Revolutionary Geometry

REVOLUTIONARY GEOMETRY: A Foundation for Nature-Based Architecture by JAMES JACOBS July 2003 page 1 | 2

Historically, simple geometric forms have been the basis for envisioning structure in architecture. It follows that a study of advanced geometric forms may provide the basis for envisioning advanced structures in architecture. There is no historical record of new geometric structural systems being revealed since the discovery of the circle, square and triangle. The geometric structural system of the fourth archetypal form, the spiral–in its 3-D form, the the helix–has been uncovered and developed over the past 25 years by the author. Helical Geometry is the study of geometry within the tetrahedron, the most fundamental of the 5 Platonic Solids of Solid Geometry.

Fig. 1 The tetrahedron, the most basic of the 5 Platonic Solids

Fig. 2 The Helical Field Geometry within the tetrahedron

Helical Geometry is the geometry of the straight twisted rod (Fig. 3), in the same way that Plane and Solid Geometry are the geometries of the straight rod. (Fig. 4) These are two distinctly different representations of distance, and so, two distinctly different approaches to understanding the geometric properties of space.

Fig. 3 The straight twisted rod of Helical Geometry

Fig. 4 The straight rod of Plane and Solid Geometry

The geometric forms of Plane and Solid Geometry continue to be the basis for the way we design and build, as well as for the way we think about the laws of nature and how nature

builds. The new geometrical system of Helical Geometry, by redefining distance in space as having a simultaneous measurable degrees-of-rotation, or twist, has profound implications for the foundations of our existing knowledge. Literally, Helical Geometry adds new meaning to our ideas of what is "rational." It is not too bold to suggest that Helical Geometry offers a foundation for advancement in all areas of knowledge as it changes the way we think about that most fundamental concept, distance in space. This article discusses Helical Geometry, its correspondence with Nature, and its incorporation of existing geometric knowledge.

Geometry’s Correspondence with Nature Geometry is an attempt to understand the source of the symmetry seen in nature, and the structural order of that symmetry in space. The search for this understanding can be approached in two ways, numerically (i.e. mathematically), or structurally. Modern science uses the mathematical modeling approach, assuming that numerical models or formulas will reveal the source of symmetry in space. The ancient Greek geometers used the structural modeling approach, assuming that structural models would reveal the source of symmetry in nature, and, express a numerical model, the formula of a mathematical theorem. The geometer Pythagoras was credited with first showing a correspondence between a geometric structure and the source of symmetry in space around 350 BC. He demonstrated that two of the properties of the source of symmetry in Nature are the right angle and four-fold rotation. The archetypal geometric form he used was the triangle containing a ninety-degree angle, a plane right-angled triangle. He showed how this family of geometric structures reveals a correspondence with nature’s symmetry in 2-dimensional space.

Fig. 5 The plane rightangled triangle's proof of correspondence with the 4-fold, 2dimensional symmetry in Nature.

Pythagoras rotated this unique type of triangle (with its one right angle) in a fourfold pattern, its longest side facing outward, and so generated the symmetrical boundaries of a perfect square. Then, by a redistribution of the triangles making up the symmetrical pattern, he showed that the remaining two sides of the right-angled triangle structure also generated the symmetry of two perfect squares. And, that these two squares of symmetry are contained within and equivalent to the symmetry of the square of the longest side. (Fig. 5) This is true of all triangles having a right-angle, and not true of any other archetypal geometric structure. (Ref: "The Ascent of Man", Jacob Bronowski)

This unique type of 2-dimensional geometrical structure having a right-angle visibly demonstrated its correspondence with nature’s symmetry in 2dimensional space. For this reason, the single unique property of this geometrical structure, its right-angle, and, the fourfold rotation required to generate the symmetry of the square, are considered to be properties of the source of the symmetry in 2-dimensional space. All plane right-angled triangles express a numerical model, the formula of the mathematical theorem which states: The square of the longest side of the plane right-angled triangle is equal to the sum of the squares of the two shortest sides, which is, c2=a2+b2, the formula of the Pythagorean Theorem, the most important theorem in all mathematics. The plane right-angled triangle expresses the Table of Natural Trigonometric Functions of Sines and Cosines, without which there would be neither Newtons’s laws of nature, nor Einstein’s Theories of Relativity. The validity of science’s natural laws and universal theories, dependent as they are on the plane right-angled triangle, speaks for the correspondence of this geometrical structure with Nature’s symmetry, and the source of symmetry in 2-dimensional space. If there were no such correspondence, then its numerical expressions of the plane rightangled triangle would not have led to subsequent mathematical descriptions corresponding with the laws of nature and universe. Helical Geometry uses a structural modeling approach similar to that of the ancient Greek geometer’s approach to understand the source of symmetry in nature. It demonstrates its correspondence with the 3-dimensional symmetry in space by the fourfold rotation of a unique geometrical structure, the helical building panel, that mimics the 3-dimensional symmetry of the natural form of a molecule-thin liquid membrane, the soap-film (Fig. 6). In addition, Helical Geometry’s unique geometrical structure shows a direct correspondence with the geometry of the plane right-angled triangle, and expresses the Table of Natural Trigonometric Functions of Sines and Cosines, but in 3-dimensional space as opposed to 2-dimensional space.

