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Figure 13: Triangular space-frames generated by connecting sphere centers of HP1 Polyhedral lattices can be created from any close-packing sphere arrangement. Figure 13 shows triangular attices generated by connecting circle centers of an octahedron-tetrahedron arrangement of close-packing spheres, HP1. Figure 14 shows triangular lattices from HP1 extracted in the form of truncated octahedrons. Figure 15 shows triangular lattices from TP3/HP3. Close-packing sphere arrangements are very stable and attices derived from them will share the same characteristics — space efficient and structurally stable with stress loads easy to calculate. Sphere lattices generated from tessellating close-packing sphere clusters extend infinitely and 3D forms extracted from these lattices are as varied as imagination allows. Whatever forms are extracted such as ‘space-frames,’ ‘perceptual lattices,’ or product designs, they will be in proportion, one with another — a harmony of space-filling structures!

Figure 13 Triangular space-frames generated by connecting sphere centers of HP1 Polyhedral lattices can be created from any close-packing sphere arrangement. Figure 13 shows triangular attices generated by connecting circle centers of an octahedron-tetrahedron arrangement of close-packing spheres, HP1. Figure 14 shows triangular lattices from HP1 extracted in the form of truncated octahedrons. Figure 15 shows triangular lattices from TP3/HP3. Close-packing sphere arrangements are very stable and attices derived from them will share the same characteristics — space efficient and structurally stable with stress loads easy to calculate. Sphere lattices generated from tessellating close-packing sphere clusters extend infinitely and 3D forms extracted from these lattices are as varied as imagination allows. Whatever forms are extracted such as ‘space-frames,’ ‘perceptual lattices,’ or product designs, they will be in proportion, one with another — a harmony of space-filling structures!

Related Figures (14)

Figure 1: Lattice structures from close-packing spheres TP3/HP3, see Figures 10,11,12,1: Close-packing circles and spheres touch each other and triangulate to form 2D and 3D cells or clusters that repeat infinitely in 2D and 3D space. From these cells and clusters many interlinking space-efficient lattices can be extracted for applications in design (for product design or packaging); in art as lattices that stimulate perception “perceptual lattices”; and in architecture as lightweight “space-frames” or polyhedra (Figure 1) aan
Figure 1: Lattice structures from close-packing spheres TP3/HP3, see Figures 10,11,12,1: Close-packing circles and spheres touch each other and triangulate to form 2D and 3D cells or clusters that repeat infinitely in 2D and 3D space. From these cells and clusters many interlinking space-efficient lattices can be extracted for applications in design (for product design or packaging); in art as lattices that stimulate perception “perceptual lattices”; and in architecture as lightweight “space-frames” or polyhedra (Figure 1) aan
Figure 2: /ris, Planet Earth, Bubbles, Frog Spawn together into a sphere whether they are the electron clouds surrounding an atom or the gelatinous material that surrounds a frog’s embryo in frogspawn, Figure 2. Beginnings of a New Geometry of Circles and Spheres and Perceptual Designs
Figure 2: /ris, Planet Earth, Bubbles, Frog Spawn together into a sphere whether they are the electron clouds surrounding an atom or the gelatinous material that surrounds a frog’s embryo in frogspawn, Figure 2. Beginnings of a New Geometry of Circles and Spheres and Perceptual Designs
In 1969, in London, England, the physicist Dr. Ensor Holiday gave me a line drawing of an Islamic geometric design based on close-packing circles. The drawing was from a collection compiled by Jules Bourgoin in 1879 CE [2], who copied it from a window grille in the Mamluk-period Sarghatmish Mosque Madrasa (1356 CE) in old Cairo, Egypt, see Figure 3. The unit symmetrical, ‘cell,’ of the design was a 45 degree right-triangle with five close-packed circles of different sizes. Clipped tangent lines at circle contact points created five regular and almost regular tessellating polygons — of sides 4, 5, 6, 7, and 8. Ensor also showed me lattices generated from the two-dimensional close-packing circle arrangement by connecting circle centers and circle contact points. Ensor and I started working together and found that the 2D lattices conjured an almost infinite number of images in the, ‘mind’s-eye,’ of those that looked at them, much like clouds in the sky or cracks on a wall — abstract shapes, images of animals, plants, people, and even as complete scenes — where any image found could be found again and again, rotated, translated, reflected — so that all sorts of patterns could be generated. I called this type of lattice, “Perceptual”. The lattices were eventually published by the Longman Publishing Group in 1973 and became the popular geometric coloring book series Altair Designs - the books are now published by Wooden Books Ltd., [3], [4].
