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Search: id:A156547
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| A156547 |
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Decimal expansion of the central angle of a regular dodecahedron. |
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+0
2
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| 7, 2, 9, 7, 2, 7, 6, 5, 6, 2, 2, 6, 9, 6, 6, 3, 6, 3, 4, 5, 4, 7, 9, 6, 6, 5, 9, 8, 1, 3, 3, 2, 0, 6, 9, 5, 3, 9, 6, 5, 0, 5, 9, 1, 4, 0, 4, 7, 7, 1, 3, 6, 9, 0, 7, 0, 8, 9, 4, 9, 4, 9, 1, 4, 6, 1, 8, 1, 8, 8, 9, 9, 6, 6, 6, 7, 6, 7, 1, 3, 8, 7, 9, 5, 4, 8, 3, 4, 0, 7, 8, 1, 9, 4, 7, 3, 5, 0, 0, 2, 0, 8, 0, 9, 5
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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If A and B are neighboring vertices of a regular dodecahedron having
center O, then the central angle AOB is this number; exact value is
arccos((1/3)*sqrt(5)). The (minimal) central angle of the other four
regular polyhedra are as follows:
tetrahedron, A156546
cube, A137914
octahedron, A019669
icosahedron, A105199.
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FORMULA
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The dodecahedron has 12 faces and 20 vertices. To find the central angle,
we need any neighboring pair of vertices. Here are all 20 vertices:
(d,d,d) where d is 1 or -1 (that's 8 vertices);
(0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2;
(d*(t-1), d*t, 0); and ((d*t,0,d*(t-1));
An example of a neighboring pair is (1,1,1) and (0,t,t-1).
Apply the usual formula for the cosine of the angle between two vectors.
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EXAMPLE
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arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees,
41.810314895778598065857916730578259531014119535901347753...
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CROSSREFS
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Sequence in context: A031026 A021936 A154176 this_sequence A003673 A021141 A147677
Adjacent sequences: A156544 A156545 A156546 this_sequence A156548 A156549 A156550
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KEYWORD
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nonn,cons
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2009
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