Fig. 6 The helical building panel's proof (#1 of 2) of correspondence with the 4-fold, 3-dimensional symmetry in Nature.

How Helical Geometry Demonstrates Correspondence with the Source of Symmetry in Nature.

A nature-based way of designing and building begins with a nature-based geometry. Helical Geometry is structural biomimicry, that is, its forms structurally mimic the natural form of a soap film. The helical form is universal, existing in every form of matter. From the microcosmic, atomic structure of crystal growth to the molecular structure of DNA to the macrocosmic spiral form of galaxies, all structure in matter mimics the source of symmetry in Nature. Uncovering the 3-dimensional geometry of the helical form in Nature, then, is to uncover the 3-dimensional source of the symmetry in nature. Soap films exemplify an important mathematical idea called a minimal surface. Soap films form minimal surfaces because the energy of surface tension in a soap film is proportional to its area. Nature always minimizes energy expenditure, so soap films minimize area. For example, the natural form of a soap film represents the surface of smallest area within a framework of Plexiglas tubes.

Fig. 7 The structural properties of the helical soap-film

Fig. 8 The helical edge of the soap-film

Structurally mimicking the properties of a soap film stretched within a framework of Plexiglas tubes demonstrates the natural basis for Helical Geometry. When a soap film is suspended within a framework of Plexiglas tubes that are strung like long cylindrical beads we are able to observe its structural properties (Fig. 7).

At first glance the warped surfaces of the soap film appear to resemble the familiar form called the hyperbolic-paraboloid, a saddle-shaped surface generated by straight rods. But a significant difference is seen under closer observation. The outer edges of the soap film, where it adheres to the Plexiglas tubes, have a helical form. The minimal surface of the soap film's edges twists around the Plexiglas tubes (Fig. 8). This is the structural property of the natural surface form of a soap film that distinguishes it from the hyperbolic-paraboloid, whose outer edges are straight rods.

Fig. 9 Structurally mimicking the helical edges of the soap-film.

Fig. 10 Structurally mimicking the soap film

Fig. 11 Matching the helical edges of 6 Helical Geometry Elements (Cos 45°)

Fig. 12 Matching the helical edges of 32 Helical Geometry Elements (COs 45°)

We can mimic this helically edged property of the natural soap film form using a framework of flat, twistable rods or struts. (Fig. 9) We can then mimic the surface form of the soap film by extending flat, twistable struts between the opposing helical edges of the framework. (Fig. 10). [Note the four kite-shaped forms that generate the full helical framework by their 4-fold rotation]. The resulting system of "soap film rigid structures" may now be seen as segments or units of linear helical structures, rather than as independent saddle-shapes. Linear helical structures are generated by matching the helical edges of the helical units. (Figs. 11, 12) These helical units are the basic elements of Helical Geometry, a structure system that mimics natural helical form. It was the fourfold rotation in 2-dimensional space of the plane right-angled triangle that generated the natural 2dimensional symmetry of the plane square in Pythagoras's demonstration of the correspondence of the plane right-angled triangle with the source of symmetry in Nature. (Fig. 5). Likewise, it is the fourfold rotation in 3-dimensional space of the kite-like helical building panel (Fig. 6), that generates the 3-dimensional helical geometric structure that mimics the symmetry of the soap film. (Fig. 7) This demonstrates the correspondence of the elements of Helical Geometry with the source of symmetry in Nature. The kite-like helical building panels are like 3-dimensional right-angled triangles.

How Helical Geometry Incorporates Previous Knowledge "Does it incorporate previous knowledge?" "Does it demonstrate undeniable correspondences with existing knowledge?" These are the crucial questions, the prerequisites, for a valid claim by any new knowledge to

an advance in the foundations of existing knowledge. Helical Geometry satisfies these two questions by demonstrating the incorporation of the knowledge represented by the plane right-angled triangle, and by its ability to generate the archetypal forms of Plane and Solid Geometry.

