In the familiar Cartesian system (when you have 2 or more axes), each axis is at right angles to every other axis. This page is about alternate coordinate systems.

The terminology I use is xMy where x is the number of axes and y the number of dimensions being measured. Cartesian coordinates are 1M1, 2M2, and 3M3 in 1, 2, or 3 dimensions.

Others’ work closest to mine (that I’ve found) is that of Kirby Urner (who knew and worked with Bucky Fuller) — his Quadrays are similar to my 4M3 (see below). In light of an email exchange with him, I better understand the difference in our outlooks. He is particularly interested in an arrangement of vectors (rather than bi-directional axes) oriented in the same way as 4M3 — Kirby notes that a caltrop has this shape.

There are two aspects of the concepts discussed here that need to be separated in order to show where my approach and Kirby’s are similar, and where they diverge.

The first is, for each dimension (line, plane, space), considering the various arrangements of axes that might be used in that dimension (the xMy arrangements discussed here). Kirby and I are both fond of the “caltrop” (4M3) arrangement of axes, as was Bucky Fuller. Kirby concentrates almost exclusively on the 4M3 axis-pattern. I’m looking to see that as one of many of the coordinate systems possible via the structures of the Platonic Solids, and how these alternate coordinate systems inter-relate.

The second is, for each of the axes in a given set, how, mathematically, the axis is used. I use number lines with coordinates extending from negative to positive infinity. Kirby uses vectors (hence the “ray” in Quadray) with non-negative magnitudes. Where I go into a “sums to zero” property of some coordinate systems, there is a related property in Kirby’s Quadrays. He shows that one can get to any point in a 3-dimensional space by going only positive distances in at most 3 of the 4 directions of the “caltrop”.

In One Dimension

I haven’t conceptualized coordinate systems beyond the Cartesian 1M1 number line and the mirrored 2M1 (The 2 axes are along the same line, going in opposite directions from the same starting point).

2M1 is one in the sequence of Zero-balanced coordinates (described below). A variation with one-directional vectors which might be called “BiRay” (instead of Quadray, as it has two vectors, not four) which has the property of covering all (linear/1D) space using only the summing of non-negative vectors.

In Two Dimensions

In Cartesian space, 2M2 has X (horizontal) and Y (vertical) axes forming a plane. Also possible is 3M2, a Zero-balanced system in two dimensions using three coordinates (shown in the image below), and a 4M2 in which both Cartesian coordinates are mirrored in the same fashion as 2M1 is in one dimension. A 3M2 variant – a “TriRay” (to add a three-vector arrangement to BiRay and Quadray) also covers all (planar/2D) space using only the summing of non-negative vectors.

In Three Dimensions

Because I’m exploring structures found in the Platonic solids, I should add here a shape Bucky frequently included when talking or writing about the Platonic solids — what he called the Vector Equilibrium (or VE). It’s formed by the (space enclosed by the) points at the centers of the edges of either the cube or the octahedron. (Another way of saying this is, the points where the edges of a nested cube and octahedron touch).

4M3 is the connecting point between Kirby’s ideas and mine. I haven’t seen him discuss applying the properties of the Quadray to a plane (2D vs 3D). He does talk a bit about the “BiRay” (though he doesn’t use that term) – needing two vectors in opposite directions to cover the 1D space.

There are three different versions of 6M3, which I identify using a subscript to the 6, and which I will write here as 6(0)M3, 6(1)M3 and 6(2)M3. The first of these, 6(0)M3 is a doubling of 3M3, each axis pairing with its mirror image (same process as with 2M1 and 4M2). 6(1)M3 has axes oriented from the center of a cuboctahedron (Bucky Fuller calls it a Vector Equilibrium, or VE) to its vertices. The 3M3, 4M3, 6(0)M3 and 6(1)M3 can all be easily lined up with the tetrahedron, octahedron, cube and VE. The more complex systems 6(2)M3, 10M3 and 15M3 that line up with the icosahedron and dodecahedron. 6(2)M3 points to the vertices of the icosahedron. The re-orientation of going from 6(1)M3 to 6(2)M3 can be thought of as taking the opposite corners of one of the square faces of the VE and pulling closer until you get two equilateral triangles. Doing all the “matching” pulls across all the other square faces completes the transformation of the VE into an icosahedron.

Zero-balanced coordinates

There is a 2M1 system, a 3M2 system and a 4M3 system in which all the coordinates add to zero for any point in (1, 2, or 3 dimensions of) space. As well, all have the property Kirby discusses, of non-negative vectors covering the relevant space.

2M1 is simply a standard Cartesian single dimension paired with its mirror opposite. Point X has coordinates (-2,2).

