The enneagram, embedded

Beer and Truss's finding: the enneagram is not a plane figure but hangs within the icosahedron. Here its nine points carry the emirp structure.

hexad 1-4-2-8-5-7 — the four emirp pairs + two palindromes

triangle 3-6-9 — the emirp dead band (an equilateral triangle)

icosahedron frame; three slate nodes mark the excluded axis

arc 5-7-1-4  arc 4-2-8-5 — the hexad's two wings, splayed 138°

The triad's surface is flat. The hexad's is not: a plane fitted through its six points leaves residuals of +0.35 (points 1, 4, 8, 5) and −0.70 (points 2, 7), against a circumradius of 2. The skew is forced; the division into two wings follows from triangulating that circuit from its centroid.

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The 3-6-9 triangle is a true equilateral triangle inscribed in the icosahedron; the six hexad points ring it; the excluded three form an axis. By Beer's symmetry argument this embedding holds on every one of the 30 edges at once.