A205769 -id:A205769 - cp's OEIS frontend

A379135 Decimal expansion of the midradius of a pentakis dodecahedron with unit shorter edge length.

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

Offset: 1

Paolo Xausa, Dec 17 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			1.4756836610416140907689600838494857255268212565695...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 2.

Cf. A379132 (surface area), A379133 (volume), A379134 (inradius), A379136 (dihedral angle).
Cf. A205769 (midradius + 1 of a truncated icosahedron with unit edge length).
Cf. A010499.

First[RealDigits[(11 + Sqrt[45])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "Midradius"], 10, 100]]

(11 + 3*sqrt(5))/12 \\ Charles R Greathouse IV, Feb 05 2025

Equals (11 + 3*sqrt(5))/12 = (11 + A010499)/12.

A377697 Decimal expansion of the midradius of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 05 2024

			2.9270509831248422723068802515484571765804637697...

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

Cf. A377694 (surface area), A377695 (volume), A377696 (circumradius), A377698 (Dehn invariant, negated).
Cf. A239798 (analogous for a regular dodecahedron).
Cf. A010499, A205769.

First[RealDigits[(5 + Sqrt[45])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Midradius"], 10, 100]]

Equals (5 + 3*sqrt(5))/4 = (5 + A010499)/4.

Equals A205769 - 1/2.

A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 06 2024

			72.60725303413392187893153397383948620117264765443...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377751 (volume), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A010527 (analogous for a regular icosahedron, with offset 1).
Cf. A002163, A002194, A375067.

First[RealDigits[3*(10*Sqrt[3] + Sqrt[25 + Sqrt[500]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "SurfaceArea"], 10, 100]]

3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) \\ Charles R Greathouse IV, Feb 05 2025

Equals 3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) = 30*A002194 + 3*sqrt(25 + 10*A002163).

Equals 30*(A002194 + A375067).

A377751 Decimal expansion of the volume of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 07 2024

			55.28773075812273923639861693886121953098664736582...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 2.

Cf. A377750 (surface area), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A102208 (analogous for a regular icosahedron).
Cf. A002163.

First[RealDigits[(125 + 43*Sqrt[5])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "Volume"], 10, 100]]

(125 + 43*sqrt(5))/4 \\ Charles R Greathouse IV, Feb 05 2025

Equals (125 + 43*sqrt(5))/4 = (125 + 43*A002163)/4.

A374883 Decimal expansion of phi(2phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Jul 22 2024

The author conjectures that this is the minimum volume of an axis-aligned bounding box which includes the shortest minimum-link circuit joining all the vertices of the cube {0,1}^3 (i.e., the closed polygonal chains consisting of exactly 6 edges visiting all the points of the set {(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)}).
In detail, such a circuit of 6 links is given by (1/2,1+phi,1/2)-((1-phi)/2,0,(1+phi)/2)-((phi+1)/2,0, (1-phi)/2)-(1/2,1+phi,1/2)-((phi+1)/2,0,(phi+1)/2)-((1-phi)/2,0,(1-phi)/2(1/2,1+phi,1/2), where phi := (1+sqrt(5))/2 (see A001622).
Then, phi*(2*phi + 1) = phi^2*(phi + 1) since phi - 1 = 1/phi.

			6.8541019662496845446137605030969...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.

Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Marco Ripà, General uncrossing covering paths inside the Axis-Aligned Bounding Box, Journal of Fundamental Mathematics and Applications, Volume 4, 2021, Number 2, Pages 154-166.

Cf. A001622, A090550, A104457, A134973, A205769, A225227, A261547, A363755, A373537, A374149, A374260.

RealDigits[3*GoldenRatio + 2, 10, 120][[1]] (* Amiram Eldar, Jul 23 2024 *)

Equals (7 + 3*sqrt(5))/2.
Equals phi^2*(phi + 1), where phi = (1 + sqrt(5))/2.
Equals A104457^2 = 2*A205769. - Hugo Pfoertner, Jul 22 2024
Equals A090550 + 1 = A134973 + 5. - Amiram Eldar, Jul 23 2024
Equals phi^4. - Stefano Spezia, May 28 2025

A377752 Decimal expansion of the circumradius of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 07 2024

			2.47801865906761553756640791226630780693649473...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A377750 (surface area), A377751 (volume), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A019881 (analogous for a regular icosahedron).
Cf. A002163.

First[RealDigits[Sqrt[58 + 18*Sqrt[5]]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "Circumradius"], 10, 100]]

sqrt(58 + 18*sqrt(5))/4 \\ Charles R Greathouse IV, Feb 05 2025

Equals sqrt(58 + 18*sqrt(5))/4 = sqrt(58 + 18*A002163)/4.

cp's OEIS Frontend

A379135 Decimal expansion of the midradius of a pentakis dodecahedron with unit shorter edge length.

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A377697 Decimal expansion of the midradius of a truncated dodecahedron with unit edge length.

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A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

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A377751 Decimal expansion of the volume of a truncated icosahedron with unit edge length.

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A374883 Decimal expansion of phi*(2*phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.

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A377752 Decimal expansion of the circumradius of a truncated icosahedron with unit edge length.

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A374883 Decimal expansion of phi(2phi + 1) (i.e., (7 + 3*sqrt(5))/2), where phi is the golden ratio.