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Quadray coordinates, also known as caltrop, tetray or Chakovian coordinates, were developed by Darrel Jarmusch and others, as another take on simplicial coordinates, a coordinate system using a
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''.
In this American transcendentalist philosophy, the regular tetrahedron of edges one, as defined by four intertangent uni-radius balls, is taken as unit of volume.
A set of familiar convex polyhedra, termed "the concentric hierarchy" is nested around it, per the above table, such that the cube has volume 3, the octahedron volume 4, rhombic dodecahedron volume 6, and cuboctahedron volume 20.
''Quadray formulas''
* Ace, Tom
* Urner, Kirby
* Urner, Kirby
the Face Centered Cubic lattice (FCC)''
on Github {{DEFAULTSORT:Quadray Coordinates Coordinate systems
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
or tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
as its basis polyhedron.Urner, Kirby. "Teaching Object-Oriented Programming with Visual FoxPro." ''FoxPro Advisor'' (Advisor Media, March, 1999), page 48 ff.
Geometric definition
The four basis (but not necessarily unit) vectors stem from the center of a 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 positively scaled without rotation (e.g. negation) and linearly combined to span conventional ''XYZ'' space, with at least one of the four coordinates unneeded (set to zero).
Pedagogical significance
A typical application might set the edges of the basis tetrahedron as unit. The tetrahedron itself may also be defined as the unit of volume (see below). 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 . The 12 permutations of define the vertices of the volume 20 cuboctahedron centered at (0,0,0,0). These vectors point from any given sphere to its 12 surrounding neighbors in the cubic close packing (CCP), equivalently the IVM (isotropic vector matrix) in Synergetics. Therefore CCP ball centers all have non-negative integer coordinates.
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''.
In this American transcendentalist philosophy, the regular tetrahedron of edges one, as defined by four intertangent uni-radius balls, is taken as unit of volume.
A set of familiar convex polyhedra, termed "the concentric hierarchy" is nested around it, per the above table, such that the cube has volume 3, the octahedron volume 4, rhombic dodecahedron volume 6, and cuboctahedron volume 20.
See also
*Barycentric coordinates (mathematics)
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The baryc ...
*Caltrop
A caltrop (also known as caltrap, galtrop, cheval trap, galthrap, galtrap, calthrop, jackrock or crow's foot'' Battle of Alesia'' (Caesar's conquest of Gaul in 52 BC), Battlefield Detectives program, (2006), rebroadcast: 2008-09-08 on History Ch ...
* Synergetics coordinates
* Synergetics
*Trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
*Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
References
External links
* Ace, Tom''Quadray formulas''
* Ace, Tom
* Urner, Kirby
* Urner, Kirby
the Face Centered Cubic lattice (FCC)''
on Github {{DEFAULTSORT:Quadray Coordinates Coordinate systems