Triangular Bipyramid

e₃ points up · unit sphere in gray · hidden edges dashed

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convex hull of D₃,₅ (the bipyramid)

the polar P(D₃,₅), a prism

the six cosine vectors

unit sphere

vertex coordinates of the polar

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cm(D₃,₅) = 1 ⁄ √5: 0.447
angle arccos(1 ⁄ √5) to the nearest direction: 63.43°
vertices of the polar at distance: √5 ≈ 2.236
faces of the bipyramid: 6 congruent triangles

The six cosine vectors are the outward unit normals of the six faces — equivalently, the six vertices of the polar prism, normalized. Each makes the angle 63.4° with the nearest element of D₃,₅.

Legend

bipyramid: convex hull of the five directions polar prism T × [−1, 1] cosine vectors