A177870 -id:A177870 - cp's OEIS frontend

A378715 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis dodecahedron.

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

Offset: 1

Paolo Xausa, Dec 07 2024

The disdyakis dodecahedron is the dual polyhedron of the truncated cuboctahedron (great rhombicuboctahedron).

			2.7066946454792287856258644383068280456984454555717...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Disdyakis dodecahedron.

Cf. A378712 (surface area), A378713 (volume), A378714 (inradius), A378393 (midradius).
Cf. A177870, A195698 and A195702 (dihedral angles of a truncated cuboctahedron (great rhombicuboctahedron)).
Cf. A002193.

First[RealDigits[ArcCos[-(71 + 12*Sqrt[2])/97], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DisdyakisDodecahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(71 + 12*sqrt(2))/97) = arccos(-(71 + 12*A002193)/97).

A378394 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal icositetrahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 30 2024

The deltoidal icositetrahedron is the dual polyhedron of the (small) rhombicuboctahedron.

			2.410613141653407606153665785465949185980362906...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Deltoidal icositetrahedron.
Index entries for transcendental numbers.

Cf. A378390 (surface area), A378391 (volume), A378392 (inradius), A378393 (midradius).

Cf. A177870 and A195702 (dihedral angles of a (small) rhombicuboctahedron).

First[RealDigits[ArcSec[Sqrt[32] - 7], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DeltoidalIcositetrahedron", "DihedralAngles"]], 10, 100]]

acos(-(4*sqrt(2) + 7)/17) \\ Charles R Greathouse IV, Feb 11 2025

Equals arcsec(4*sqrt(2) - 7) = arcsec(A010487 - 7).

Equals arccos(-(4*sqrt(2) + 7)/17) = arccos(-(A010487 + 7)/17).

A384473 Decimal expansion of the middle interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Offset: 3

Stefano Spezia, May 30 2025

			108.366120162561467008046935277164429896133431...

Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Alexander Bogomolny, Approximate Construction of Regular Pentagon by A. Dürer.
Stefano Spezia, Albrecht Dürer's approximate construction of the regular pentagon.
Wikipedia, Albrecht Dürer.

Cf. A228719, A384474 (in radians).
Cf. A002194, A019824, A072097, A177870.
Cf. A384475, A384476, A384477, A384478.

RealDigits[(3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]])180/Pi,10,100][[1]] (* or *)
RealDigits[(Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])])180/Pi,10,100][[1]]

Equals 135 - 180*arcsin(sqrt(3)*sin(Pi/12))/Pi.
Equals (Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))))*180/Pi.
Equals (540 - 2*A384475 - A384477)/2.
A384475 < this constant < A384477.

A384474 Decimal expansion of the middle interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, May 30 2025

			1.891345594448510418687173478952739199024779225...

Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Alexander Bogomolny, Approximate Construction of Regular Pentagon by A. Dürer.
Stefano Spezia, Albrecht Dürer's approximate construction of the regular pentagon.
Wikipedia, Albrecht Dürer.

Cf. A228719, A384473 (in degrees).
Cf. A002194, A019824, A177870.
Cf. A384475, A384476, A384477, A384478.

RealDigits[3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]],10,100][[1]] (* or *)
RealDigits[Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])],10,100][[1]]

Equals 3*Pi/4 - arcsin(sqrt(3)*sin(Pi/12)).
Equals Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))).
Equals (3*Pi - 2*A384476 - A384478)/2.
A384476 < this constant < A384478.

A232716 Decimal expansion of the ratio of the length of the boundary of any parbelos to the length of the boundary of its associated arbelos: (sqrt(2) + log(1 + sqrt(2))) / Pi.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Nov 28 2013

Same as decimal expansion of P/Pi, where P is the Universal parabolic constant (A103710). - Jonathan Sondow, Jan 19 2015

