The Octahedron, Two Pyramids Proved
Advanced companion to Octa Meets Herself, trading the character frame for the real vocabulary: an octahedron has 8 triangular faces, 6 vertices, and 12 edges, satisfying Euler's formula V minus E plus F equals 2. At each vertex four 60-degree angles meet for 240 degrees total, a 120-degree angular defect. Swapping its face and vertex counts (8 and 6) produces a cube (6 faces, 8 vertices): the octahedron and cube are dual solids. Written as a two-voice duet between the shape's upper and lower square-based pyramid halves.
[UPPER PYRAMID] Four triangles lean up from a square base, meeting at one sharp crown, [LOWER PYRAMID] four more triangles lean down from that same square, meeting facing down,
[UPPER PYRAMID] Every face is equilateral, sixty degrees at every corner it owns, [LOWER PYRAMID] matching triangles below, the same sixty degrees, nothing else it condones,
[UPPER PYRAMID] Press the two square bases together, let the middle square dissolve, [LOWER PYRAMID] eight triangular faces remain, the square seam gone, the shape resolved,
[UPPER PYRAMID] Count six sharp points total, four around the middle, one up top, [LOWER PYRAMID] four around that same middle, one more point below, six points, full stop,
[UPPER PYRAMID] Twelve edges frame the eight faces, run the count and see it hold, [LOWER PYRAMID] six vertices minus twelve edges plus eight faces, Euler's rule retold,
[UPPER PYRAMID] At each middle point, four triangles meet, sixty degrees times four, [LOWER PYRAMID] two hundred forty total, a full turn short by a hundred twenty more,
[BOTH] Swap our faces for corners and a cube stands in our place, eight faces trade for six, six vertices trade for eight, a dual face to face, the octahedron and the cube, mirrors built from the same equation, two pyramids joined at the seam, geometry's own reflection.
Key Concepts: An octahedron has exactly 8 triangular faces, 6 vertices, and 12 edges, satisfying Euler's formula V − E + F = 2 | At each vertex, four 60-degree angles meet for 240 degrees, a 120-degree angular defect below a full 360-degree turn | Swapping the octahedron's face and vertex counts (8 and 6) produces a cube (6 faces, 8 vertices) — the octahedron and cube are dual solids
Characters: none — advanced companion sung about the real solid directly, paired with Octa Meets Herself, keeping the original's two-voice pyramid-halves duet structure
Key Words: octahedron, vertex, edge, face, Euler's formula, angular defect, dual solid, cube
Advanced framing note: The original Octa Meets Herself joins two pyramids for a younger audience. This companion keeps that same two-voice structure and adds the vertex/edge count, Euler's formula, the angular-defect fact, and the real octahedron-cube duality the original explicitly defers as "beyond this age range."