Introduction

The following article is about the type of geometry we could all relate to in grade school, the one made with wood sticks that we could see and touch and ponder and knew as triangles, squares, circles, pyramids, cubes and spheres. Einstein referred to it as “practical geometry”.

The following article is about the type of geometry we could all relate to in grade school, the one made with wood sticks that we could see and touch and ponder and knew as triangles, squares, circles, pyramids, cubes and spheres. Einstein referred to it as “practical geometry”.

The article is not about the modern mathematicians’ “pure
axiomatic geometry”. Pure axiomatic
geometry is an abstract language of logically deduced propositions that
Einstein described as: “not sure as far as it refers to reality, and as far as
it is sure it does not refer to reality.”
In fact, the basic tenet of “pure axiomatic geometry” is that it does

*not*refer to reality.
Pure axiomatic geometry does not originate from real
geometric structures that can be seen, touched and pondered, but from logically
deduced propositions or “voluntary creations of human mind” as Einstein
said.

Practical geometry has, by contrast, been considered “valid
beyond any doubt” since antiquity because it is based on real geometric
constructions and not the so-called “mental constructions” of the pure
axiomatic geometry that has dominated our inquiry into the nature of reality
for the past one-hundred years.

This article is about a new “practical geometry”, a
practical geometric system of structures called the “Spiral Unified Field
System” that has been overlooked since antiquity, and so has remained unknown
to mathematicians throughout recorded history.
The “practical geometries” of antiquity are all composed of straight
sticks. They are

*linear*geometric systems. The new “practical geometry” is composed of flat, straight twistable sticks. It is a*linear-helical*geometric system. This simply means that the new Spiral Unified Field System is made up of straight, flat twistable sticks in place of the straight sticks of linear geometric systems.
Of course, linear-helical geometry generates a different
looking system of geometric forms than linear geometry. The elements of linear practical geometry:
squares, triangles and circles, generate the closed geometries of pyramids,
cubes and spheres. The Spiral Unified
Fields generate an open geometry composed of spiral-shaped field structures
that precisely model the natural helical form of molecular structure, such as
that found in all biological cell membranes.

Being a

*practical**geometry*system, what the Spiral Unified Field System reveals can be considered; as all practical geometry has since antiquity, to be valid beyond any doubt. What is shown in this article are the ways in which the Spiral Unified Field System validates itself as the geometric system of the spiral unified fields that underlie everything. It shows us a way to logically and rationally understand the invisible structure system of the spiral unified fields that underlie physical reality itself, and also, logically and rationally shows us the validity of our common intuitive sense of connection with the invisible creative intelligence underlying the whole reality.
Finally, the Spiral Unified Field System shows us that the
most basic concept of mathematics, the straight-line concept to which all
mathematical geometry can be traced according to Einstein, is an incorrect
concept. Space is perceived as curved
because its curvature is the effect of the invisible, dynamic spiral unified
fields underlying reality. “Both string
theory and twistor theory are attempts to understand this fundamental structure
of universe” to quote Professor Sir Roger Penrose, the creator of Twistor
theory.

************

Artist/sculptor/researcher James Jacobs began his task of developing his Spiral Unified Field Coordinate System by

*structurally*transforming the geometric form of the system of plane right-angle triangles from which the Pythagorean theorem is derived.
Alternatively, Albert Einstein began his task of developing his relativity theories by

*mathematically*transforming the Pythagorean theorem with the addition of a few mathematical terms. The mathematically transformed theorem Einstein used was:
ds^2 = dx_1^2 + dx_2^2 + dx_3^4 + dx_4^2, which
has the form of the ordinary Pythagorean theorem in four dimensions, except for
the pure imaginary value “x_4”.

Axiomatic mathematics, which is founded on imagined or mental constructs, has proven inadequate for all unifying "theories of everything" for over 100 years due to the lack of a physical geometric foundation.

Accordingly, the structurally transformed geometric form Jacobs constructed was accomplished as follows:

Two right-angle triangles (minus the hypotenuse) are fixed at a right angle as shown at the bottom left in the photo. Next, the two longer legs of the right-angle triangles are squeezed together and joined. When squeezed together and joined all the edges of the framework twist. This helical edged framework can then generate a surface form that models that of a molecule-thin liquid membrane, the natural catenoid minimal surface form with its characteristic helical edges.

