Units and Unity
Bradford Hansen-Smith
wholemovement@gmail.com
Abstract
We use the symbol of circle as a unit, but know little of circle unity. Folding
circles demonstrates qualities belonging to unity of the whole. Unity is not
constructible, but can be experientially realized in transforming the circle
through folding. Traditional 2-D & 3-D geometry and fundamental math
concepts are revealed in the folds of the circle. Within every polygon and
random shape is a circle-pattern of organized movement that can be revealed
through a proportionally ordered folding process.
Introduction
Unity is the constant inherent to all relationships, a place where
countless units are generated and primary context where everything is
interrelated by unity of the Whole. Construction of “parts-to-whole” in
education is always partial. A linear progression towards synthesis is based
on predetermined parts that can only generate more complexity or simplified
generalizations. With insistence on the importance of units, the magnitude and
unexplainable dynamic nature of unity has become marginalized. The idea of
a universal building block misinterprets the nature of unity, where
individualized parts are interdependent, multifunctional, and intertransformable within the contextual whole.
The approach of “Whole-to-parts” provides context first, giving balance
to “parts-to-whole” construction where endless generations of parts suggest
we will always fall short. The circle unit is uniquely both whole and part,
revealing qualities that go unrecognized because we do not fold circles.
Systems of formal geometry reflecting alignment to order and patterns
observed in the phenomenal world are the same patterns revealed through a
principled and systematic folding of circles.
Geometry measures and classifies parts using symbols for constructing
and controlling predictable relationships. Expanding understanding is
achieved through the experience of folding circles and observing unit parts as
they are revealed in context with other parts. Folding the circle generates
structural dynamics with proportional gain through a patterned organization of
variables principled to the first fold giving consistency throughout all
subsequent folding, forming, reforming, transforming, and joining multiple
circles.
First
Unity is a quality of wholeness. The circle, while being used primarily as
a unit symbol for nothing, has a longstanding tradition tied to the concept of
spherical unity where nothing can be added and nothing taken away. This
means everything is endlessly in process of being revealed; this primes our
curiosity keeping us alert towards discovery. Euclid first defined a point, “…is
that which has no part.” This suggests unity, without dimension or size. Using
point unity he defines lines, planes, angles, and then defines a circle by the
relationship of points. The circle ends up being an image symbol that
represents nothing, yet fundamental to construction of everything. Tradition
has destroyed unity by cutting through the sphere to demonstrate great and
lessor circles that we draw on a flat plane. Further along we truncate circles to
polygons that are used to construct static polyhedra that approximate
spherical unity destroyed in the first place.
By compressing the sphere to a circle disk in space a transformation
takes place without destroying or minimizing the whole for the sake of parts;
nothing is added or taken away. The first fold of the circle decompresses both
2-D and 3-D relationships revealing abstracted math concepts. Drawing
circles for construction is a gesture towards control. Folding circles is a
dynamic process revealing units within unity.
There is no tradition for folding circles. When cutting the image of the
circle from the paper it becomes the 3-D volumetric equivalent of a
compressed sphere in circle form without any loss of 2-D information. The
concentric nature of infinite self-alignment of the circle suggests consistency
of both part and whole without scale where the circle is to itself center. Every
tangent line drawn to a circle is in context a chord to a larger circle, yet
appears detached in relationship to smaller aligned circles without point of
contact. Through folding the circle countless parts and complex systems of
geometric organization are generated by the creases. Throughout folding,
reforming, and joining multiples, the circle remains whole. The circle does not
change from being a circle; it is the perception of the folder that changes
about the nature of the circle.
A shift occurs, from putting parts together to observing the generations
of parts in an ordered and organized system where all parts are inner-related
within one location. To realize the greatest potential from the circle all parts
work together in a variety of transformational relationships. Creased lines,
points of intersection, areas of spatial movement, angles of symmetry are all
multifunctional. Removing any element from the circle diminishes potential for
all individual parts, thereby decreasing the transformational possibilities and
reducing the circle expression to a part that is no longer whole. The circle
reveals patterns in multiple forms of expression. If all this information was not
in the circle/sphere to begin with, it could not be folded, nor could it be
traditionally constructed in images using compass and straight edge. It must
be there first, inherent within the circle. The folded circle is the compass and
the straight edge.
