhedral arrays
also withstand torques. FIG. 22 is a perspective view of a slice of
an array Q showing a torque T applied to the array and, most
directly, to the polyhedron H. Although not shown, the connections
between the H polyhedron H and its neighboring polyhedrons help in
dissipating and resisting the torque T, thus allowing the array Q
to withstand it. As shown, the torque T is sufficient to move the
polyhedron H, i.e., to rotate it approximately 60 degrees, but the
array Q remains intact.
[0086] The examples of FIGS. 21 and 22 are illustrative of ideal
forces and ideal torques. In practice, nearly every loading of an
array would include both forces and torques. Also, the examples
describe loadings applied to boundaries of the array, such as a
surface, edge or corner of the array. In practice, other loadings
may also be experienced.
Bias Direction Array
[0087] As described above, one embodiment includes connections
oriented on the bias. Relative to a face of the array to which a
load is applied (or resolved to apply) in a normal direction, the
bias directions are defined to mean directions intersecting the
face of the array at an angle. In specific implementations, the
bias directions can form angles of about 30 to about 60 degrees
with the face of the array. Even better attributes are recognized
with bias directions of about 40 to about 50 degrees.
[0088] Typically, the bias directions extend at about 45 degrees
relative to the face of the array, but can be adjusted in
orientation to the applied load. Thus, the bias direction array is
in direct contrast with an orthogonal array in which the
connections extend at approximately 90 degrees or zero degrees
relative to the face of the array. Compared to other orientations,
connections extending along bias directions provide robust
resistance to shear.
[0089] FIG. 1E is a perspective view of a bias direction array 10.
The array 10 is comprised of multiple polyhedrons 12 arranged in a
generally repeating pattern, with connecting elements 14, 16, 18
interconnecting the polyhedrons 12 and extending along bias
directions.
[0090] In the array 10, which is shown at equilibrium, the
polyhedrons 12 are spaced apart and do not contact each other. In
FIG. 1E, the array 10 is shown as having three layers for
convenience of illustration, with some parts of highly obscured
layers omitted for clarity, but as described elsewhere, it could be
extended in any direction indefinitely.
[0091] The polyhedrons 12 in the array 10 have the same general
orientation. In the example of FIG. 1E, the polyhedrons 12 are
regular icosahedrons. As shown, the icosahedrons in the bias
direction array 10 have a "face up" orientation, i.e., one of the
triangular faces is oriented vertically upward.
[0092] In the array 10, there is a lower layer 20 of polyhedrons
generally occupying a first plane, an intermediate layer 22 above
the lower layer 20 and generally occupying a second plane and an
upper layer 24 above the intermediate layer 22 and generally
occupying a third plane. As best seen from the right side of FIG.
1, the right front face of the array is slanted, i.e., biased, from
the protruding lower layer 20 in a rearward direction to the
recessed upper layer 24.
[0093] Each of the polyhedrons 12 in the lower layer 20 and in the
upper layer 24 is an edge polyhedron, i.e. a polyhedron occupying
an edge position that defines an edge of the array 10. For the
intermediate layer 22, the visible edge 12 are along the front
right face (five in number) and along the front left face (two in
number). Each edge polyhedron has fewer connections to adjacent
polyhedrons than an interior polyhedron.
[0094] Although not clearly visible in FIG. 1E, there are interior
polyhedrons in the intermediate layer 22 rearward of the visible
edge polyhedrons. Each interior polyhedron has a connection to each
of twelve surrounding polyhedrons, including the six surrounding
polyhedrons in the same layer, three polyhedrons in the immediately
adjacent upper layer and three polyhedrons in the immediately
adjacent lower layer, as depicted in the illustrated
orientation.
[0095] The connections 14, 16, 18 are shown in FIGS. 1E and 2
schematically as lines solely for purposes of clear illustration.
Of course, each connection can take the form of any suitable
connection as is described elsewhere in this application. Referring
to FIG. 2, which is side view of a portion of the array 10 shown in
FIG. 1E, the connections visible for the edge polyhedrons are
shown. Each edge polyhedron has at least three connections to
adjacent polyhedrons if it is part of two edges, five connections
if it is part of one edge and six connections if it is part of a
face.
