Tetrahedron as Fourth-Dimension Model

The tetrahedron based coordinate system from Synergetics generalizes to
any number of dimensions easily.

Almost everyone who has written anything about this says vectors from
the origin of the coordinate system in the directions of the
tetrahedron's vertexes should be added so they only get one point in
three dimensions, but they don't add vectors pointed in the directions
of the cube's vertexes in the Cartesian coordinate system. The
coordinate axes are perpendicular to the planar facets of the cube from
the center of volume of the cube in the Cartesian system. The
coordinates refer to movements of the planes.

Here are some quotes from Synergetics.

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.

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 XYZthree-
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.)

http://bfi.org/node/574

Cliff Nelson

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