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Transformational Geometry


by Bradford Hansen-Smith

I define geometry as WholemovementTM in that “geo” the earth is spherical; the sphere is Whole. “Metry” is measure and about movement, of one to another. Thus comprehensively the word geometry means WholemovementTM. Transformation is about change. Transformational geometry is changing forms through movement of the Whole.

Compressing any two of an infinite number of opposite points along any axis of the sphere, reforms the sphere to a circle disk at right angle to the direction of compression. There has been a transformational movement without destroying unity or losing the Whole of spherical information.

Webster’s collegiate Dictionary 5th edition 1936
Transformation
1. To change the form of speed:
a. To change in outward shape or semblance.
b. To change in structure or composition; rarely, to transmute.
c. To change in nature, disposition, heart, or the like; convert.


2. To change (a current) in potential or in type.
3. To change the form of, as an algebraic expression of geometrical figure, without altering the meaning or value.
4. To change (one form of energy) to another.

Wikipedia
In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and explicitly using matrix theory. For example, in 3D computer graphics, the operations of moving, scaling, or rotating a 3D model are commonly called transformations. In 2D computer graphics, transformations commonly available include rotation, translation, reflection and shear (usually referred to as skew).

Websters online dictionary
1. A qualitative change.
2. A function that changes the position or direction of the axes of a coordinate system.
3. The act of transforming; "a photograph is a translation of a scene onto a two-dimensional surface".
in mathematics, "transformation" refers to a variety of different operations that can be performed using linear algebra. For example, in 3D computer graphics, the operations of moving, scaling, or rotating a 3D model are commonly called transformations.



Transformation is about movement, which always brings about form change. WholemovementTM is a transformational concept since it is through the movement of the Whole where differentiation of individualization is generated and change occurs. Transformation is a process about movement and change, shifting symmetries and properties of relationships.

The form of a seed changes through the tree into the fruit to another seed. A single spherical ovum transforms into a complex biological system of many different expressions. The sphere compressed to a circle does not lose unity, but there is a form distortion.

This site refers to two examples of different forms of a hands-on demonstration about the transformational process; one is folding circles, www.wholemovement.com and the other is weaving rigid sticks together into a spherical movement system, www.stickweaving.com. Both are concrete examples of transformational systems. These are not about the mathematics of transformational geometry, though that information is inherent in the movement of the systems. Mathematics is an abstracted symbolic language, as are these words, to describe changes that naturally occur as an ongoing result of complex. Interactions. The actual transformations between forms shows an infinite number of shifting spatial relationships, of which we recognize only a few coherent forms. I am far more excited about seeing what is between the stop action and generalizations of mathematics.

Both of these systems demonstrate the spatial experience of one form changing into another form commonly separated by their outer properties. My exploration is to understand structural patterns, the unity of movement that allows one form to change to another. How does shifting of energy change emotional forms of expression? How does the pitch and tambour of a specific note cause the rearranging of particles? What is the difference between form and the design of form, the reforming of design, the structural pattern that allows change, the transforming of generations?

These two forms are based on the same geometry, a consistency of pattern; the folded creases in the circle, and the weaving pattern of sets of ridged sticks. The folded or woven pattern does not change, but the reformations of the relationships are continually changing allowing for a great diversity of systems transformation. The circle allows infinite variety of changing forms through patterned movement of what is never less than the circle.

Below are some examples of transformational systems.

Example A) StickweavingTM transformational system:

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One system showing four transformational stopping points through a movement of continuously changing forms. There are no fixed fulcrum points in this net of bamboo sticks. This allows three directions of movement occurring simultaneously where there are few constraints in the relationships between each individual woven joint.

Example B) Here is a transformational system of wooden blocks hinged together with nylon cord. It is painted to show cross sections of the closest packing of spheres seen in the polyhedron of one octahedron with two tetrahedra on opposite ends. The circles show unexpected rearrangements in relationship to the space between spheres as the space opens up by movement of the bisections as they are rearranged.

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Example C) A transformational system using 12 folded circles:

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transformational geometry

In this arrangement of 12 reformed circles (9”paper plates) in 4 groups of 3 each, formed to the pattern of the octahedron. The corners of the triangles open and curl back where two of the three points of each triangle connect and form an open circle. A wonderful biological-like transformation has taken place through a simple movement of folded parts of circles. There has been no cutting of the circles; they are simply transformed through folding and joined into a system that itself changes forms. It moves back and forth between an octahedron and tetrahedron pattern.
Example D) A transformational system using 8 folded circles:

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©2007 Bradford Hansen-Smith. All rights reserved.
WholemovementTM and StickweavingTM are trademarks of Bradford Hansen-Smith / Wholemovement Publications.
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