A389281 - OEIS

%I #16 Oct 27 2025 11:39:57

%S 3,0,4,2,8,0,8,8,7,2,7,0,2,4,7,0,3,0,3,9,7,5,7,8,5,4,8,1,7,2,4,9,4,8,

%T 8,2,9,8,6,6,8,8,5,3,0,2,1,1,2,8,4,2,4,0,2,5,1,8,4,8,6,2,4,9,3,1,8,5,

%U 2,7,9,9,1,9,5,3,8,5,0,8,6,2,7,2,5,1,7,6,2,4

%N Decimal expansion of the largest dihedral angle, in radians, in an augmented truncated dodecahedron (Johnson solid J_68).

%C This is the dihedral angle between a triangular face and a square face at the edge where the pentagonal cupola (Johnson solid J_5) and truncated dodecahedron parts of the solid meet.

%C Also the analogous dihedral angle in a parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron and triaugmented truncated dodecahedron (Johnson solids J_69, J_70 and J_71, respectively).

%H Paolo Xausa, <a href="/A389281/b389281.txt">Table of n, a(n) for n = 1..10000</a>

%H Polytope Wiki, <a href="https://polytope.miraheze.org/wiki/Augmented_truncated_dodecahedron">Augmented truncated dodecahedron</a>.

%H Polytope Wiki, <a href="https://polytope.miraheze.org/wiki/Metabiaugmented_truncated_dodecahedron">Metabiaugmented truncated dodecahedron</a>.

%H Polytope Wiki, <a href="https://polytope.miraheze.org/wiki/Parabiaugmented_truncated_dodecahedron">Parabiaugmented truncated dodecahedron</a>.

%H Polytope Wiki, <a href="https://polytope.miraheze.org/wiki/Triaugmented_truncated_dodecahedron">Triaugmented truncated dodecahedron</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Augmented_truncated_dodecahedron">Augmented truncated dodecahedron</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metabiaugmented_truncated_dodecahedron">Metabiaugmented truncated dodecahedron</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Parabiaugmented_truncated_dodecahedron">Parabiaugmented truncated dodecahedron</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Triaugmented_truncated_dodecahedron">Triaugmented truncated dodecahedron</a>.

%F Equals arccos(-sqrt((23 + 3*sqrt(5))/30)) = arccos(-sqrt((23 + A010499)/30)).

%e 3.0428088727024703039757854817249488298668853021128...

%t First[RealDigits[ArcCos[-Sqrt[(23 + Sqrt[45])/30]], 10, 100]] (* or *)

%t First[RealDigits[Max[PolyhedronData["J68", "DihedralAngles"]], 10, 100]]

%Y Cf. other J_68 dihedral angles: A137218, A344075, A377995, A377996, A389851.

%Y Cf. A386464 (J_68 volume), A386465 (J_68 surface area).

%Y Cf. A386466 (J_69 and J_70 volume), A386543 (J_69 and J_70 surface area).

%Y Cf. A386544 (J_71 volume), A386545 (J_71 surface area).

%Y Cf. A010499.

%K nonn,cons,easy

%O 1,1

%A _Paolo Xausa_, Oct 27 2025