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Sunday, 22 July 2012 05:25

Fractal Unity

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This is a continuation from June's exploration and part of an ongoing investigation using only the nine creases of the tetrahedron in systems development. I am finding relationships in folding circles similar to fractal growth observed in nature and in algorithmic produced fractal images. Growing systems are always limited to the needs of larger systems where survival is supported by serving the larger context that is itself evolving. There are no whole systems, only systems serving larger systems. Every system functions in alignment to a larger context of purpose that is reflected in every part of all systems, showing unity throughout. There are always unique and identifiable evolving patterns directing the recognition of individuals, families, regions, cultures, kingdoms, phylum classifying similar formations, functions and individualized parts. Geometrically the context is the circle compression of spherical unity.

This ongoing model shows a systematic development using one reformations and a few variations from the numerous possibilities in the tetrahedron net as folded in the circle.

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 Above) Each individual unit is a reformed circle. Two similar parts can be combined in three of the same ways (right side.) I choose the above middle on the right since it was most like a line with two end points, points expanded to circles.


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Above left) Arrangement of six of the above units in a tetrahedron pattern.

Above right) Thirty units, (sixty circles) forms the icosidodecahedron.


Below) I decided to double the single units making a set using four circles. There are two basic variations; one with open sides folded over and the other with sides not folded over (bobby pins are holding them together.)

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Below) are three views of the icosahedron pattern formed using both variations of double units showing the spherical icosidodecahedron formed in a great rhombicosidodecahedron arrangement of squares hexagons and decagons. It is lopsided because long units were used on one side and short units on the other side distinguishing polar division.

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Each cell is a multiple of unity; each circle reformed is consistently whole. Growing changing systems are always incomplete expressions of unity. Each circle is unity carrying potential for countless possibilities of reformations directed by specific design limitations of a given system. Multiple circles give a unique and individual expression to any association of  parts. The unlimited potential lies within each component part as a unique expression. The unity inherent in each part insures unity through out. This is a primary fractal process of self-similarity towards finite expression of infinite potential.

Putting parts together gives form to synthetic unity, an idea about unity that is decidedly different than unity inherent in the circle that is not constructed. Both are important, it taking one to understand the other. By self-alignment in folding the circle ( information is generated that is principled and directive to all subsequent fractal development. This is not figured out mathematically, it grows geometrically through a principled, patterned process.

The regularity of the icosidodecahedron pattern gives stability to the arranging of parts. The form is changed through layering recursive stages introducing human choice as a factor allowing variations to alter the forms in unexpected ways, always in line with specific pattern, not unlike what happens in genetic recombining. This self-referencing system allows for spontaneous and consistent choices purposefully made where unexpected growth can occurs.


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Above) Three views where one half of the hexagon openings have been developed on one side showing the pentagons of the dodecahedron. The other side requires a different configuration to accommodate the difference in the two units initially used.


Below) shows further growth by modifying the single unit to conform to what is there. Again variations of reformation are necessary to accommodate each half. I have left the band around the middle open developing the polar ends.

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Below) more layers of units are consistently added to areas that will easily accept them in a way that looks like natural growth coming from within. There is now greater variations in reforming the circle units giving greater complexity to the system.

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Below) more layers are added expanding what is already formed moving more towards a spherical uniformity as the middle  becomes increasingly filled out.

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Above) the sphere as it now stands where all units on the outer layer are the same reconfigurations. I have left one decagon open to the inside as you might find with any growing systems where there is a primary connection between the inside and outside through all stages of growth. The outside spherical configuration is getting more spherical and complex. It remains open much like a sieve, without filling in the open space. I don’t know if I will continue developing this to see where it goes or get on with exploring some new formations that have come up during this process. Most likely I will continue to explore both.


The circle is Whole and part, something no other shape or form can demonstrate. There is unity throughout because every part starts with and ends with unity, giving consistency to the variety and differences of interrelated transformational parts. I understand the circle as pattern for all fractal self-similar processes.




Read 63033 times Last modified on Tuesday, 24 July 2012 08:14
Bradford Hansen-Smith

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