The Mathematics Education for the Future Project

Proceeding of the International Conference

Mathematics Education in a Connected World


Alan Rogerson

Sep 16-21, 2015

Grand Hotel Baia Verde, Cataina, Sicily, Italy

Units and Unity……pp. 143-149

All rights reserved, © Bradford Hansen-Smith 2015



Units and Unity

Bradford Hansen-Smith

This email address is being protected from spambots. You need JavaScript enabled to view it.


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.


         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 inter-transformable 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.


         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.


         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 circle- pattern 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.








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 in-formation 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.


         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 in-formation. 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.

Consider the word Geometry; from the Greek meaning "earth measure," or the measurement of things of the earth. Geo means earth, and metry is measure. The earth is spherical. The sphere is the only form we know to be inherently Whole. Measure is about movement. So we find the word geometry comprehensively means Wholemovement.

Wholemovement is self-referencing; the movement of the Whole to itself. Wholemovement is an inclusive concept. It gives description to the process of folding circles.

The context giving this word-shift importance is how language continually changes to keep up with human experience. We have been off this planet measuring things on the moon. We are now measuring universes that 10 years ago we did not know even existed. While the geometry of patterns of movement and relationships do not appear to change, our experience and understanding of the forms of geometries and mechanical workings of our universe does change. Yet we have made the meaning static by continuing to define geometry as it was understood twenty-five hundred years ago. It is important to update words to reflect our experiential level of understanding; otherwise we disconnect, limiting what is fully available to us.