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.
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'.
Four rotations in the four planes of the four hexagonal cross sections of the cub-octahedron (VE) will not give gimbal lock.
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. 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.