[0096] In FIG. 2, the connections 14 connect adjacent polyhedrons
in the plane of the face that are within the same layer (as
illustrated in the same horizontal row). The connections 16 extend
into the page at an angle and connect polyhedrons within one layer
to polyhedrons in a layer above or a layer below. The connections
18 also interconnect polyhedrons in different layers. The mutually
orthogonal directions of the connections 14, 16, and 18 is most
clearly seen on the coordinate axes shown at the right of FIG.
2.
[0097] For the specific implementation of FIGS. 1E and 2, the
connections 14, 16, 18 interconnect the polyhedrons 12 at
connection locations positioned on edges of the polyhedrons. FIGS.
3, 4, 5 and 6 are side views of an interior polyhedron 12 of the
array 10 showing the connections to the twelve surrounding
polyhedrons in dashed lines. Considering the "pentagon" sides of
FIGS. 3 and 4 to be top and bottom sides, six of the twelve
connections can be seen from either of these sides. The remaining
sides, called the "front" and "back" sides for convenience, each
show four connections.
[0098] The polyhedrons 12 in the example of FIG. 1 are regular
icosahedrons, but any other suitable polyhedron meeting the
criteria described elsewhere in this application could substituted
in place of a regular icosahedron, either at all locations or at
fewer than all locations
Radial Array
[0099] FIG. 7 is a perspective view of a portion of a radial array
110. As described above, the radial connections 114 between
polyhedrons 112 extend generally uniformly at right angles relative
to the respective polyhedron's center.
[0100] As shown in FIG. 7, there are twelve edge polyhedrons
surrounding the single interior polyhedron 113. In this example,
there are three edge polyhedrons in a lower layer, six polyhedrons
in the intermediate layer (three edge polyhedrons and the interior
polyhedron being visible), and three edge polyhedrons in the upper
layer. Overall, the twelve edge polyhedrons occupy the same
positions as the vertices of a cubo octahedron.
[0101] As shown in FIG. 7, the connections 114 in the radial array
connect different planes. Stated differently, each connection has a
first end connected at a first plane on a first of the polyhedrons,
and a second end connected at a second plane extending and
translating orthogonally onto a second of the polyhedrons. In the
specific implementation shown in FIG. 7, the connections 114
terminate at faces of the polyhedrons. Thus, each connection does
not extend between closest faces of adjacent polyhedrons, but
instead can be said to "bend" to connect a next-closest face.
[0102] In the radial array, the polyhedrons in the different layers
as shown in FIG. 7 are oriented in face to face contact. Thus, in
response to an influence, one or more aligned polyhedrons may be
forced together and into face-to-face contact. Polyhedrons in
face-to-face contact provide a strong and stable configuration
(similar to a column) for resisting compressive forces applied to
the array. Also, the radial array is particularly resistant to
torsional loads applied to the array, even a torque applied to a
single peripheral polyhedron, as the loaded polyhedron's
connections to adjacent polyhedrons help withstand and spread the
torque.
[0103] FIG. 8 is a perspective view of another implementation of a
radial array 210, shown from a different perspective than FIG. 7,
in which the polyhedrons can transform in shape in response to an
influence. For example, one or more of the polyhedrons 212 can
transform in shape, i.e. morph or skew, from its equilibrium shape
as shown in FIG. 14B without breaking. FIG. 14C shows one of the
polyhedrons 212 after it has been transformed from its equilibrium
configuration by a generally single-axis tension force C applied to
two opposite faces of the icosahedron. The resulting transformed
icosahedron 212' has been elongated in the directions of the force
C. In FIG. 14D, the resulting transformed polyhedron 212'' is the
result of a tension force D acting on the polyhedron, which causes
it to elongate in the directions of that force as well as in
directions E perpendicular to the force D. The same transformed
polyhedron 212'' results if the force is applied in the same plane
as the force D, except rotated 90 degrees (i.e., if the force is
applied along a direction normal to the page).
