81018 — THE RESEARCH EXPEDITION
Expedition index · Next: G2 · G1 Analysis
G1 — First Certified Coordinate
The tetrahedron–octahedron construction is where the geometric record becomes something we can reconstruct independently.
Status: M — Mathematically established.
A regular tetrahedron, with its six edges bisected and the six midpoint points connected, decomposes into four congruent regular tetrahedra and one central regular octahedron. Each resulting solid has edge length one-half that of the original tetrahedron.
O · Origin
The historical starting point was a Geometry 101 classroom exercise using physical models of the Platonic solids. It was a math manipulative, not a cosmological theory.
M · Exact construction
Take a regular tetrahedron with vertices
A=(1,1,1), B=(1,-1,-1), C=(-1,1,-1), D=(-1,-1,1)
Its edge length is 2√2. Bisect all six edges. The six midpoint coordinates are
(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-1,0), (-1,0,0)
These six points form a regular octahedron of edge length √2, exactly one-half the original tetrahedral edge. The four corner regions are congruent regular tetrahedra of the same edge length √2.
Volume check
Vlarge = 8/3
4 Vsmall tetra = 4/3
Voctahedron = 2/3
4/3 + 2/3 = 8/3
The volumes account exactly for the original tetrahedron.
Q · How far within can we go?
This question can now be separated from the construction itself. The construction is mathematical; its physical interpretation remains open.
What G1 does not establish
- That tetrahedra or octahedra are fundamental physical constituents.
- A connection to the Planck scale.
- The 202-notation scale architecture.
- A cosmological interpretation.
Earlier versions of G1: [1] Geometrical [b] 3D [3] Practical
Notes to coders: “Continue to G2 — The Factor of 2 →”