E8's Central Slice: The Cuboctahedron as a 3D Projection of 240 Roots — E8 Intelligence Research

FINDING: The cuboctahedron (Vector Equilibrium) is the 3D central slice of the E8 root system's 240 vertices, and its 12 vertices correspond to the 12 roots of the A2⊕A2⊕A2 subalgebra, while the full E8 projects to 3D via eigenvector-based slicing. | MATH: E8 has 240 roots in 8D; its 3D projection via eigenvectors yields a cuboctahedron at the central slice (z=0). The cuboctahedron has 12 vertices, 24 edges, 8 triangular + 6 square faces. Its vertex figure is the rhombic dodecahedron. The ratio of circumradius to edge length is √2/2 ≈ 0.7071. The cuboctahedron's dual is the rhombic dodecahedron (12 rhombic faces, 14 vertices). The E8 root lattice's Voronoi cell is the Gosset polytope 4_21, whose 3D central section is the cuboctahedron. | CONNECTION: The cuboctahedron is the vector equilibrium — all 12 vertices equidistant from center (radius = edge length), giving perfect isotropic vector matrix. This is the geometric embodiment of the golden ratio family: the ratio of the cuboctahedro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com