[0104] The polyhedrons 212 can be provided with predetermined
features to facilitate the transformations in shape. In the
specific implementation of FIG. 8, the polyhedrons 212 are
icosahedrons formed of multiple pieces. Twelve of the faces of the
icosahedron have a connection element, such as a protruding post or
hub 222, for connecting the polyhedron to other polyhedrons. The
remaining eight faces of the icosahedron are formed of three
overlapping portions arranged to slide or "shutter" over each
other, thus allowing the icosahedron to transform in shape. In
addition, the connections 214 have ends that are rotatably
connected to respective faces of the polyhedrons to which they are
connected, thus allowing the connections 214 to pivot relative to
the polyhedrons. For the connection 216, it can be seen that the
ends 218, 220 are pivotably connected to respective hubs 222, 224,
as one example.
[0105] FIGS. 9, 10, 11, 12 and 13 are side views of an interior
polyhedron 212 with the connections 214 and their general
orientation shown (but in a reduced length for clarity of
illustration).
[0106] FIG. 14A is an exploded view the polyhedron 212 shown with
twelve connections 214, i.e., suitable for positioning it as an
interior polyhedron of an array. The polyhedron 212 is constructed
of 12 interconnecting components 230 that together form a generally
regular icosahedron with 20 faces with overlapping portions. As
shown, each of the twelve components has one of the twelve hubs 222
of the icosahedron, several of which are visible in the drawing.
The assembled components 230 are movable relative to each other to
allow the icosahedron to transform in shape.
Attributes of Arrays
[0107] The polyhedral arrays described herein are
"omni-extensible," which is to say that an existing array can be
extended in any direction by joining additional polyhedrons with
additional connections. It is not necessary to first disassemble
the existing array before extending it. In the same way that an
existing array can be extended, it also possible to join two or
more arrays into a single larger array.
[0108] When the polyhedral arrays are configured in a regular
pattern, the equilibrium locations of the polyhedrons in the array
are unique and thus can be specified, e.g., using a Cartesian
coordinates or other similar system. Thus, the position of each
polyhedron in the array can be described as being "addressable." In
some implementations, at least some portion of the array is capable
of carrying or holding electric charge, and the polyhedrons or
connections in this portion of the array can be individually
addressed to store data and/or to convey signals or transmit
power.
[0109] As shown in the examples, the polyhedral arrays at
equilibrium have spaces separating the polyhedrons. As also
described, additional spaces at regularly repeating intervals can
also be designed into the array. Depending upon the scale at which
the array is constructed, as well as the selected length(s) of the
connections relative to the size of the polyhedrons, the resulting
array can be designed to provide a desired permeability suitable
for a particular application. Thus, an array configured as a
building material to be installed in a generally horizontal
orientation in an exposed environment can be configured for
permeability to rain and storm water, yet still provide the
structural strength to support the expected loads for the
anticipated design life.
[0110] As shown in some of the application examples described
below, the configuration of the polyhedral arrays allows for them
to interface with other structures, such as, e.g., conventional
building materials. Many common conventional building materials
have a generally rectangular prism shape, such as conventional
bricks, dimensional lumber, plywood and other types of sheet
material, etc., which is defined by 90 degree angles. Although most
examples of the polyhedral arrays do not terminate at edges
defining 90 degree angles, the same mutually orthogonal connection
locations of the polyhedrons, some of which are free for edge
polyhedrons, can be used for dimensionally reliable and stable
attachment to other adjacent materials and structures.
[0111] In the described examples, the polyhedral arrays are shown
as comprising polyhedrons of the same general size, which is a
typical configuration. In some arrays, however, it is possible to
mix polyhedrons of different types, or to substitute polyhedrons of
different sizes. For example, an array can be formed where the
majority of polyhedrons are regular icosahedrons having a unit
size, and a larger icosahedron having a size that is a geometric
multiple of the unit size is substituted into the array
periodically.
[0112] A polyhedral array can be configured to be isotropic, i.e.,
to have the same properties in all directions. Alternatively, the
array can have a predetermined anisotropy, e.g., to address a
particular requirement of the specific implementation. Also, the
polyhedrons and the connections can be isotropic or
anisotropic.
