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## Two Points and More

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.

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

*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.

* 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.

*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.

*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.

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.

*Above/Below)* Two views of the same helical model

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

*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.

* 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.