The structurally surfaced framework has transformed the plane right-angle triangle. The plane 45 degree angle of the right-angle triangle is transformed into the 45 degrees of twist or helical rotation along the blue edge (A to D) of the Spiral Unified Field model. The ratio of the distance A-D to the distance A-B is .7071 to 1; .7071 being the cosine of the 45 degrees of twist along the distance A-D.

When four of the Spiral Unified Field models are self-organized by matching their 45 degree twisted or helical edges they model the Spiral Unified Field Coordinate System of the cosine 45 degrees. The length and degrees of twist of the x, y coordinates (blue)are variable. The length of the bounding helical edges of all unified field models are constant while the degrees of twist along the bounding helical edges of all unified field models vary between 0 degrees and 90 degrees.

The Spiral Unified Field Coordinate System is generated by varying the length of the x, y coordinates from a 1:1 ratio with the bounding edges of the field to a 1:0 ratio of the x, y coordinates to the bounding edges of the field. The 1:1 ratio, a plane, is shown on the far left in the photo. The 1:0 ratio , a vertical bundling of the bounding edges, is shown on the far right in the photo.

The variable x, y coordinates of the Spiral Unified Field Coordinate System vary in terms of both the length and the degrees of twist along the length of the x, y coordinates in correspondence with the cosines of the natural trigonometric functions. As the length of the x, y coordinates reduce from 1 to 0, the degrees of twist increase from 0 degrees to 90 degrees.

Each of the various unified spiral fields structurally and numerically model a "unit" of a specific linear spiraling field. Shown in the photo are self-organizing linear spiral field compositions generated by matching the twisted bounding edges of six different spiral unified field "units" to create seamless, continuous, three-unit spiral field "waveforms".

Twistor-String theory implies twisting-vibrating elements. The Spiral Unified Field Coordinate System "waveform" compositions structurally and numerically model the twisting-vibrating elements implied by Twistor-String theory.

In addition, The "waveforms" are found to self-organize; logically intersecting and connecting to generate meaningful "waveform" compositions as shown in the following photos.

Six cosine 45 degrees spiral unified field "waveforms"; three left-handed twisting (white) and three right-handed twisting (black), are shown to self organize; intersecting to generate a six-pointed form embodying a perfect circle.

This self-organizing composition is shown in the next photo independent of the spiral unified field "waveforms" that generated it.

The photo shows the six-pointed form embodying a perfect circle. The left-handed and right-handed twisting elements of the six unified field "waveforms" that generated the form remain.

When four of these six-pointed forms self-organize by the matching of of their twisted elements another interesting form is generated as shown in the following photos.

An eight-pointed form is generated by the self-organizing intersection of four of the six-pointed forms shown in the previous photo. Since six intersecting spiral unified field "waveforms" generated the four six-pointed forms that here self-organize to generate this eight-pointed form it follows that six times four, or twenty-four intersecting spiral unified field "waveforms" self-organize to generate this eight-pointed form.

The form on the left in this photo shows the form generated by the self-organizing of twenty-four intersecting spiral unified field "waveforms": the cube, the octahedron, and the perfect sphere are structurally generated within the form of the twenty-four intersecting spiral unified field "waveforms".

Another self-organizing form of the Spiral Unified Field Coordinate System (cosine 45 degrees) models the natural self-organizing structural order in the cell membranes in all biological systems. The following scientific paper led to this discovery.

Dr. KATHRYN M. M^{c}GRATH
Department of Chemistry
University of Otago Dunedin, New Zealand |

**Last updated march 2003**

**Molecular self-assembly, complex fluids and hierarchical solid formation**

Many molecules have the ability to aggregate into large complexes when placed in solution and/or controlled externally by variations in temperature, pressure or electric or magnetic field strengths. This aggregation process is termed molecular self-assembly. Examples of where self-assembly is found naturally are in cell membranes in all biological systems, oil-recovery, cosmetics and dairy products.

