Three Dimensional Polyhedral Array
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.
This application is a § 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).
FIELDThis application relates to arrays, and in particular, arrays of materials comprised of a plurality of geometric polyhedrons and a connection network interconnecting the polyhedrons.
BACKGROUNDChoices 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.
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.
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.
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.
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 (C60) molecule (commonly known as the “Buckyball”) due to its unusual geometry that may have unique useful properties.
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.
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.
SUMMARYDescribed 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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
In specific implementations, at least some of the polyhedrons are icosahedrons.
In specific implementations, the plurality of polyhedrons includes at least one interior polyhedron connected to twelve adjacent polyhedrons.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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.
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).
For example, as shown in
Similarly, as shown in
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
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
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.
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.
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.
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
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.
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 LoadingAs 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.
In addition to withstanding forces, the polyhedral arrays also withstand torques.
The examples of
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.
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.
In the array 10, which is shown at equilibrium, the polyhedrons 12 are spaced apart and do not contact each other. In
The polyhedrons 12 in the array 10 have the same general orientation. In the example of
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
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.
Although not clearly visible in
The connections 14, 16, 18 are shown in
In
For the specific implementation of
The polyhedrons 12 in the example of
As shown in
As shown in
In the radial array, the polyhedrons in the different layers as shown in
The polyhedrons 212 can be provided with predetermined features to facilitate the transformations in shape. In the specific implementation of
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 C60 molecule is a truncated icosahedron, and the B12 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 ImplementationsAccording to one specific implementation as shown in
Referring to
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.
As best shown in
As also shown in
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.
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.
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
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
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.
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.
As another example,
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.
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.
ApplicationsIn 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.
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.
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.
Type: Application
Filed: Aug 15, 2006
Publication Date: Feb 21, 2008
Inventor: Samuel J. Lanahan (Portland, OR)
Application Number: 11/579,307
International Classification: E04H 1/00 (20060101);