In 1969, in London, England, the physicist Dr. Ensor Holiday gave me a line drawing of an Islamic geometric design based on close-packing circles. The drawing was from a collection compiled by Jules Bourgoin in 1879 CE [2], who copied it from a window grille in the Mamluk-period Sarghatmish Mosque Madrasa (1356 CE) in old Cairo, Egypt, see Figure 3. The unit symmetrical, ‘cell,’ of the design was a 45 degree right-triangle with five close-packed circles of different sizes. Clipped tangent lines at circle contact points created five regular and almost regular tessellating polygons — of sides 4, 5, 6, 7, and 8. Ensor also showed me lattices generated from the two-dimensional close-packing circle arrangement by connecting circle centers and circle contact points. Ensor and I started working together and found that the 2D lattices conjured an almost infinite number of images in the, ‘mind’s-eye,’ of those that looked at them, much like clouds in the sky or cracks on a wall — abstract shapes, images of animals, plants, people, and even as complete scenes — where any image found could be found again and again, rotated, translated, reflected — so that all sorts of patterns could be generated. I called this type of lattice, “Perceptual”. The lattices were eventually published by the Longman Publishing Group in 1973 and became the popular geometric coloring book series Altair Designs - the books are now published by Wooden Books Ltd., [3], [4].
Figure 4: 4 Close-Packing Circle Sequence generating four Close-Packs, also see Figure 9 The Dynamic Sphere Geometry presented here positions circles or spheres within, or on, the symmetrical boundaries of a tessellating cell such as rectangle or a triangular prism. From a ‘start’ arrangement, circles or spheres are allowed to change size and position according to a rule set that generates close-packing cells and clusters in sequences. The geometry is based on algorithmic steps that change circle and sphere sizes and positions with an analytic that identifies new close-packing arrangements. Imposed parameters give a ‘priority of growth’ to circles or spheres as well as an equivalent of a dynamic and a direction of growth. The geometry can be directed to ‘hunt’, indefinitely, for new and unique geometrical arrangements and for unique correspondences in two- and three-dimensional space. In this first example, see Figure 4, circle centers are fixed and lie at the corners of a unit right-isosceles triangle, ABC, of a square. S1.1 is close- packing ‘start’ position. Circle R2 is given the priority of growth forcing Ri to reduce in size. As circle Ri gives way it allows R; to grow. Ro grows and triangulates when it touches R; (it already touches R:) creating a ‘close-pack’ $1.2. R2 continues to grow until it reaches its maximum size determined by the symmetrical boundary BC of the unit triangle, a ‘close-pack’ S1.3. R;3 then takes the priority of growth until reaching its maximum at AB, a ‘close-pack’ $1.4. The wavy arrows represent transitions as circles change size and move through non-close-packing positions (Transitions). Dots represent circle centers.
Figure 4: 4 Close-Packing Circle Sequence generating four Close-Packs, also see Figure 9 The Dynamic Sphere Geometry presented here positions circles or spheres within, or on, the symmetrical boundaries of a tessellating cell such as rectangle or a triangular prism. From a ‘start’ arrangement, circles or spheres are allowed to change size and position according to a rule set that generates close-packing cells and clusters in sequences. The geometry is based on algorithmic steps that change circle and sphere sizes and positions with an analytic that identifies new close-packing arrangements. Imposed parameters give a ‘priority of growth’ to circles or spheres as well as an equivalent of a dynamic and a direction of growth. The geometry can be directed to ‘hunt’, indefinitely, for new and unique geometrical arrangements and for unique correspondences in two- and three-dimensional space. In this first example, see Figure 4, circle centers are fixed and lie at the corners of a unit right-isosceles triangle, ABC, of a square. S1.1 is close- packing ‘start’ position. Circle R2 is given the priority of growth forcing Ri to reduce in size. As circle Ri gives way it allows R; to grow. Ro grows and triangulates when it touches R; (it already touches R:) creating a ‘close-pack’ $1.2. R2 continues to grow until it reaches its maximum size determined by the symmetrical boundary BC of the unit triangle, a ‘close-pack’ S1.3. R;3 then takes the priority of growth until reaching its maximum at AB, a ‘close-pack’ $1.4. The wavy arrows represent transitions as circles change size and move through non-close-packing positions (Transitions). Dots represent circle centers.
