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Tetrahedron as Fourth-Dimension Model

Discussion in 'Electronic Design' started by Clifford Nelson, Jun 23, 2007.

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  1. 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

    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
     
  2. Jasen

    Jasen Guest

    They have it backwards the vectors in the direction of the
    tetrahedrons faces should be subtracted, (it's exactly the same thing)

    BTW: the vectors used for the cartesion system are in the direction of
    the vertices of an octahedron, dunno where you dreames this cube
    business up :^)

    Bye.
    Jasen
     
  3. Each coordinate fixes the location of a plane in Synergetics. The
    intersections of the planes define the six edges of a regular
    tetrahedron, a four-dimensional point.
    Each coordinate fixes the location of a plane. The intersection of the
    planes is an edge length zero cube, a three-dimensional point.
    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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