[0113] The arrays can be configured to have different portions
exhibiting different properties. For example, the polyhedrons
occupying the edge positions in an array could be formed of a
material more resistant to environmental conditions, or
specifically adapted for receiving a thin covering layer or
attaching to a conventional adjacent structure. Polyhedrons having
different physical properties, including elasticity, density,
melting point, strength in compression, etc., to name a few, could
be substituted in the array to achieve a desired result. Moreover,
connections can be adapted in the same way. The properties of the
connections can be varied according to their location in the array,
their orientation relative to the expected load, etc. In addition,
connections can be designed to exhibit different properties along
their length.
[0114] As an example, the relative rigidities of the polyhedral
elements and the connections elements can be tailored for a given
application. In addition, individual instances of the same type of
element, e.g., the connections, may be provided with varying
rigidities to provide a desired anisotropy to the array. For
example, connections extending in the "z" direction may be made
less or more rigid than the interconnecting members in the "x" and
"y" directions where the anticipated loading configuration differs
in the "z" direction as compared to the "x" and "y" directions.
[0115] In some cases, the polyhedral arrays deviate from perfect
regularity yet still have the overall function and behavior of a
regular array. In some cases, a polyhedron or a connection may be
missing, or there may be a foreign object or impurity present in
the array instead of one of the polyhedrons. Indeed, certain
implementations warrant slight departures from perfect regularity
to account for specific local conditions.
[0116] In general, the polyhedrons and connections may be made of
any material suitable for the particular application. For large
scale applications, familiar materials such as metals, alloys,
composites, plastics and others may be used.
[0117] It is also possible to provide very small scale arrays, such
as at the nanoscale or microscale. Some molecular forms have
specific polyhedral geometry. As two examples, the C.sub.60
molecule is a truncated icosahedron, and the B.sub.12 molecule is
an icosahedron. Work in the area of these and similar molecules is
represented by U.S. Pat. No. 6,531,107 entitled "Fabrication of
Molecular Nanosystems," U.S. Pat. No. 6,841,456 and U.S. Pat. No.
6,965,026 entitled "Nanoscale Faceted Polyhedra" (these references
are incorporated herein by reference). Assembly techniques, such as
atomic force microscopy or self assembly, may be used to provide
suitable molecules as polyhedral elements. Further, and not
appreciated until now, the polyhedral molecules can be arranged
into predetermined arrays, such as bias direction arrays or radial
arrays, with connections designed as described above and formed of
molecules, ligands or ligatures to give the resulting arrays
overall properties useful in the design and fabrication of larger
arrays (such as lattices) and objects.
Specific Implementations
[0118] According to one specific implementation as shown in FIGS.
15, 16 and 17, polyhedral arrays can be constructed using
mechanical elements, such as icosahedral elements 250 and connector
elements 252.
[0119] FIG. 15 is a perspective view of a portion or "chunk" of a
bias direction array 254. As shown, the icosahedral elements 250
and the connector elements 252 are arranged generally as shown for
the bias direction array of FIG. 2, i.e., with the connector
elements 252 extending on the bias relative to the planes defined
by the layers of icosahedral elements 250.
[0120] Referring to FIG. 16, each icosahedral element 250 has
twelve connection locations 256 positioned about the outer
periphery of the icosahedral element, eight of which are visible in
FIG. 16. The twelve connection locations 256 are arranged in six
pairs, i.e., three sets of oppositely arranged pairs, and each of
these three sets has an orientation mutually orthogonal to the
other two sets.
[0121] In the illustrated embodiment, each of the connection
locations 256 is positioned within an opening in a projection 258
protruding slightly from the icosahedral element's outer surface
and extending across one of its edges.
[0122] As best shown in FIG. 17, each connector element 252 has a
first end 260, an opposite second end 262 and a body 264 between
the ends 260, 262. Each end 260, 262 is shaped to connect with one
of the connection locations 256 by insertion of the end into one of
the openings defined in the projections 258. Once inserted, the end
260 or 262 terminates near the edge of the icosahedral element 250
(which is covered by the projection 258).
