Post by Jon"The fact that we cannot visualize points in spaces of dimension higher than
three has of course nothing whatsoever to do with their existence."
- Murray R. Spiegel, Vector Analysis
Do 'points in spaces of dimension higher than three' have any practical
utility, or are they just a sort of metaphysical parascientific tangent of
math?
[snip]
Quotes from Synergetics to show what Bucky Fuller meant by 4-D.
http://www.rwgrayprojects.com/synergetics/s04/figs/f6501.html
Fig. 465.01 Four Axes of Vector Equilibrium with Rotating Wheels or
Triangular Cams:
A. The four axes of the vector equilibrium suggesting a four-dimensional
system. In the contraction of the "jitterbug" from vector equilibrium to
the octahedron, the triangles rotate about these axes.
"The four axes of the vector equilibrium suggesting a four-dimensional
system".
527.71 Substance Is Systemic
527.711 People think of a point as the most primitive thing with which
to initiate geometrical conceptioning. A point is a microevent of
minutiae too meager, they say, to be dignified with dimensionality:
Ergo, they assume a point to be only an "imaginary fix." But speaking in
the experiential language of science, whatever is optically
point-to-able is a substance, and every substance has insideness and
outsideness -- ergo, is systemic: Ergo, all point-to-ables can never be
less than the minimum system: the tetrahedron. Points always amplify
optically to be identifiable as systemic polyhedra.
527.712 All conceptual consideration is inherently four-dimensional.
Thus the primitive is a priori four-dimensional, being always comprised
of the four planes of reference of the tetrahedron. There can never be
any less than four primitive dimensions. Any one of the stars or
point-to-able "points" is a system- ultratunable, tunable, or
infratunable but inherently four-dimensional.
966.20 Tetrahedron as Fourth-Dimension Model: Since the outset of
humanity's preoccupation exclusively with the XYZ coordinate system,
mathematicians have been accustomed to figuring the area of a triangle
as a product of the base and one-half its perpendicular altitude. And
the volume of the tetrahedron is arrived at by multiplying the area of
the base triangle by one-third of its perpendicular altitude. But the
tetrahedron has four uniquely symmetrical enclosing planes, and its
dimensions may be arrived at by the use of perpendicular heights above
any one of its four possible bases. That's what the fourth-dimension
system is: it is produced by the angular and size data arrived at by
measuring the four perpendicular distances between the tetrahedral
centers of volume and the centers of area of the four faces of the
tetrahedron.
966.21 As in the calculation of the area of a triangle, its altitude is
taken as that of the triangle's apex above the triangular baseline (or
its extensions); so with the tetrahedron, its altitude is taken as that
of the perpendicular height of the tetrahedron's vertex above the plane
of its base triangle (or that plane's extension outside the
tetrahedron's triangular base). The four obtuse central angles of
convergence of the four perpendiculars to the four triangular midfaces
of the regular tetrahedron pass convergently through the center of
tetrahedral volume at 109° 28'.
962.04 In synergetics there are four axial systems: ABCD. There is a
maximum set of four planes nonparallel to one another but
omnisymmetrically mutually intercepting. These are the four sets of the
unique planes always comprising the isotropic vector matrix. The four
planes of the tetrahedron can never be parallel to one another. The
synergetics ABCD-four-dimensional and the conventional
XYZ-three-dimensional systems.
962.03 In the XYZ system, three planes interact at 90 degrees (three
dimensions). In synergetics, four planes interact at 60 degrees (four
dimensions). re symmetrically intercoordinate. XYZ coordinate systems
cannot rationally accommodate and directly articulate angular
acceleration; and they can only awkwardly, rectilinearly articulate
linear acceleration events.
(Footnote 4: It was a mathematical requirement of XYZ rectilinear
coordination that in order to demonstrate four-dimensionality, a fourth
perpendicular to a fourth planar facet of the symmetric system must be
found--which fourth symmetrical plane of the system is not parallel to
one of the already-established three planes of symmetry of the system.
The tetrahedron, as synergetics' minimum structural system, has four
symmetrically interarrayed planes of symmetry--ergo, has four unique
perpendiculars--ergo, has four dimensions.)
Four rotations in the four planes of the four hexagonal cross sections
of the cub-octahedron (VE) will not give gimbal lock.
http://en.wikipedia.org/wiki/Gimbal_lock
Give me a fourth gimbal for Christmas meant give me a fourth dimension.
http://www.hq.nasa.gov/alsj/gimbals.html
You can divide space into closest packed equal edge length cubes and
label the planes of squares that are perpendicular to three directions
X, Y and Z. There are ninety degree angles between X, Y and Z
directions. The XYZ Cartesian coordinate system uses three-tuples of
numbers (x,y,z), the intersection of three planes, to represent the
location of a point in three-dimensional space.
You can closest pack equal diameter spheres instead of cubes and label
the planes of spheres that are perpendicular to four directions ABCD.
Each plane is divided into a grid of equilateral triangles when the
centers of the spheres are connected to the centers of their neighbors
by vectors and the spheres removed. The angle between any two directions
A, B, C, and D, is ArcCos[-1/3], approximately 109 degrees 28 minutes.
The Synergetics coordinate system uses four-tuples of numbers (a,b,c,d),
the intersections of four planes, to represent the location of a point
in four-dimensional space, a regular tetrahedron.
The edge length of the regular tetrahedron (a,b,c,d) is the absolute
value of a+b+c+d. The total a+b+c+d can be positive, negative or zero.
Negative values of a+b+c+d are tetrahedrons that are upside down and
inside out.
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
Cliff Nelson
Dry your tears, there's more fun for your ears,"Forward Into The Past"
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