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Jun 24, 2007, 9:11:19 AM6/24/07

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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.

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.)

Cliff Nelson

Dry your tears, there's more fun for your ears,

"Forward Into The Past" 2 PM to 5 PM, Sundays,

California time,

http://www.geocities.com/forwardintothepast/

Don't be a square or a blockhead; see:

http://bfi.org/node/574

http://library.wolfram.com/infocenter/search/?search_results=1;search_per

son_id=607

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