Figure 7: Tessellating cells and circle connecting lattices of the second sequence
Figure 7: Tessellating cells and circle connecting lattices of the second sequence
Figure 6: Circle Ratios of the second sequence Changing Start Points
Figure 6: Circle Ratios of the second sequence Changing Start Points
Figure 8: Any cell can be used to generate a new sequence Any start position can be selected to generate a close-packing sequence. In this case Cell R1.6 has beer selected and rs given a priority of growth, see Figure 8.
Figure 8: Any cell can be used to generate a new sequence Any start position can be selected to generate a close-packing sequence. In this case Cell R1.6 has beer selected and rs given a priority of growth, see Figure 8.
Figure 10: Triangular prism close-packing sphere sequences — dots show sphere centers: TP1; TP2; TP3. Three-dimensional sphere sequences develop in a way similar to circle sequences. Sphere close-packin; clusters can be generated within 3D space tessellating structures, such as triangular, or hexagonal prism where the prisms are equivalent to the 2D constraining symmetries such as squares and equilateral triangles Figure 10 shows three sphere packings generated by the geometry where spheres change size and positiot within an equilateral triangular prism. The TP1 cluster represents the start position and does not fully close pack. Clusters TP2 and TP3 are generated from TP1 and do close pack. Arrangement TP3 is a 3D cluste corresponding to the ‘golden ratio’ cell R1.6, Figure 5. Figure 11 shows another three sphere cluster generated by the geometry this time within a hexagonal prism. The HP1 cluster is a tetrahedron/octahedrot arrangement of spheres; HP2 cluster is a cube packing of spheres; the HP3 cluster can be considered as | hexagonal development of TP3. See Figure 12 for different views and sphere proportions of the TP3/HP. clusters.
Figure 10: Triangular prism close-packing sphere sequences — dots show sphere centers: TP1; TP2; TP3. Three-dimensional sphere sequences develop in a way similar to circle sequences. Sphere close-packin; clusters can be generated within 3D space tessellating structures, such as triangular, or hexagonal prism where the prisms are equivalent to the 2D constraining symmetries such as squares and equilateral triangles Figure 10 shows three sphere packings generated by the geometry where spheres change size and positiot within an equilateral triangular prism. The TP1 cluster represents the start position and does not fully close pack. Clusters TP2 and TP3 are generated from TP1 and do close pack. Arrangement TP3 is a 3D cluste corresponding to the ‘golden ratio’ cell R1.6, Figure 5. Figure 11 shows another three sphere cluster generated by the geometry this time within a hexagonal prism. The HP1 cluster is a tetrahedron/octahedrot arrangement of spheres; HP2 cluster is a cube packing of spheres; the HP3 cluster can be considered as | hexagonal development of TP3. See Figure 12 for different views and sphere proportions of the TP3/HP. clusters.
Figure 11: Hexagonal prism close-packing sequence — dots show sphere centers: HP1; HP2; HP3
Figure 11: Hexagonal prism close-packing sequence — dots show sphere centers: HP1; HP2; HP3
Figure 14: Triangular space-frames forming truncated octahedrons by connecting sphere centers of HP 1
Figure 14: Triangular space-frames forming truncated octahedrons by connecting sphere centers of HP 1
Figure 15: Triangular space-frames generated by connecting sphere centers of TP3/HP3. Once a space filling lattice is generated by the geometry then 3D forms of all sorts can be extracted from it.
Figure 15: Triangular space-frames generated by connecting sphere centers of TP3/HP3. Once a space filling lattice is generated by the geometry then 3D forms of all sorts can be extracted from it.
Related topics:
Architectural GeometrySpace filling designsGolden Ratioarchitectural proportion
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