REVOLUTIONARY GEOMETRY: A Foundation for Nature-Based Architecture
by JAMES JACOBS
July 2003
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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.
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Fig. 8 The helical edge
of the soap-film
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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.
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Fig. 10
Structurally mimicking the soap film
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Fig. 11
Matching the helical edges of 6 Helical Geometry Elements
(Cos 45°)
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Fig. 12
Matching the helical edges of 32 Helical Geometry Elements
(COs 45°)
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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 2-dimensional 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
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Fig. 14
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Cosine distance (blue)
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The 45° twisting Cosine distance
of the helical building panel (left), and the Plane Right-angled
Triangle (right).
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Fig. 15
Helical Right Triangle (left),
Plane Right Triangle (right)
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Fig. 16
The angle of plane rotation vs. the angle of helical rotation.
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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 right-angled triangle to
the longest edge of the helical building panel (Fig. 15), and the
horizontal leg of the plane right-angled 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 3-dimensional 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 right-angled 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.
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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)
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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.
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