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Tuesday, 18 September 2012 04:40

Two Points and More

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Exploring more about three diameters I again wondered about one diameter and the first fold (continuation from last month) and what else I have not seen. So I go over it one more time.

 

Marking two points on the circumference and touching them together creasing the circle in half forms a line of symmetry, the diameter.  By connecting the 4 points showing 6 lines of distance between them reveals a kite shape. Four points and six edges is the number ten; 6 touching points and 4 spheres in a non-centered spherical order. There are eight triangle associations, six of them are right triangles. There is a lot of mathematical information revealed in this one fold of the circle. 

 

All this changes when rotating around the diameter lifting the circle from the flat plane forming two open planes and two closed planes defined by the lines forming the kite shape. In rotating around the diameter in both directions there are two congruent tetrahedra, one inside out of the other.

 

 

Below) Development of kite shape by touching any two points and folding in half.

 

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Ok, the question comes up; what if we fold the circle and mark two points after the fold?  

 

 

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Above) Fold the circle in half without marking the first two points. Mark two points arbitrary anywhere on the circumference, one on each side of the diameter. Draw the straight lines between the four points. There is no kite shape; it is a quadrilateral figure without symmetry, even though the circle is in half. We still have 8 triangles, but only two are right triangles. The quadrilaterals have varying edge lengths depending on where the points are located. It remains a tetrahedron without congruent edges or angles, except for the two right angles where the diameter functions as the hypotenuse.

 

 

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                    Above ) Two circles folded on the diameter showing a symmetrical and balanced tetrahedra formed to different proportions reflecting the two points arbitrarily placed on the circumference. Each forms a differently proportioned kite shape where the diameter is a perpendicular bisector of the distance between the two points. The four outside chords have been creased to better see the tetrahedron in a traditional straight edge form. This adds more elements for spatial reconfiguring not possible with other forms of modeling.

 

 

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                    Above) Two examples of adding two arbitrary points on the circumference, one on each side of the diameter, connecting them with straight lines forming a quadrilateral. Each is a different proportion resulting from the arbitrary placement of the two points. Again the outside chords have been creased making it easier see the tetrahedron. Here the two circles are folded on the diameter forming the tetrahedra showing them to be irregular without symmetry.

 

 

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Above left)  The folded diameter is no longer the axis but becomes a variable edge length where as the line crossing perpendicular through the diameter is now the functional axial crease. It still shows dual tetrahedra but in different proportions.

Above right)  The two tetrahedra remain irregular without symmetry. The reciprocal axial function is not at right angle but on some diagonal running through the diameter.  Each circle shows the dual direction of movement in two different directions making the possibility four different tetrahedra.

 

Below) Four differently proportioned tetrahedral systems are arranged in a 2-frequency tetrahedron pattern, four circles each. Using the full circle changes the form when compared to traditional polygon constructions. Each of these tetrahedron arrangements has many different possible positions since the four rotational axes are in alignment and will go from open position, lower left, to collapsed flat in the opposite direction.

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There are a lot of formal considerations and diverse information, reformation, and choices to be explored in that first fold of one diameter and four points. The more I look the more there is; for anyone to discover. There are images of other reforming the circle with one diameter on Wholemovement Facebook page.

 

 

The following images are a couple models left overs from last months explorations and a few more that came from this months exploring spirals and conical helices limited to using six creases in the circle, using the same reconfigurations. I will go further into spirals to give a context to theses and spirals in past entries in a future blog. In the mean time here are some recent models. They are all variation of the same folded unit exploring a few things that came to mind. It is always interesting that the circle reveals the geometric form of modeling as well as the more organic and biological forms we observe in some of the more obvious forms in nature. One model has additional circles reformed using more creases to give some specific forming towards  realizing some biological expression.

 


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Above/Below) Two views of the same helical model

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                      Below) 25 circles with decreasing diameters from large to small.

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Above/Below) Two views of a spiral helix that suggested a fish-like creature, so more circles were added as an example of how spirals function on a pattern/form level in complex systems.

 

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Above)  A conical helix stack of 25 units before it is reformed to half of the double-end spiral pictured above.

 

These images are part of an ongoing exploration into the spiral. There countless possibilities using any number of different reconfigurations where each formation reveals different aspects of spiral systems. The spiral is a pattern of movement through the concentric nature of circle/sphere unity.  Movement is both into and out-from the circle as its own center. There will be more about spirals and helices in some future blog.

Thursday, 23 August 2012 07:34

June, July Grows Into August

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June’s exploration started in May and moved into July expanding into Aug before finding resolution. Each open leaf-like growth left from last month has become a closed bud-like form. This seems relatively stable as a model so I will leave it as such. Were it growing in the world it might well bring forth flowers that themselves would have seeds closed within, and so it goes. Through three month of development I have reached what appears to be the limitation of the material and the direction of the form of this model.

 

Below) Two views of present state.

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Below) Various stages of growth over the last three months starting with the tetrahedron and ending at the outer boundary of tetrahedra waiting for another season.

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Below) Details showing build up of last layer of tetrahedra folded from one crease in the circle.

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For instructions on how to fold a tetrahedron from one crease go to:

https://www.facebook.com/wholemovement/photos   and look for “Fold a Tetrahedron” and “One Fold Circle.”

 

This last layer of 115 circles, each folded to a tetrahedron using only one diameter crease, got me thinking about all I have seen revealed in one folded crease in the circle, the diameter. There must be much more that I have not seen. The vector equilibrium is fundamental to spherical packing and often the first model we make by joining four circles with three diameters each,  joining them on the straight edges. Knowing that curving the circle will reflect similar configurations that come from the folded triangular grid, it then seemed reasonable to assume the spherical vector equilibrium can be made using only one fold in each of the four circles. I reasoned since the first fold is a tetrahedron pattern and four tetrahedra in the same orientation sharing the same center point is the first centered system in the closes packing of spheres, called the vector equilibrium, traditionally the cuboctahedron. So I folded four circle one time and joined them together.

 

Below) The circle folded in half. Next is folding the diameter to itself dividing the circle into two cone shapes that are attached by the diameter fold. It is held together using a bobby pin.

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Above) Two circles with the diameter touching itself full length forms four cones, two sets of two cones each.

Below) To the left shows the two sets of two cones, two circles attached using bobby pins. To the right show the two sets of two cones each attached in the same way resulting in the spherical vector equilibrium. It is not accurate in circumference measurement from point-to-point (this can be achieved by moving the bobby pins to be equally spaced) but each point is equal distance to the center. This model shows the primary function of proportional triangular division inherent in the first fold of the circle. There are 6 square relationships between 8 cones pulled into triangulation.

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The vector equilibrium is the primary spherically centered system of twelve equally spaced spheres around a center sphere. This model demonstrates again the comprehensive, economical, and interrelated nature of folding circles, unique beyond all forms of modeling. What happens first in the circle is principle to all subsequent development. We have seen with the spherical model over the last three months, starting with one fold in the circle as a tetrahedron pattern of movement, and now ending with tetrahedra that are formed from one fold in the circle.


The next thing I fold will no doubt start with the same tetrahedron pattern of movement.  

 

(To see more about one fold in the circle go to http://wholemovement.com/blog/item/112-alignment-in-milwaukee