Time Is Not the Fourth Dimension: The Tetrahedral–Octahedral Lattice as the Junction Layer of Four-Dimensional Space

The phrase "fourth dimension" is used in two incompatible senses: Minkowski's time coordinate and a fourth spatial direction in the sense of Kaluza and Klein. We take the second sense and treat time as a count of lattice updates rather than a geometric axis. Higher dimension is understood constructively: a larger space is the closure of a structure built from cells of the smaller one. With this reading, three exact geometric facts follow. (1) The tetrahedral–octahedral tessellation of space (TOT, vertex set = face-centred cubic lattice) is precisely the layer w = 0 of the four-dimensional 16-cell honeycomb (Delaunay tessellation of the lattice D₄). Its octahedra are the equatorial sections of 4D cells centred in our space; its tetrahedra are the shared walls of 4D cells centred half a step above and below, at w = ±½. This explains the 1 : 2 ratio of octahedra to tetrahedra per node, and gives the identity R²_Oct = R²_Tet + (R_Oct/2)². (2) A self-similarity with ratio √2 is impossible for any three-dimensional lattice, but exists for D₄ (multiplication by the Hurwitz quaternion 1 + i). It turns every hole of the lattice into a node and maps the edge onto the octaradius. No √2-similarity of D₄ maps our three-dimensional layer onto itself (this follows from the index argument), and for all 1152 quaternionic maps x ↦ r·x·u, r·x̄·u the layer is tilted by 45° or 90° into the fourth direction. A √2 step therefore cannot be made inside three-dimensional space. (3) Under 4D duality a vertex corresponds to a three-dimensional cell, which gives a geometric reason why a point cannot shrink below a cell. We state what is and what is not claimed, list tests, and record the relation to earlier work. Contact: Yevhenii Stakhov · [email protected] · https://theoryof.space