# Tetrahedron as Fourth-Dimension Model Game

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### Clifford Nelson

Jun 24, 2007, 9:11:19 AM6/24/07
to
I'd like to see a game like Rise of Nations Gold and Civilization III
Complete that is not played in FlatLand. Computers are fast enough to
represent a playing grid that is not flat (moving from tile to tile).
Real time strategy at very slow speed with a pause button. A
four-dimensional grid that is easy to see and understand by use of the
Synergetics coordinate system.

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

Almost everyone who has written anything about tetrahedral coordinate
systems says vectors from the origin of the coordinate system in the
directions of the tetrahedron's vertexes should be added so they only
end up with one point in three dimensions, and they don't add vectors
pointed in the directions of the cube's vertexes in the
three-dimensional 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 three-dimensional Cartesian system. Each coordinate
fixes a plane.

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