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File:Right square pyramid & its polar dual about their common intersphere.png

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Summary

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English:
  • A right square pyramid (red):
    • side/base edge length ratio: √2, i.e. with a 60° angle between the horizontal base plane and each side edge; base edge length: 2(√3)(1+2√2)/7; side edge length: 2(√3)(4+√2)/7; body height: 3(4+√2)/7;
    • coordinates of original pyramid’s vertices: S[0, 0, 2]; T[(√3)(4+√2)/7, 0, (2-3√2)/7]; U[0, (√3)(4+√2)/7, (2-3√2)/7]; V[-(√3)(4+√2)/7, 0, (2-3√2)/7]; W[0, -(√3)(4+√2)/7, (2-3√2)/7].
  • Its polar dual — a right square pyramid (blue):
    • side/base edge length ratio: (3+√2)/2; base edge length: √3; side edge length: (√3)(3+√2)/2; body height: 3(1+√2)/2;
    • coordinates of dual pyramid’s vertices: J[0, 0, -(2+3√2)/2]; K[(√3)/2, (√3)/2, 1/2]; L[-(√3)/2, (√3)/2, 1/2]; M[-(√3)/2, -(√3)/2, 1/2]; N[(√3)/2, -(√3)/2, 1/2].
  • Their (unit) common intersphere (which is also the circumsphere of their intersection) — great circles and circle arcs (thin dashed black).
  • The intersection of the two right square pyramids — half a distorted cuboctahedron (gray):
    • coordinates of original side edges’ points of tangency to the intersphere: A[(√3)/2, 0, 1/2]; B[0, (√3)/2, 1/2]; C[-(√3)/2, 0, 1/2]; D[0, -(√3)/2, 1/2];
    • coordinates of original base edges’ points of tangency to the intersphere, i.e. their midpoints: E[(√3)(4+√2)/14, (√3)(4+√2)/14, (2-3√2)/7]; F[-(√3)(4+√2)/14, (√3)(4+√2)/14, (2-3√2)/7]; G[-(√3)(4+√2)/14, -(√3)(4+√2)/14, (2-3√2)/7]; H[(√3)(4+√2)/14, -(√3)(4+√2)/14, (2-3√2)/7].
  • And 3 orthonormal axes — straight arrows, with marks for integers (black).
Date
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Author JavBol

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Right square pyramid & its polar dual about their common intersphere

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