Folding the Circle
Through systematic folding the concept of unity is experienced that is
not possible in any other form. Folding the circle in half is a spherical pattern
of movement. This first fold is principle to all subsequent folding, to reforming
and joining all flat plane surfaces. One fold of the circle shows every part
interconnected to all other parts, both folded and yet to be folded, nothing is
constructed. Each part shares in common an individual relationship to the
circle. Since we do not fold circles we have missed the dynamic
interrelationship between units and unity; being left with fragmented images of
relationships without context.
We are first taught units in numbers by adding, subtracting, multiplying
and dividing. Folding the circle shows the opposite; unity in division creates a
multiplicity of parts that can then be added and subtracted. The first fold of
division shows the circle and straight line without separation. This diameter
relationship between touching any two points on the circumference reveals a
right angle function; such as opposite edges of the tetrahedron, the form of the
Cartesian grid, and the structural nature of the Pythagorean triplet. They are
reflections from the right angle sphere/circle compression. This right-angled
first fold division shows a ratio of one whole two parts. There are three primary
symmetries that can be proportionally folded from this first ratio of 1:2; they
are 3:6, 4:8, and 5:10. Through principled folding comes familiarity with these
symmetries and the formation of individual polyhedra leading to identifying
polygons and 2-D circle-patterned systems. In discussing this first fold
generalizations can be made where the economy of using symbols opens the
abstract world of mathematics to deeper exploration into expressing what
otherwise would be difficult. Complexity is grounded in the simplicity of first
action in context, dividing into unity creating multiples that diverge and
converge, organized by the completeness of the circle shape itself.
Symmetry
Touching two of an infinite number of points on the circumference folds
the circle in half. This shows one diameter from infinite possibilities of which
no two are the same. This one fold is a spherical pattern of movement rotating
around a unique diameter/axis that at the same time reveals two congruent
and reciprocal tetrahedra. From the 1:2 ratio is generated the 3:6 symmetry
which is primary before 4:8 and 5:10.
Consistent to the 1:2 ratio we fold the circle twice more in that ratio
dividing it into thirds generating three diameters defining a hexagon
relationship in 3:6 symmetry. Again folding the circle to the same ratio 1:2
four times generates four diameters with eight equal sectors in a 4:8
symmetry. Folding 1:2 five times reveals five diameters in a 5:10 symmetry.
These three proportional relationships through consistent folding can be
expanded to higher frequency grids within the circle. The three symmetries
are transformational one to the other through commonality within circle unity. If
this were not so they could not be constructed individually.
Above) The 3:6, 4:8, 5:10 symmetries folded in the circle have been increased
to smaller divisions in an eight-frequency grid matrix. Alternate areas in each
circle have been colored to show regularity of pattern in each triangular grid.
The 3:6 and 4:8 grids are similar to an octave in music, where each diameter
is divided into eight equal segments. The 5:10 grid is unique in proportional
division creating greater complexity. Each fold is a chord resulting from the
self-organizing proportions generated from the first fold in half. The individual
relationship of touching any two points on a plane will always be at right angle
half way between those two points.
Folding Circle-Pattern
From any point using a random shape paper or polygon we can fold
these same three symmetrical grids because they are first foldable in the
circle. They all start from a divisional movement of one shape in two parts.
This 1:2 ratio is primary to all folding, consistent to the circle. Unity in a circlepattern of organized hexagonal, pentagonal, and square relationships of
triangular division, can be folded anywhere from any size or shape of paper.
Symmetry is not tied to regular polygons, but is a quality of balanced
proportional movement observable and demonstrably using any polygon or
any random irregular part.
Using any scrap of paper make one fold. Pick an arbitrary point
anywhere on that folded line as a pivot point for the next two folds. Fold one
part of the crease from one side of the pivot point over to where the folded and
unfolded angles look equal. Do not crease yet; turn the paper over and fold
over the unfolded part to line up with the previously folded edge; slid back and
forth as necessary to get edges even, where the adjacent angles at the pivot
point are congruent. Crease when the edges are even on both sides. This
forms one point and two edges of a triangle. Open the circle finding three
creased intersecting lines through a single point with six equal central angle
sections. This can be anywhere on the paper depending on the placement of
the first fold and selection of point.