[0123] As also shown in FIG. 17, each end 260, 262 can be tapered
to provide a proper fit within the opening and orientation relative
to the polyhedron. Also, each end 260, 262 can have retaining
elements 268 extending alongside and positioned to project into
edge apertures 266 provided in the icosahedral element 250 to
assist in retaining the connector elements 252 in place.
[0124] The body 264 of the connector can have a spring-shaped
construction as shown. Other geometries are, of course, also
possible. The spring-shaped body allows the connector to constrain
the icosahedral elements 250 to which it is attached, yet can
resiliently deform when those icosahedral elements are subjected to
a force or a torque.
[0125] In the specific embodiment, the icosahedral elements 250 are
made of a plastic and have a hollow construction. The connector
elements 252 are also made of a plastic.
[0126] According to some implementations, providing polyhedrons of
a hollow construction can be achieved by providing polyhedron
halves having specific geometries. Referring to a specific
implementation for the icosahedron shown in FIG. 18, each
icosahedral element half 270 can be formed such that when the
halves are mated together, the junction follows a geodesic saw
tooth-shaped equator 272.
[0127] FIG. 19A is a side elevation view of another implementation
of an array 280, also referred to as a lattice. In the array 280,
there is at least one layer in the array that includes open spaces
of a predetermined size at and location at equilibrium. These open
spaces are defined by regions that are large enough to accommodate
at least one additional polyhedron, but are vacant. Arrays with
predetermined open spaces offer advantages, such as lighter weight
per unit area or volume than corresponding filled arrays
(including, e.g., the bias direction array of FIG. 1).
[0128] FIGS. 19B and 19C are left side elevation and top plan
views, respectively, of the array 280. For clarity of illustration,
the connections between the individual polyhedrons have been
omitted.
[0129] In the array 280, which is also referred to herein as a
"trigonal" lattice, the repeating unit 282 is seven polyhedrons,
including three connected polyhedrons 284 in the lower layer, three
connected polyhedrons 286 in the upper layer, and a single
polyhedron 288 in the intermediate layer connected to each of the
polyhedrons 284, 286 in the lower and upper layers. From FIG. 19C,
it can be seen that there are eight repeating units 282 shown.
[0130] In the drawings, the spacing between adjacent repeating
units 282 has been exaggerated slightly for clarity. In practice,
the space between polyhedrons in the lower and upper layers that
are adjacent to each other, but of different repeating units, can
be the same as the space separating adjacent polyhedrons within the
same repeating unit.
[0131] The single polyhedron 288 in the intermediate layer is not
connected to other intermediate layer polyhedrons. Thus, there is
no connection between the intermediate layer polyhedron 288 and an
adjacent intermediate layer polyhedron, such as the intermediate
layer polyhedron 290. Within the intermediate layer, predetermined
recesses or spaces S are defined between the polyhedrons in that
layer (and are bounded by the polyhedrons in the lower and upper
layers). The spaces S generally occur according to a periodic
pattern. The size and frequency of spaces within the array can be
selectively determined according to properties desired for the
specific application of the array.
[0132] FIG. 19D is a side elevation view of a repeating unit 282'
for a trigonal lattice, which is similar to the repeating unit 282
in geometry, but in this specific implementation is formed of
mechanical polyhedron elements 250 and connector elements 252.
[0133] As another example, FIG. 20A is a front elevation view of an
array 292 according to another implementation. In the array 292,
there is a three-layer repeating unit 294 of twelve polyhedrons,
including a central polyhedron surrounded by and connected to six
polyhedrons in the intermediate layer, and connected to three
polyhedrons in the lower level and three polyhedrons in the upper
layer.
[0134] FIGS. 20B and 20C are left side elevation and top plan
views, respectively, of the array 292. For clarity of illustration,
the connections between the individual polyhedrons have been
omitted.
[0135] As shown, the array 292 as illustrated includes two layers
of repeating units 294 with six repeating units per layer, for a
total of twelve repeating units. Referring to the figures, assuming
the exterior polyhedrons of the repeating unit 294 are vertices,
the repeating unit is a cubo-octahedron. Thus, the array 292 can be
described as a "cubo-octahedron" lattice.
