Quadray coordinates, also known as tetray coordinates or Chakovian coordinates, were developed by David Chako, Tom Ace, Kirby Urner, Darrel Jarmusch et al., as another take on simplicial coordinates, a coordinate system using the simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

 or tetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 as its basis polyhedron.

Geometric definition

The four basis vectors stem from the origin of the regular tetrahedron and go to its four corners. Their coordinate addresses are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. These may be scaled and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded (set to zero) in any given quadrant.

The normalization scheme is somewhat unusual in keeping all coordinates non-negative. Typical of coordinate systems of this type (a, a, a, a) is an identity vector and may be added to normalize a result. To negate (1,0,0,0), write (−1, 0, 0, 0) then add (1, 1, 1, 1) to get (0, 1, 1, 1).

Pedagogical significance

A typical application might set the edges of the basis tetrahedron as unit, with the quadrays considered unit on some other scale. The tetrahedron itself may also be defined as the unit of volume, although the infrastructure does not demand using this setting.

The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0), and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively, given the starting tetrahedron of unit volume.

For example, given A, B, C, D as (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) respectively, the vertices of an octahedron with the same edge length and volume four would be A + B, A + C, A + D, B + C, B + D, C + D or all eight permutations of {1,1,0,0}. The vertices of the volume 20 cuboctahedron are all 12 permutations of {2,1,1,0}.

Shape Volume Vertex Inventory (sum of Quadrays) Tetrahedron 1 A,B,C,D Inverse Tetrahedron 1 E,F,G,H = B+C+D, A+C+D, A+B+D, A+B+C Duo-Tet Cube 3 A through H Octahedron 4 I,J,K,L,M,N = A+B, A+C, A+D, B+C, B+D, C+D Rhombic Dodecahedron 6 A through N Cuboctahedron 20 O,P,Q,R,S,T = I+J, I+K, I+L, I+M, N+J, N+K; U,V,W,X,Y,Z = N+L, N+M, J+L, L+M, M+K, K+J

If one now calls this volume "4D" as in "four-dimensional" or "four-directional" we have primed the pump for an understanding of R. Buckminster Fuller's "4D geometry," or Synergetics

Synergetics (Fuller)

Synergetics is the empirical study of systems in transformation, with an emphasis on total system behavior unpredicted by the behavior of any isolated components, including humanity’s role as both participant and observer....

.

See also

• Synergetics
  Synergetics (Fuller)
  Synergetics is the empirical study of systems in transformation, with an emphasis on total system behavior unpredicted by the behavior of any isolated components, including humanity’s role as both participant and observer....
  
  
• Barycentric coordinates (mathematics)
  Barycentric coordinates (mathematics)
  In geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates...
  
  
• Trilinear coordinates
  Trilinear coordinates
  In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...
  
  
• Synergetics coordinates
  Synergetics coordinates
  Synergetics coordinates is Clifford Nelson's attempt to describe, from another mathematical point of view, Buckminster Fuller's '60 degree coordinate system' for understanding nature...
  
  

External links

• Ace, Tom. Quadray formulas
• Ace, Tom. 4D-Quadray correspondence
• Urner, Kirby. posting to Math Forum, Feb 6, 1999