As is often the case with thought experiments, detours are encountered. Here in particular, the abstract of negative volume has delivered me to a question of divisibility, dimension, and tessellation.
The concept of classical volume holds that without manifesting any physical attributes, it may exist merely as a defined region of emptiness. Ancillary to this point, it seems intuitive to myself that in a division of space, all space is deserving of equal representation.
With this as context, I offer the equal division of space as delineated by rays of coincident origin and mutual angular distribution.
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A singular ray is unable to divide space (Fig. 1.). Two rays of coincident origin and opposite direction equally divide space forming an angle between them of 1.0π radians (180°) (Fig. 2.).
In the case of three rays, the conventional spatial construct is that of an orthogonal reference frame and coincident origin (Fig. 3.). While the smallest angle between any two rays is 0.5π radians (90°), the explementary angle is 1.5π radians (270°). Thus, this configuration lacks angular consistency throughout. However, within this construct a regular polyhedral can be created by extruding two, dimensional rays, along a third (Fig. 4.); this for each of the three rays, and for each at the same displacement. If the subsequent planes are trimmed to one another, the resulting defined volume is a cube. And of great significance for the cube is its ability to honeycomb or tile space (Fig. 5.). The same process using variable displacements results in a cuboid, and again space can be tiled (Fig. 6.). For these primitive objects, perhaps space filling is a fundamental attribute; offering a measure of validity to the dimensional construct within which they reside.
In the case of four rays (Fig. 7.), each originating from the centroid of a regular tetrahedron, and each extending through respective vertices, the criteria set forth is met with a resulting angle of 0.608173448…π radians (cos-1(-1/3) or 109.471220634491…°). This arrangement defines a non-orthogonal system, capable of defining four, equally valid, spatial dimensions. Here it seems there is nothing to preclude us from defining more than three spatial rays except our intransigence with orthogonality; and perhaps this concept of mutual division. Could our familiar orthogonal arrangement represents only a special case of greater generality? If the extrusion technique described prior is applied to four rays, each having consistent displacement, the result is recognizable as a regular concave rhombic dodecahedron (Fig. 8.). And, as with the cube and cuboid, it tiles space. This will also be true for a polyhedron of variable extrusion displacements along each of the four rays that yields an irregular concave rhombic dodecahedron (Fig. 9.). While these objects are definable using only three dimensions, the extrusion method, as applied to an equal division of space (in this case four), offers an alternative perspective.
At this time, I’m unable to fully reconcile equal spatial division by five, *six, seven, or eight rays. It seems unlikely that rays of any greater quantity could be placed in an arrangement of mutual angular displacement.
*Note that while six rays can offer equal division, the rays are not mutually consistent in their angular arrangement to one another (Fig. 10.). e.g., while the angle between a ray and it’s opposite is 1.0π radians (180°), the angle between a ray and it’s adjacent is 0.5π radians (90°).
