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'''Synergetics coordinates''' is Clifford Nelson's attempt to describe, from another mathematical point of view, [[Biography:Buckminster Fuller|Buckminster Fuller]]'s '60 degree coordinate system' for understanding nature.  [[Synergetics (Fuller)|Synergetics]] is the word Fuller used to label his approach to [[Mathematics|mathematics]].<ref>
Clifford Nelson, [http://library.wolfram.com/infocenter/MathSource/600/ Buckminster Fuller Notebooks]
</ref>

==Geometric definition==
A [[Philosophy:Theory of forms|system]] of synergetics coordinates uses only one type of [[Simplex|simplex]] ([[Triangle|triangle]], [[Tetrahedron|tetrahedron]], pentachoron, ..., n-simplex) as [[Space|space]] units, and in fact uses a regular simplex, rather like Cartesian coordinates use hypercubes (square, [[Cube|cube]], [[Tesseract|tesseract]], ..., n-cube.)

[[Image:Synergetics coordinate plane.jpg|right|thumb|Synergetics coordinates in two dimensions]]

The n Synergetics coordinates axes are perpendicular to the n defining geometric objects that define a regular simplex; 2 end points for line segments, 3 lines for triangles, 4 planes for tetrahedrons etc.. The angles between the directions of the coordinate axes are Arc Cosine (-1/(n-1)). The coordinates can be positive or negative or zero and so can their sum. The sum of the n coordinates is the edge length of the regular simplex defined by moving the n  geometric objects in increments of the height of the n-1 dimensional regular simplex that has an edge length of one. If the sum of the n coordinates is negative the triangle (n = 3) or tetrahedron (n = 4) is upside down and inside out.

==Algebraic examples==
Regular [[Triangular coordinates|triangular coordinates]] are in a [[Lattice graph|grid]] of equilateral triangles and are of the form <math>(x,y,z)</math> such that <math>x,y,z</math> are equal to or greater than 0.

Regular tetrahedral coordinates are in a [[Euclidean space|Euclidean 3-space]] 'grid' of equilateral tetrahedra and are of the form <math>(w,x,y,z)</math> such that <math>w,x,y,z</math> are equal to or greater than 0.

==See also==
*[[Argand system]]
*Barycentric coordinates (mathematics)
*[[Trilinear coordinates]]
*[[Quadray coordinates]]

== Notes ==
{{Reflist}}

== References ==
* Stan Dolan, 'Man versus Computer,' ''Mathematical Gazette'', volume 91, number 522 (November 2007), pages 469–480.
* R. Buckminster Fuller, ''Synergetics:  Explorations in the Geometry of Thinking'' (2 vols.), Vol. 2, Section 203.09 and Section 986.205. 
[http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966.20 Sec. 966.20];
[http://www.rwgrayprojects.com/synergetics/s09/p8700.html#987.010 Sec. 987.011]; Vol. 1, [http://www.rwgrayprojects.com/synergetics/s04/p0000.html Sec. 400.011] and [http://www.rwgrayprojects.com/synergetics/s04/figs/f0101.html Fig. 401.01].
* [http://www.wikieducator.org/User:KirbyUrner/Collaborations#Quadray_Coordinates ''Quadray Coordinates'' on WikiEducator]
* {{MathWorld|title=Synergetics Coordinates|urlname=SynergeticsCoordinates}}

== External links ==
* Clifford J. Nelson. [http://mysite.verizon.net/cjnelson9/index.htm Synergetics Coordinates]

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[[Category:Coordinate systems]]

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