Are humans shaped according to helical geometry (re: Picasso�s cubism)? Carl:
http://www.ecotecture.com/library_eco/ecotechnology/Jacobs_helicalgeom_1a.html REVOLUTIONARY GEOMETRY: A Foundation for Nature-Based Architecture
by JAMES JACOBS
July 2003
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.
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.
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.
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 2-dimensional 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
right-angled 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.