2M1 coordinate system

3M2 has 3 axes oriented on a plane from the center of a triangle towards its three corners (or towards the centers of its three sides). Here is a graphic representing the 3M2 system. Coordinates are given in the diagram below for points p, q, and y. Point x is precisely between p and q – hence, its coordinates can be determined by averaging the coordinates of p (0,3,-3) and q (2,-3,1). So x is at (1,0,-1).

3M2 Coordinate System

4M3 has 4 axes oriented in 3D space from the center of a tetrahedron to its four vertices (or towards the centers of its four faces). No diagram here (for now) – see Quadrays or Google caltrop.

My intuition tells me (but it is beyond my visualization capacity) that there are similar coordinate systems (where the measure of all the axes sum to zero) in higher dimensions – 5M4, 6M5 and so on. The sequence can be described easily – pull the center point in a direction perpendicular to all the existing axes, to get a symmetric pattern in the next higher dimension. Pull the center point two axes in opposite directions (2M1) to get three axes pointing to the corners of a triangle (3M2). Pull its center point to get axes pointing to the corners of a tetrahedron (4M3), and so forth.

Geometric relationships in the Coordinate Systems

The 3M3 and 6(0)M3 coordinate systems (Cartesian x,y,z) appear in the vectors from the

  • center of a tetrahedron to the centers of its edges
  • center of an octahedron to its vertices
  • center of a cube to the centers of its faces
  • center of a VE to the centers of its square faces

The 4M3 coordinate system (4 coordinates that sum to zero) appears in the vectors from the

  • center of a tetrahedron to its vertices (negative direction is to the centers of its faces)
  • center of an octahedron to the centers of its faces
  • center of a cube to its vertices
  • center of a VE to the centers of its triangular faces

The 6(1)M3 coordinate system (6 coordinates, out of which 4 sets of 3 form 3M2 coordinate planes) appears in the vectors from the

  • center of an octahedron to the centers of its edges
  • center of a cube to the centers of its edges
  • center of a VE to its vertices

Planes perpendicular to the axes of the 4M3 coordinate system are where the 3M2 subsets of 6(1)M3, as mentioned above, occur.

The relationship between 6(1)M3 and 6(2)M3 is interesting. If you take two opposite corners of any one of the square faces of a VE and pull them together (or push them apart) so that the square becomes two triangles (either change will echo in the other 5 squares), you get an icosahedron (with 20 triangular faces).

The 6(2)M3 coordinate system appears in the vectors from the

  • center of of an icosahedron to its vertices
  • center of a dodecahedron to the centers of its faces

The 10M3 coordinate system appears in the vectors from the

  • center of an icosahedron to the centers of its faces
  • center of a dodecahedron to its vertices

The 15M3 coordinate system appears in the vectors from the

  • center of either icosahedron or dodecahedron to the centers of its edges – in other words, where the nested shapes touch.

Mathematical relationships between the Coordinate Systems

I’ve not had a mathematician double-check and OK what I’ve written below, so consider it a draft.

The connection between 3M3‘s (x,y,z) coordinates and 4M3 (let’s use (p,q,r,s) ) is (leaving out a constant conversion factor to “equalize” lengths):

  • p=x-y-z,
  • q=y-x-z,
  • r=z-y-x,
  • s=x+y+z.

The connection between 3M3 and 6(1)M3 (let’s use (i,j,k,l,m,n) ) is (leaving out a different conversion factor):

  • i=x+y,
  • j=x-y,
  • k=x+z,
  • l=x-z,
  • m=y+z,
  • n=y-z.

These 4 and 6 vectors each have a negative, e.g., -p=-x+y+z, -i=-x-y.

In sum, the 8 possibilities of combining 1 each of +/-x, +/-y, and +/-z map directly to +/- p,q,r,s. The 12 possibilities of combining any two from the same set (+/-x, +/-y, and +/-z) map directly to +/- i,j,k,l,m,n.

I haven’t tried to work out similar relationships to 6(2)M3, 10M3, or 15M3.

Connections Elsewhere

Fuller gave the name to the Vector Equilibrium because he perceived it to be the shape right at the balance point between being inward focused and outward focused (this is a rough approximation of his idea) – at an equilibrium point between imploding and exploding. The pattern of its edges and the struts between center and vertices ARE an octet truss. Flooring made with an octet truss has some rather interesting engineering attributes – such as the way it distributes a load put at any point on its surface.

Kirby Urner has worked diligently for decades to bring some of Buckminster Fuller’s ideas to a wider audience. He uses the term Quadrays (or Quadray Coordinates) to describe something similar to what I call 4M3. More info on this can be found at or on Wikipedia.

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