A377849 -id:A377849 - cp's OEIS frontend

A377805 Decimal expansion of the volume of a snub dodecahedron with unit edge length.

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

Offset: 2

Paolo Xausa, Nov 09 2024

			37.616649962733362975777673671302714340355289873...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.
Index entries for algebraic numbers, degree 12.

Cf. A377804 (surface area), A377806 (circumradius), A377807 (midradius).
Cf. A102769 (analogous for a regular dodecahedron).
Cf. A001622, A090550, A104457, A134946, A377849.

First[RealDigits[((3*GoldenRatio + 1)*#*(# + 1) - GoldenRatio/6 - 2)/Sqrt[3*#^2 - GoldenRatio^2], 10, 100]] & [Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "Volume"], 10, 100]]

Equals ((3*phi + 1)*xi*(xi + 1) - phi/6 - 2)/sqrt(3*xi^2 - phi^2) = (A090550*xi*(xi + 1) - A134946 - 2)/sqrt(3*xi^2 - A104457), where phi = A001622 and xi = A377849.

Equals the largest real root of 2176782336*x^12 - 3195335070720*x^10 + 162223191936000*x^8 + 1030526618040000*x^6 + 6152923794150000*x^4 - 182124351550575000*x^2 + 187445810737515625.

A379892 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentagonal hexecontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 10 2025

The pentagonal hexecontahedron is the dual polyhedron of the snub dodecahedron.

			2.6734732271767846682790703348957917197870317502693...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.
Index entries for transcendental numbers.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379891 (midradius).
Cf. A377997 and A377998 (dihedral angles of a snub dodecahedron).
Cf. A377849.

First[RealDigits[ArcCos[#/(# - 2)] & [Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["PentagonalHexecontahedron", "DihedralAngles"]], 10, 100]]

acos(polrootsreal(209*x^6 - 94*x^5 - 137*x^4 + 100*x^3 - 9*x^2 - 6*x + 1)[1]) \\ Charles R Greathouse IV, Feb 10 2025

Equals arccos(A377849/(A377849 - 2)).

Equals arccos(c), where c is the smallest real root of 209*x^6 - 94*x^5 - 137*x^4 + 100*x^3 - 9*x^2 - 6*x + 1.

A377807 Decimal expansion of the midradius of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 10 2024

			2.0970538352520879924039590523482862400308397305810...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.
Index entries for algebraic numbers, degree 12.

Cf. A377804 (surface area), A377805 (volume), A377806 (circumradius).
Cf. A239798 (analogous for a regular dodecahedron).
Cf. A377849.

First[RealDigits[Sqrt[1/(1 - Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1])]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "Midradius"], 10, 100]]

Equals sqrt(1/(1 - A377849))/2.

Equals the real root closest to 2 of 4096*x^12 - 21504*x^10 + 16384*x^8 - 4672*x^6 + 624*x^4 - 40*x^2 + 1.

A377806 Decimal expansion of the circumradius of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 10 2024

			2.1558373751156397018366290766930582770168512187748...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.
Index entries for algebraic numbers, degree 12.

Cf. A377804 (surface area), A377805 (volume), A377807 (midradius).
Cf. A179296 (analogous for a regular dodecahedron).
Cf. A377849.

First[RealDigits[Sqrt[1 + 1/(1 - Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1])]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "Circumradius"], 10, 100]]

Equals sqrt(1 + 1/(1 - A377849))/2.

Equals the real root closest to 2 of 4096*x^12 - 27648*x^10 + 47104*x^8 - 35776*x^6 + 13872*x^4 -2696*x^2 + 209.

A380002 Decimal expansion of long/short edge length ratio of a pentagonal hexecontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 11 2025

			1.749852566736202791676446693655921179649815818590...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379890 (midradius), A379892 (dihedral angle), A380003 and A380004 (face internal angles).

Cf. A377849.

First[RealDigits[(1 + #)/(2 - #^2) & [Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]], 10, 100]] (* or *)
First[RealDigits[1/Divide @@ PolyhedronData["PentagonalHexecontahedron", "EdgeLengths"], 10, 100]]

Equals (1 + xi)/(2 - xi^2), where xi = A377849.

Equals the largest real root of 31*x^6 - 122*x^5 + 177*x^4 - 128*x^3 + 51*x^2 - 11*x + 1.

A377997 Decimal expansion of the dihedral angle, in radians, between triangular faces in a snub dodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 15 2024

			2.8654006883447286076046073417336569141190096726652...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.

Cf. A377804, A377805, A377806, A377807, A377849, A377998.

First[RealDigits[Pi - ArcCos[2*Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]/3 + 1/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["SnubDodecahedron", "DihedralAngles"]], 10, 100]]

Equals Pi - arccos((2/3)*A377849 + 1/3).

Equals Pi - arccos(c), where c is the largest real root of 729*x^6 + 486*x^5 - 729*x^4 - 756*x^3 + 63*x^2 + 270*x + 1.

A377998 Decimal expansion of the dihedral angle, in radians, between triangular and pentagonal faces in a snub dodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 15 2024

			2.669130633625756107707940935718208230518703745355...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.

Cf. A001622, A377804, A377805, A377806, A377807, A377849, A377997.

First[RealDigits[Pi - ArcCos[Root[91125*#^12 - 668250*#^10 + 2006775*#^8 - 2735100*#^6 + 1768275*#^4 - 502410*#^2 + 43681&, 7]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["SnubDodecahedron", "DihedralAngles"]], 10, 100]]

Equals Pi - arccos(sqrt((12*phi - (4*phi + 8)*xi*(xi + 1) + 19)/15)), where phi = A001622 and xi = A377849.

Equals Pi - arccos(c), where c is the real root closest to 9/10 of 91125*x^12 - 668250*x^10 + 2006775*x^8 - 2735100*x^6 + 1768275*x^4 - 502410*x^2 + 43681.

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A377805 Decimal expansion of the volume of a snub dodecahedron with unit edge length.

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A379892 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentagonal hexecontahedron.

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A377807 Decimal expansion of the midradius of a snub dodecahedron with unit edge length.

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A377806 Decimal expansion of the circumradius of a snub dodecahedron with unit edge length.

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A380002 Decimal expansion of long/short edge length ratio of a pentagonal hexecontahedron.

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A377997 Decimal expansion of the dihedral angle, in radians, between triangular faces in a snub dodecahedron.

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A377998 Decimal expansion of the dihedral angle, in radians, between triangular and pentagonal faces in a snub dodecahedron.

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