This is the third post exploring the alignment of concentric circles as spherical shells starting with folding the circle in half. Folding less than half of the circle causing misalignment that will locate an off-center position and peripheral distortion. We will continue with the center off-center concentric circles and how they relate to polyhedra.

The touching points are not connected except in relationship to the circle as a dynamic system. Aligning two points on the circle boundary will provide consistency in continued development of folding and combining circles with a common spherical center. We see this in the symmetries of the three primary spherical systems: 3-6, 4-8, and 5-10 (below.)

Concentric circles are self-organizing and self-generating structural pattern. The circumference and center are simply the largest and smallest definable circle units. Movement into and out from each circle boundary is a right angle movement. This is reflected in movement by touching any two points forming a crease at right angle to the movement between points. Centering of the circle comes from this alignment of the circle to itself.

Below left) are equally spaced concentric circles. The intervals are arbitrarily ½ " apart; consistency of intervals is important. This will give an idea of spherical shells as the circle is reformed.

Below right) shows two arbitrarily placed points on the circumference showing concentric circles from each point. The kite shape is straight-line connecting of intervals between the circles as they intersect. This is the same as folding the circle in half. An interference pattern is inherent between two locations of concentric circles in a right angle relationship.

Above) Combining the touching points and the center point of the circle show the interference pattern of three center locations. Three is structural, two by themselves are not. Alternate concentric rings are shaded to make them easier to see. The right angle intersection of the kite shape comes from the intervals inherently patterned to circle movement. Folding the circle in half shows alignment of the circle/sphere context.

Below left) concentric circles with intersection of three folded diameters.

Below right) Using the same 1/2” intervals and shading alternate rings shows the interference pattern with three diameters. Each of the six points is a local center point with concentric circles. Only two rings of each have been shaded.

Above left) Six primary points of the tetrahedron net (a two-frequency triangle) where the center points have two concentric rings each. Expanding the circles to all fifteen points would have been too dense to clearly see the net.

Above right) This wave pattern shows the 13 primary points of intersection forming the hexagon star pattern. Each diameter is divided into four equal sections where each point is a center point. The level of complexity is determined by establishing design constraints, in this case two shaded rings around each local center point.

Above) Three diameters have been divided into eight equal sections, a consistent development in folding the circle. There are twenty-four creases with two concentric rings around each of the nineteen primary points forming a limited interference pattern. The intervals and shading of rings is consistently 1/2" to keep it simple. Each small white triangle interval is a point of three intersecting creases that coincide with circle intersections. This is a fractal design, often called a pattern.

The interference patterns of concentric circles reveal the polyhedral nets that are inherent in the dynamic ordering of circle division. This is where the polyhedral forms relate directly to circles and to spherical packing.

Below left) A tetrahedron unit folded from the tetrahedron net (above with six centers) indicates four spheres in closest packed order. In this case the edge length is an eight-frequency division of a two frequency tetrahedron. A single tetrahedron unit does not exist in spherical order, only as an organization of four spheres that have been truncated.

Below right) A two-frequency polyhedral form of four tetrahedral units showing ten spheres with the open octahedron space. The edges shows a sixteen-frequency division.

Above) The octahedron relationship is a formed unit of two open and joined tetrahedra. It shows six vertex points, six tangent spheres in spherical packing. The concentric circles defined on each face wrap around showing the spherical pattern in a polyhedral form.

Below left) Adding eight tetrahedra and an octahedron to the vector equilibrium sphere we see spherical packing by filling intervals between the thirteen points with concentric rings. There is an unseen interference pattern created between the concentric circles of the centered vector equilibrium sphere and the thirteen local-centered spheres in the closest packed order.

Above right) A polyhedral representation of spherical packing of the tetrahedron/octahedron matrix showing eight equal divisions of each unit edge. This association of polyhedra shows patterned spherical origin in the form of truncated spheres.

Above left) The octahedron is opened on three edges with two points joined forming a dual pentagon cap arrangement of four triangle faces and one open triangle plane; five triangle planes around the two opposite vertex points, ten triangle planes. Seven spheres are tangent on the plane surface, but unlike the matrix above, there is interior spherical distortion by reduction of the radial measure.

(Above right) By adding two open tetrahedra to the octahedron net in a tetrahedron pattern and bringing edges together an icosahedron is formed: sixteen triangle faces and four open triangle planes. This is a non-centered arrangement of twelve tangent circles on the surface planes, but again spherical distortion occurs on the inside. It is not an expression of spherical order.

Below) Another reconfiguration showing two circle each reformed to 1/2 a tetrahedron using the 12 creases (shown above.) Each reformed circle reveals the square dividing through the octahedron within the tetrahedron.

Above) The two units joined on open square faces form the tetrahedron. The concentric circles surface design shows a higher frequency division of each edge. More creases generate more complex reformations, still keeping wrap around consistency of surface design.

(Below left) This model of the off-center folding of the vector equilibrium from last month will be used to look at how the center off-center come together.

(Below right) Eight tetrahedra, from above, fill each of the open tetrahedron spaces. The tetrahedra come together at the center point showing a consistency of outward facing triangles that form the open square relationships. The completion of spherical packing has been drawn to indicate an interference pattern of interpenetrating spherical shells.

Above) Two views of octahedra filling in the six open square spaces This forms a large regular octahedron that begins to approach the off-center periphery. There is a lot of information about the interrelationships between spheres, circles and polyhedra in this system.

Continuing higher frequency development suggest the off-centered periphery will eventually be absorbed and become aligned to the circle center in a polyhedral form. Every location of local phenomena within circle/sphere unity is center, albeit local. It is then a matter of consistent higher frequency development towards bringing the center and off-center into alignment as one.

We will go more into that next month as this exploration continues.