The two primary categories of molecules that are able to undergo molecular self-assembly are surfactants (surface active agents) and polymers (including proteins, DNA and sugar chains). Both are found widely in Nature and are used extensively in industrial and commercial processes. Molecular self-assembly occurs in a variety of environments, being both driven and used for control by the unique combinations of components. Molecules experience a driving force resulting in aggregation such that the interaction energy is minimized. This results in the formation of a vast and diverse array of geometrical forms, worthy of study in their own right but also opening the possibility of microstructured reaction environments. The formation of these patterns in fluids are also seen in many solid materials, albeit on very different lengths scales. Some examples of this include the skeleton of diatoms, synthesized as a silicate, calcium carbonate deposition in sea urchins and the intricate three-dimensional patterns formed by zeolites.

**Modelling of Bicontinuous Cubic Phases using Minimal Surfaces**

Amphiphilic
bicontinuous cubic phases have been likened to the infinite periodic minimal
surfaces in differential geometry. Single crystal crystallographic data
has been obtained for the known bicontinuous cubic phases. Theoretical
diffraction patterns are being obtained modelling these phases as minimal
surfaces.

**Small angle X-ray diffraction patterns of the Im3m, and Pn3m bicontinuous cubic phases. Obtained on the Princeton 1K CCD detector, using a rotating copper anode source.**

**Infinitely periodic minimal surfaces, primitive (P), and double diamond (D), corresponding to Im3m, and Pn3m, respectively.**

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Here is a clearer view of the infinitely periodic minimal surface of the double diamond (D) shown above that was obtained from single crystal crystallographic data and modeled as a minimal surface. The mathematical minimal surface, unlike the natural minimal surface of the molecule-thin liquid membrane form of the catenoid, lacks the self-organizing property; the catenoid's characteristic twisted or helical edges visibly evident in the natural minimal surface of a molecule-thin liquid membrane.

The self-organizing spiral unified field "waveforms"; cosine 45 degrees, generates this structural numerical model of the structural order in the cell membranes of all biological systems. The twisted elements of the spiral unified fields model the twisting-vibrating elements implied by Twistor-String theory.

The Technological Foundation

for a

Global Renaissance Project

The Renaissance that gave us
all of the products that surround us today began around the 14

^{th}and 15^{th}centuries, or about five to six hundred years ago. The technological foundation for this rebirth of the Arts and Sciences was the uncovering of 3-dimensional Technical Drafting.
Leonardo Da Vinci is credited
with uncovering the means and techniques that gave birth to 3-dimensional
Technical Drafting; the numerically proportional graphic modeling of things
creatively imagined in 3-D perspective on a 2-D surface.

It was not uncommon for
artists around the 15

^{th}century to make distant objects in their paintings to appear smaller to give a sense of 3-dimensional perspective to their paintings. Da Vinci being familiar with the system of unique numerical structures; the family of right-angle triangles, incorporated two identical right-angle triangles in a back-to-back arrangement which formed a triangle. Da Vinci’s insight thus made 3-dimensional perspective on a 2-dimensional surface numerically treatable, which gave birth to Technical Drafting.
The uncovering of this
numerically treatable 3-D graphic imaging provided the tool for all of the
design and engineering of the millions of products that fill today’s manmade
environment. It represented the “tool of
knowledge” for 3-D perspective and the technological foundation for the rebirth
of the Arts and Sciences called the Renaissance.

Today, and for the past 100
years we have needed a new “tool of knowledge”; a visible concrete model of
Albert Einstein’s proven existence of curved space-time; the structural order
that underlies everything in universe. This new “tool of knowledge” is the required
technological foundation for a rebirth of the Arts and Sciences; a Global
Renaissance. A Global Renaissance that
produces a manmade environment that is in structural harmony with the
structural order throughout Nature.

Imagine the effect of a
global rebirth of the Arts and Sciences that produces a manmade environment in
structural harmony with everything in Nature, including you and me. That’s the promise of the new “tool of
knowledge” as we begin applying it in the Arts and Sciences; in how we see,
think and build.

What is true of the effect of
the global rebirth of the Arts is also true for the Sciences. Nikola Tesla said it best,

*“the day science begins to study non-physical phenomena, it will make more progress in one decade than in all the previous centuries of its existence.**To understand the true nature of the universe, one must think in terms of energy, frequency and vibration.”*
The new “tool of knowledge”;
the “Spiral Unified Field Coordinate System”, structurally and numerically
models the non-physical phenomena; the invisible, dynamic spiral structural
order underlying everything in universe, in terms of waveforms having energy,
wavelength and vibrational frequency.