A fractal 'Sierpinski tetrahedron' metal sculpture in Breckenridge, Minnesota
credit: Breckenridge Senior High Mathletes

A tetrahedral system for defining co-ordinates in all-space has been proposed by me in 1996 in Tetra Space Co-ordinates. My proposal was roughly time coincident but independent from discussions and mathematical elaborations on 'Quadray coordinates' taking place between David Chako, Tom Ace and Kirby Urner. The outline of their ideas and some links can be found in a Wikipedia article titled Quadray Coordinates.

The idea of basing all-space coordinates on tetrahedral geometry instead of the Cartesian right-angle X-Y-Z axes goes back to Buckminster Fuller. It is conceptually sound but it needs mathematical elaboration to become useful for general application. The maths should be simple and of immediate applicability, which is quite a challenge. We have all been brought up with a system of coordinates that today is in exclusive use, leaving little space for a different idea...

Cartesian coordinates

Today's Cartesian co-ordinates proceed from a point in space, giving origin to three directions or vectors, the X, Y and Z axes, which are arranged in 90-degree co-ordination. Those axes define one of the angles of a cubic section of space. By establishing distances along each one of the three axes, a point in space can be determined. Those co-ordinates work well in a setting of space where a cube or cuboid-shaped volume is to be described, however it gets more complicated when describing all-space, that is, any conceivable direction of the spherical extension we call space. In combination with the subdivision of a complete circle into 360 degrees, where 90 degrees is exactly one quarter, we may, in the Cartesian system, define the space around us with reference to six vectors: up, down, east, west, north, south. Each of those vectors is separated by 90 degrees from its four nearest neighbors, and by 180 degrees from its negative 'twin'.

Cartesian Coordinates - image credit: rab3d - Blender

To describe any direction in space, we must first establish a horizon, cutting the spherical volume in two. Choosing one of the two half-spheres, we can define which way is up (above our horizon) or down (below our horizon). A direction or exact vector can now be defined with reference to the 360 degree subdivision of the circumference of the horizon, and an elevation from horizon towards the vertical expressed in degrees. For increased accuracy, each degree may be subdivided into 60 minutes and each minute into 60 seconds.

The system worked rather well as long as we stayed on the surface of the earth, navigating the seas, where the horizon is naturally established by the existing vertical (downward directed) vector of gravitation. It is not optimal however for describing all-space.

So pervasive has the Cartesian coordinate system become in our culture, that its X-Y-Z axes have taken on the meaning of "dimensions". Each one of the right-angle axes of the system is thought to denote a dimension with physical meaning. We are describing our world as "three dimensional".

I suspect that in reality, physical extension (in all directions) should be called a dimension of physical space, and time should probably also have the distinction of being a dimension. There is no compelling reason to see the axes of a coordinate system as inherent in the make up of the physical universe and call them dimensions.

A different geometry of 'dimensions'

In a space setting, tetrahedral co-ordinates have the potential to considerably simplify orientation and navigation. Tetrahedral co-ordinates are based on four principal axes. For the sake of convenient description, the four tetrahedral axes are defined as having their origin at the central point of an imaginary tetrahedron, each one exiting at one of the four vertices. (This is different from my original proposal, where the tetrahedral axes were said to exit at the center of a face. Note that both systems are possible.) Our imaginary tetrahedron may be situated at any preferred point of observation. It may be oriented with reference to an external fixed point, such as a distant star system or a galaxy, or with regard to a vehicle (satellite, space ship) in free space. It may also co-rotate with a sphere such as a planet or a star, for instance the sun.

Without the need for an artificial horizon, we can now define any imaginable direction in space - starting from the point of origin - by reference to three of those tetrahedral axes. Since we are dealing with a natively spherical system of orientation, our old subdivision of the circle into degrees, minutes and seconds is no longer an optimal system for describing the relation of any vector to the three principal tetrahedral axes that delimit the "sector" containing the direction we wish to describe. Note that the entire sphere has been subdivided into four sectors, each one delimited by three of the tetrahedral axes. For the sake of simplicity, each sector may be seen as corresponding to one of the faces of the tetrahedron. In reality however, we are describing points on the surface of a sphere.

Expressed in our present 360-degree-system, the angle between each one of the four tetrahedral axes is an awkward and only approximately accurate 109.4712 degrees. Therefore, in order to fully utilize the potential of this tetrahedral system of all-space orientation, we will need a different subdivision from the 360-degree horizon and the 90-degree "right" angle of the old X-Y-Z Cartesian system.

Two possibilities come to mind...