Incorporating the Knowledge of the Plane Right-Angled Triangle You can point at the shortest edge of any helical building panel and say, "The length of this shortest edge is the cosine of the degrees of rotation or twist along its length (Fig. 13)." Likewise, you can point at the horizontal leg of a plane right-angled triangle and say, "The length of this shortest edge is the cosine of the degrees of rotation relative to the hypotenuse (Fig. 14)." Helical Geometry incorporates the knowledge of the plane right-angled triangle, transforming it into a 3-dimensional concept.

Fig. 13 Fig. 14 Cosine distance (blue) The 45° twisting Cosine distance of the helical building panel (left), and the Plane Rightangled Triangle (right).

Hypotenuse (blue) Fig. 15 Helical Right Triangle (left), Plane Right Triangle (right)

Fig. 16 The angle of plane rotation vs. the angle of helical rotation.

Helical Geometry’s multiple kite-like helical building panels each represents a cosine ranging from 0 to 90 degrees. The kite-like helical panel represents the number of degrees of rotation by the amount of twist over the length of its shortest edge. And, its length is the cosine of the number of degrees of rotation or twist along the shortest edge. So, for example, the helical building panel in Figures 13 and 14, has 45 degrees of twist or rotation along the shortest edge. And, the length of this shortest edge is the cosine of 45°, or, .7071 in relation to the constant length of the longest edge, which is 1. This trigonometric relationship is true for all the kite-like helical building panels: As the length of the shortest edge decreases the amount of twist, or degrees of rotation, increases, and, the numerical relationship is the same as that of the Table of Natural Trigonometric Functions for Sines and Cosines. This table of functions is derived from the 2-D system of structures, the plane right-angled triangle. We can liken the hypotenuse of the plane rightangled triangle to the longest edge of the helical building panel (Fig. 15), and the horizontal leg of the plane rightangled triangle to the shortest edge of the helical building panel (Figs. 13, 14). The difference is that the plane right-angled triangle expresses its degrees of rotation as a plane angle of rotation between 0 and 90 degrees, a 2-dimensional representation, while the helical building panel expresses its degrees of rotation over a distance, a helical angle of rotation or twist, a 3dimensional representation. (Fig. 16) This is why the helical building panels can be called "3-dimensional right-

angled triangles", or "Helical Right Triangles", just as the 2-dimensional rightangled triangles are called Plane Right Triangles.

Incorporating the Archetypal Structures of Plane and Solid Geometry Imagine that we had never seen the geometric structures of Plane and Solid Geometry, never seen a circle, square, triangle, sphere, cube or pyramid. If all we knew was Helical Geometry it would reveal to us all of these archetypal geometric structures. And this stands to reason. If Helical Geometry represents an advance in geometrical knowledge, then, it should be able to generate in its geometrical configurations the geometrical structures of Plane and Solid Geometry, the geometries that historically preceded Helical Geometry. Helical Geometry is a 'field geometry', meaning its helical building panels with lattice surfaces represent a geometrical system of fields having helical, saddle-shaped form. Using the building panels of Helical Geometry we can make models in which the panels intersect one another, matching their helical edges at the intersections. Matching the helical edges means that we are following the "logical rules of connection" inherent in the Helical Geometry panels or fields. A logical connection is one in which the degrees of twist and direction (left or right), and, the corresponding length at the connection of two intersecting or edge-connecting helical panels, match or coincide. Following these logical rules of connection a configuration can be constructed in which 96 kite-like panels intersect to generate the outward form of two interpenetrating tetrahedrons. Within this configuration of logically intersecting helical fields of form can be seen an empty space. It is a space that is defined by the inner surfaces of the helical fields of form. Looking closer we can see that within the configuration of intersecting fields of helical form has been generated the plane geometry of the circle, square and triangle; and the solid geometry of the sphere, cube and pyramid (Figs. 17, 18). So, while Helical Geometry cannot be generated from Plane or Solid Geometry, Plane and Solid Geometry can be generated from Helical Geometry. Helical Geometry, then, can be said to be "prior to" Plane and Solid Geometry, which is to say it incorporates existing geometrical knowledge and represents an advance in fundamental geometrical knowledge.

Figs. 17, 18 Within Helical Geometry's intersecting tetrahedrons (on right in photos), is generated the archetypal forms of Plane and Solid Geometry (on left)

Helical Geometry represents a foundation for advancement in architectural knowledge. It promises the foundation for visionary applications of advanced architectural design that is organic, ecological and evolutionary. It is a nature-based geometry that embodies a synthesis of form and function. Its varied structural forms promise sensitivity to the environment, simplicity in application and economics, and natural elegance. Helical Geometry is learned by modeling, by creating constructions using multitudes of one or more types of the kite-like building panels of Helical Geometry. How Helical Geometry has been used to date in modeling and constructions, and instructions on how to fabricate helical building panels for modeling and construction will be subjects of future writings.