According to Wadim Zudilin, Campbell's formula (see below) follows from results of Borwein, Borwein, Glasser, Wan (2011): Take n=-2, s=1/4 in equations (4) and (20) to see that the formula is about evaluating K_{-2,1/4}. Take r=-1/2, s=1/4 in (76) to see that K_{-2,1/4} = cos(Pi/4)-K_{0,1/4}/16. Finally, use (51) and (52) to conclude that K_{0,1/4} = 2G_{1/4} = 2*log(1+sqrt(2)). - Jonathan Sondow, Sep 03 2016

			0.730708084248143098345459389970990137736723287291660275735498...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Borwein, J. M. Borwein, M. L. Glasser, J. G. Wan, Moments of Ramanujan's generalized elliptic integrals and extensions of Catalan's constant, J. Math. Anal. Appl., 384 (2) (2011), 478-496.
M. Hajja, Review Zbl 1291.51018, zbMATH 2015.
M. Hajja, Review Zbl 1291.51016, zbMATH 2015.
J. Sondow, The parbelos, a parabolic analog of the arbelos, arXiv 2012, Amer. Math. Monthly, 120 (2013), 929-935.
E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv:1210.5580 [math.MG], 2012-2013; Amer. Math. Monthly, 121 (2014), 438-443.

Reciprocal of A232717. Ratio of areas is A177870.

Cf. A000796, A103710.

R:= RealField(); (Sqrt(2) + Log(1 + Sqrt(2)))/Pi(R); // G. C. Greubel, Feb 02 2018

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/Pi,10,100]

(sqrt(2) + log(1 + sqrt(2)))/Pi \\ G. C. Greubel, Feb 02 2018

Equals A103710 / A000796.

Empirical: equals 3F2([-1/2,1/4,3/4],[1/2,1],1). - John M. Campbell, Aug 27 2016

A387320 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 27 2025

This is the dihedral angle between triangular faces in the antiprism part of the solid.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.687150505637070622058237671034217872408094243788...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square bicupola.
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square bicupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387321, A387322, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A002193.

First[RealDigits[ArcCos[(1 - Sqrt[8 + Sqrt[32]])/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos((1 - 2*sqrt(2 + sqrt(2)))/3) = arccos((1 - 2*sqrt(2 + A002193))/3).

A387321 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 27 2025

This is the dihedral angle between adjacent triangular faces at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.6412090003740395440214510528511358326798716782548...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square bicupola.
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square bicupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387322, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A002193, A010487, A195696.

First[RealDigits[ArcSec[Sqrt[3]] + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]],2], 10, 100]]

Equals arcsec(sqrt(3)) + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A195696 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A195696 + A387323.

A387322 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and a square face at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.4712905456469785754732547961552537994857493330886...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square bicupola.
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square bicupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A003881, A002193, A010487.

First[RealDigits[Pi/4 + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]], 4], 10, 100]]

Equals Pi/4 + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A003881 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A003881 + A387323.

A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and the octagonal face.

			1.6858923822495302658575939503353780784364569832448...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387322.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A002193, A003881, A010487, A195696.

First[RealDigits[ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A387321 - A195696 = A387322 - A003881.

A232715 Decimal expansion of the ratio of the area of a parbelos to the area of its associated arbelos: 4/(3*Pi).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Nov 28 2013

Also distance from the diameter of unit semicircle to its centroid. - Franck Maminirina Ramaharo, Oct 22 2018

			0.424413181578387562050356702326704965425225721974550529993779...

J. Sondow, The parbelos, a parabolic analog of the arbelos, arXiv 2012, Amer. Math. Monthly 120 (2013) 929-935.
E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv 2012.
Index entries for transcendental numbers

Reciprocal of A177870. Ratio of lengths of boundaries is A232716.

Mathematica
```
RealDigits[4/(3 Pi),10,100]
```

4/Pi/3 \\ Charles R Greathouse IV, Oct 01 2022

cp's OEIS Frontend

A378715 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis dodecahedron.

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A378394 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal icositetrahedron.

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A384473 Decimal expansion of the middle interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

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A384474 Decimal expansion of the middle interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

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A232716 Decimal expansion of the ratio of the length of the boundary of any parbelos to the length of the boundary of its associated arbelos: (sqrt(2) + log(1 + sqrt(2))) / Pi.

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A387320 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387321 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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