Refold back to the triangle. Fold the pivot point to any place on one
folded edge and crease. This crease ends at a point on the adjacent edge.
Unfold and bring pivot point to the new point on the adjacent edge, crease
again. There are now two symmetrical intersecting and congruent right-angle
triangles. When opened to the flat paper there is a hexagon star where the
hypotenuse of the triangles are six radii revealing a circle-pattern from a
randomly placed line and point without any circle shape.
Refold to the creases. Now fold from the pivot point, the two sides of the
triangle bringing the two edges together. This divides the 60° angle in half
making two 30° angles. Open the folds out to flat paper, there are six
diameters, twelve 30° central angles in a hexagon star division. This is exactly
what you get folding a circle to a 4-frequency diameter grid where the three
primary diameters are divided in four equal parts. The second set of three
diameters are divided unequally and are at right angle to the first three
diameters, half way between the star points. They function as bisecting or
secondary diameters. This hexagon pattern, now a 3:6:12 symmetry, can be
reformed showing the pentagon, square, and triangle arrangements.
Refold to the creases of the two straight edges of the 60° triangle. Line
up between the two end points of the hypotenuse of the two intersecting right
triangles, making a creased line connecting those two points. This line shows
the third leg to this equilateral triangle. Opened to the flat paper to find the
creases in a regular hexagon where each equilateral triangle is bisected three
times. Each line connecting opposite star points is the same length diameter
to an unformed circle. To show this circle-pattern in circle form refold to the
triangle, inscribe an arc above the last folded crease and cut away the
irregular perimeter between the two end points. Open the paper showing the
same creased folds now in a circle; the same as if we folded starting with the
circle. The circle-pattern is foldable using any shape.
Hexagon, 3:6
Square, 4:8
Pentagon, 5:10
The hexagon, the square, and the pentagon patterns are expressed by
folding differently proportioned angles on an arbitrary first fold with a randomly
placed pivot point on an irregular piece of paper.
Measuring angles and the paper shape are unimportant in folding these
symmetries. Numbers describe proportionally what is already there. Accuracy
of proportions in folding is important, but not expected at first; improvement
comes with familiar in seeing proportional similarities and differences. This
helps eye-brain-hand coordination useful in all disciplines. By discussing these
observations information is revealed that allows deeper understanding of
traditional geometry and math concepts. The more we fold, the more information is revealed, the more there is to observe.
Above) Each folded grid has been extended and alternate areas in each net
have been colored to emphasize the triangulated nature.
Above) The 3:6 net has been reformed to a pentagon star, a square half
octahedron, and to the triangle in a tetrahedron configuration.
Above) Two reconfigurations of the 4:8 net using the same scrap of paper.
Reforming of the three girds is a subtractive process by folding parts in
leaving only what is required for a given configuration. Folding out is adding
what is already there having been folded in. The triangulated nature of the
grids allow for this kind of transformation regardless of perimeter shape.
Conclusion
There are three primary proportioned symmetries that are unified within
the circle. The pattern of movement when folding circles reveals structural
organization demonstrating both 3-D and 2-D, revealing abstracted functions
foundational to formal math. Proportional folding the circle demonstrates unity
that is the generator for the organization of individualized parts. Any irregular
part of the circle plane inherently carries circle-pattern unity. A unit part by
itself has no value, and without units unity has no expression. Parts and whole
are each within the other.
It is time to begin folding circles along with drawing symbols used to
represent them, if we are to expect a different outcome from the math we are
teaching. Folding circles is cost effective with an unexpected high yield of information. Starting from unity will greatly enhance our understanding of the
circle as a self–referencing, self-generating pattern without limitation of
boundary or scale. More importantly folding circles provides a hands-on and
comprehensive experience of unity that is unique to only the circle.
Will folding circles facilitate learning math? Will it change the way we
think to explain geometry and the universe? Do we ignore it or incorporate it?
Maybe it has no relevance to the problems that face our world, but then
maybe it does. Words without experience of doing carry little understanding;
folding circles is an experience in doing and discovery.