Two Dimensions

(2)    The idea of moving the three lines of a triangle or the four planes of a tetrahedron to address a point can be seen by starting with a paper ruled with equilateral triangles, and using three numbers representing the perpendicular displacement of the three defining lines of an equiangular triangle that is positioned at the center of the paper {0,0,0}. The reference triangle has an edge length of zero and an area of zero. All three of the vertices of the reference triangle are at the same place on the plane.

Weisstein, Eric W. "Triangular Grid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TriangularGrid.html

(3)    If all three defining lines of the triangle move one unit away from the center of the origin triangle you get the triangle {1,1,1}. A unit here (for perpendicular displacement) is the height of an equilateral triangle of edge length one.

(4)    The vertices of {1,1,1} are:

(5)    They can be converted to perpendicular 90 degree coordinates by the function StoP. StoP is short for Synergetics coordinates to perpendicular coordinates. The function StoP converts Synergetics coordinates to coordinates with mutually perpendicular axes. The Synergetics coordinates must add up to zero so that they are a regular simplex of edge length zero.

(6)    Coordinates with perpendicular axes are usually called Cartesian coordinates, named after René Descartes, even though Descartes is said to have expected the use of oblique coordinate axes instead of perpendicular.

(7)    If the perpendicular displacement of the defining lines is toward the center of the triangle, you get {-1,-1,-1}, an upside down inside out (negative) triangle.

(8)    The edge length of triangle and NegativeTriangle defined above equals three.

Created by Wolfram Mathematica 6.0  (17 March 2008)