[0136] Referring to the figures, each repeating unit 294 is
connected to any adjacent repeating unit(s) in the same horizontal
"row" or vertical "column." Among the interconnected repeating
units 294 that comprise the array, however, there are predefined
spaces T that occur at each intersection of eight repeating units
294. In addition to the full spaces T, there are additional smaller
spaces U between at each intersection of four repeating units.
Applications
[0137] FIG. 23 shows a polyhedral array implemented as a core or
substrate 402 of a catalytic converter 404 for a vehicle, e.g., an
automobile. The individual polyhedrons can be spaced apart as
necessary to provide the proper air flow through the catalytic
converter. Also, the properties of the polyhedron provide for
substantial surface area per volume, which increases the area for
chemical reactions to occur.
[0138] FIG. 24 shows a polyhedral array implemented as an
artificial reef 406 in a marine setting. In the artificial reef
406, the polyhedrons have open faces and hollow interiors that
permit the reef to sink and allow water and organisms to circulate
through the structure. Also, the polyhedrons can be configured to
enclose guest nutrients that become available to organisms in the
water as through an active delivery system or a passive erosion of
the artificial reef.
[0139] FIG. 25 shows a heat exchanger, such as an air conditioning
unit, with the heat exchange elements 410 implemented as polyhedral
arrays. The heat exchange elements 410 can be sized as desired to
provide the desired flow rates and mixing in the air flow passages
and in the heat exchange passages, as well as to achieve other
performance goals.
[0140] FIG. 26 shows a polyhedral array implemented as an
artificial biological material, in this case an artificial human
bicep muscle portion 412. The muscle portion 412 can be configured
in accordance with predetermined requirements, e.g., tensile
strength, flexibility, elasticity, desired change in shape through
the associated limb's range of motion, etc.
[0141] FIG. 27 shows a polyhedral array implemented as a
multi-layer construction material 418. The material 418 can be
configured to have a road surface portion 420 capable of
withstanding vehicle weights. The road surface portion 420 can be
porous to provide drainage, but if covered, an integrated drainage
portion 422 can be included to drain away storm water, etc. from
the road surface portion 420. On the adjacent sidewalk, the
material 418 can have a plant support portion 424. Optionally, the
array of the plant support portion can be configured for integrated
irrigation of plants. At a curb portion 426 separating the road
surface portion 420 from an adjacent sidewalk portion 428, the
array can be configured for integrated charge and/or signal
carrying capability to provide illumination (e.g., warning lights,
street lights, traffic signal lights) and/or information (warnings,
street names, etc.). Similar to the road surface portion 420, the
array of the sidewalk portion 428 can be left exposed to provide a
porous construction or it can serve as a base for receiving a solid
outer layer.
[0142] FIG. 28 shows a polyhedral array implemented as storm water
and erosion control structures, such as a multi-level culvert or
canal wall 430 construction. Also shown is a polyhedral array
implemented with earthen layers as part of a reinforced flood wall
432 adjacent one side of the canal wall 430. On the other side of
the canal, another polyhedral array implemented as part of a
vehicle and/or pedestrian path 434, together with other earthen
layers.
[0143] FIG. 29 shows a house 435 under construction with polyhedral
arrays implemented as exterior building panels 436, interior
high-strength safe room panels 438, interior panels 440 and roofing
material 442, to name just a few. All of the polyhedral arrays as
implemented are configured for compatibility with conventional
dimensional lumber framing and other conventional building
materials.
[0144] In addition to bias direction arrays and radial arrays,
other array configurations are of course also possible, provided
that the polyhedrons are discrete from each other at equilibrium
and the connections interconnect the polyhedrons and constrain
their movement to a desired degree in the resulting array.
[0145] The terms and expressions that have been employed in the
foregoing specification are used as terms of description and not of
limitation, and are not intended to exclude equivalents of the
features shown and described or portions of them. In view of the
many possible embodiments to which the disclosed principles may be
applied, it should be recognized that the illustrated embodiments
are only preferred examples and should not be taken as limiting in
scope. Rather, the scope is defined by the following claims. I
therefore claim all that comes within the scope and spirit of these
claims.
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