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Introduction to the “Dynamic Sphere Geometry,” – see descriptive text below:The following are extracts from my latest book, “3D Thinking,” to be published by Thames and Hudson Ltd, Spring 2017. Fig 1 shows a “first sequence” generated by the close-packing sphere geometry. See further descriptions below:
Animation 1: An animation of the “first” sequence of seven close-packing sphere arrangements generated by the “Dynamic Close-Packing Sphere,” geometry appears below:
Animation 2: The animation below shows the “Golden Rectangle Sphere Cluster,” that corresponds with Cell RTR 1.6 of the first sequence – where the cluster will tessellate infinitely in 3D space. Three planes of close-packing spheres within Golden rectangle cells and three planes of hexagonal packing – for a further description see below.
CLOSE-PACKING SPHERES: Close-Packing spheres are unique structures in that spheres touch each other and their centers triangulate. Close-packings of equal sized spheres have been known for millennia – to scientists, geometricians, and engineers, from Plato and DaVinci, to modern day chemists and molecular physicists. The manufacturers of almost anything circular or spherical use close packings to most efficiently pack their products, from nano-sized hollow spheres used in solar cells and lithium batteries, to packing tennis balls and optical cables. Structurally close-packing spheres and circles are stable and space efficient whilst their centers and contact points create infinite space-filling lattices. Molecules and atomic structures tend to follow close-packing arrangements and their lattices form many types of polyhedra. Their lattices make some of the most space efficient structures possible.
DYNAMIC CLOSE-PACKING SPHERES: As with all the geometries described in “3D Thinking” there are latent possibilities for close-packing circles and spheres. In the past three-dimensional applications have been primarily linked to close-packing, equal-sized, circles and spheres. What then, if we consider close-packing spheres, and circles, of different sizes, and what if we allow them to move from one close-packing relationship to another?
The “Dynamic Sphere Geometry” is based on algorithmic steps and defined parameters by which circles, or spheres, are allowed to change size and position within imposed limitations of symmetry and momentum. The parameters of the geometry can be changed, new limits imposed, new symmetries, non-symmetries, and the like, can be created to explore a broad range of sphere arrangements. The geometry can serve to “hunt”, indefinitely, for new and unique geometrical arrangements and for unique correspondences in three-dimensional space.
A FIRST 2D SEQUENCE: The Dynamic Close-Packing Sphere Geometry is a little complex – when considering spheres moving around, and changing size, in three dimensions; but in two dimensions the dynamics should be much easier to understand – making the transition to thinking in three-dimensions a lot easier. We will start, then, in two dimensions with just one set of algorithms; and then look at some of the surprising, “close packing,” arrangements generated.
(i) The building blocks of the geometry, in the first case, are defined as circles or spheres that constantly change position and size, from infinitely small to infinitely large, according to characteristics equivalent to the laws of momentum. (ii) The disposition of the system is towards a “conservation of energy,” and to “stability.” This is represented by a disposition towards, “close-packing”. However, the “momentum” of the geometry is such that circles or spheres can transform through unstable, “non-close-packing,” arrangements, towards new, “close-packing,” arrangements, and then on again. (iii) The boundaries of the system are represented by lines or planes of symmetry – so either circle or sphere centers fall on lines or planes of symmetry; or their centers and circumferences are contained within lines or planes of symmetry. Also, circles or spheres preserve their own surface/circumference integrity – where circumferences are not allowed to overlap or penetrate each other. If a circle or sphere is increasing in size it will force other circles or spheres into reducing their size, and/or into changing their position. (iv) The dynamics of the system are such that circles or spheres with the primary momentum dominate their space until they have reached the “limits” of their growth – which is determined by the limits of symmetry or, “close-packing,” imposed. Within this set of rules, and limits, circles and spheres only build momentum and grow into significance if dominating circles or spheres have reached their limits, or if there is an immediate close-packing, “need,” for their existence. (v) Any rule applied to the system (methodology) can be changed.
A 2D START-UP SEQUENCE :
We will start by looking at a “start-up” sequence. See, the illustration below. The constraining symmetry is that of a rectangle – of which a square is a special case. Cell RTR1.A represents one “cell” of a plane of close-packing spheres / circles contained within a square symmetry. According to the principles of the geometry, this particular arrangement just represents a moment in time when “close-packing” has occurred. In the example we can say that the four corner circles, r2, have the momentum – and will therefore continue to grow towards a new and different close-packing relationship. Other limitations imposed for this start-up sequence are that the r1 circle retains contact with one set of opposite sides of the containing rectangular symmetry – and that the vertices of the rectangles maintain contact with a circumscribed circle (dashed line). The start-up sequence includes a moment of transition as the first close-packing arrangement and the addition of a “latent” circle R3. Cell RTR 1.A, transforms to the next “close packing” arrangement, Cell RTR 1.B and then to Cell RTR 1.C.
Fig 2: Show a “Start Up” sequence:
A FIRST SEQUENCE
The “first sequence” as shown in Fig 1 on this page, generates seven close packing arrangements that tessellate as shown in the animation below:
Animation 3: Shows how the first sequence tessellates across a two dimensional plane:
In the “first sequence” the circumscribed circle (dashed lines in Fig 1) is used to control the relative proportions of the cells and is shown surrounding Cell RTR1.1. The first sequence continues where r3, (blue), is given the first priority of growth and grows through the close-packing cells, RTR1.2, RTR1.3, and RTR1.4, where it reaches its maximum growth by touching the top and bottom points of the rectangular symmetry. In Cell RTR1.4 r2 is squeezed to zero after r4 (green) is given a priority of growth, growing through close-packings RTR 1.5, RTR 1.6, and RTR 1.7, where it reaches it’s maximum growth. To complete the possibilities within this cell sequence r2 can be given a priority of growth; but this is not shown. RTR1.2 circles are in whole number relationships. Cells RTR1.1 and RTR1.4 will generate a lattice of squares and octagons – a semi-regular tessellation. Cell RTR1.7 will generate a square and equilateral triangle lattice – a second semi-regular tessellation on a 2D plane. Cell RTR1.6 generates a tessellating rectangular lattice – the Golden Rectangle: r1 = 2; r2 = 0; r3 = √5 – 1; r4 = 1; AB/AC = (√5 + 1)/2. The cluster below shows the “Golden Sphere Cluster” that corresponds to Cell RTR1.6 – with three planes of the Golden Ratio and three planes of hexagonal ratio.
Animation 2: The “Golden Sphere” cluster:
If alternating rings of six r4 spheres are removed the clusters will rotate, one against another.
Fig 3: Show an example of a lattice structure derived from a Golden Sphere lattice: