A391653 Decimal expansion of the volume of a Dürer's solid with unit shorter edge length.

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

Offset

2,2

Comments

This truncated triangular trapezohedron (composed of six mirror-symmetric pentagonal faces and two equilateral triangular faces) is the solid depicted in Albrecht Dürer's 1514 engraving "Melencolia I".

The value of this constant is based on the interpretation that the internal angles of the pentagonal faces are exactly 72, 108 and 126 degrees, and that all the vertices of the solid lay on a sphere (see Schreiber and Weisstein links).

Links

Paolo Xausa, Table of n, a(n) for n = 2..10000

Peter Schreiber, A New Hypothesis on Dürer's Enigmatic Polyhedron in His Copper Engraving “Melencolia I”, Historia Mathematica, Volume 26, Issue 4, November 1999, pp. 369-377.

Eric Weisstein's World of Mathematics, Dürer's Solid.

Wikipedia, Rhomboederstumpf (in German; contains explicit formulas).

Wikipedia, Truncated triangular trapezohedron.

Formula

Equals (5/3)*sqrt(38 + 17*sqrt(5)) = (5/3)*sqrt(38 + 17*A002163).

Equals the largest real root of 81*x^4 - 17100*x^2 - 625.

Example

14.530920634737691132451313443693890411912343971189...

Mathematica

First[RealDigits[5*Sqrt[38 + 17*Sqrt[5]]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DuererSolid", "Volume"], 10, 100]]

Crossrefs

Cf. A391654 (surface area), A391655 (circumradius).

Cf. dihedral angles: A238238, A391656, A391657.

Cf. A080992, A002163.

Sequence in context: A019743 A010663 A385445 * A016494 A248144 A171870

Adjacent sequences: A391650 A391651 A391652 * A391654 A391655 A391656

Keyword

nonn,cons,easy

Author

Paolo Xausa, Dec 15 2025

Status

approved