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Decimal expansion of the minimum surface index of a closed cylinder.
+10
5
5, 5, 3, 5, 8, 1, 0, 4, 4, 5, 9, 3, 2, 0, 8, 5, 2, 5, 7, 2, 9, 0, 4, 1, 1, 6, 3, 8, 7, 6, 6, 9, 6, 4, 0, 8, 0, 3, 4, 0, 1, 0, 8, 7, 9, 8, 6, 4, 7, 8, 5, 4, 9, 8, 8, 1, 7, 6, 5, 2, 6, 9, 2, 0, 0, 0, 4, 3, 0, 3, 8, 3, 1, 5, 1, 2, 7, 9, 8, 9, 5, 1, 9, 6, 5, 9, 2, 8, 4, 3, 5, 9, 1, 6, 0, 8, 1, 3, 5, 4, 0, 4, 1, 6, 7
COMMENTS
Equivalently, the minimum surface area among all closed cylinders (cans) enclosing a unit volume. It is attained when the cylinder is right circular and its height matches the diameter of its base. For evident reasons, this value should be called the 'cannery constant'. Its determination is a nice exercise in elementary analysis.
For a general definition of surface index, see A232808.
FORMULA
Equals (54*Pi)^(1/3) = 3*(2*Pi)^(1/3).
EXAMPLE
5.53581044593208525729041163876696408034010879864785498817652692...
MATHEMATICA
RealDigits[3 * Surd[2 * Pi, 3], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)
Decimal expansion of the minimum surface index of a half-open cylinder.
+10
5
4, 3, 9, 3, 7, 7, 5, 6, 6, 2, 6, 8, 4, 5, 6, 9, 7, 8, 9, 0, 6, 0, 4, 2, 7, 5, 8, 1, 7, 9, 1, 3, 7, 1, 1, 7, 5, 2, 1, 5, 7, 9, 0, 5, 6, 6, 8, 8, 3, 8, 1, 1, 5, 2, 3, 0, 7, 1, 7, 6, 7, 8, 1, 1, 4, 1, 5, 1, 8, 0, 9, 4, 9, 3, 9, 4, 8, 5, 8, 7, 9, 8, 4, 3, 7, 0, 4, 6, 2, 0, 3, 4, 6, 1, 0, 6, 1, 4, 6, 2, 6, 9, 2, 5, 1
COMMENTS
Equivalently, the minimum external surface area among all pots or cups (half-open cylinders) delimiting a unit volume. It is attained when the cylinder is right circular and its height matches the radius of its base. We might call it the 'cooking pot constant'. It minimizes the material needed to make the pot (and also its cooling rate). Its determination is a nice exercise in elementary analysis.
For a general definition of surface index, see A232808.
EXAMPLE
4.3937756626845697890604275817913711752157905668838115230717678...
MATHEMATICA
RealDigits[3 * Surd[Pi, 3], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)
Decimal expansion of the minimum surface index of a closed cone.
+10
5
6, 0, 9, 2, 9, 4, 7, 7, 8, 5, 3, 7, 9, 5, 5, 5, 6, 0, 3, 4, 3, 6, 3, 1, 6, 0, 0, 8, 9, 4, 8, 7, 5, 6, 9, 3, 6, 9, 1, 0, 7, 6, 1, 5, 1, 5, 0, 4, 7, 2, 0, 7, 1, 7, 3, 2, 1, 3, 0, 5, 2, 1, 5, 2, 9, 0, 1, 1, 2, 8, 0, 7, 0, 2, 4, 1, 1, 7, 9, 6, 0, 6, 9, 6, 2, 2, 1, 2, 0, 4, 7, 0, 7, 6, 5, 6, 3, 9, 1, 8, 0, 0, 3, 2, 6
COMMENTS
Equivalently, minimum surface area among all closed cones enclosing a unit volume. It is attained when the cone is right circular and its height equals the diameter of its base multiplied by sqrt(2). Considering one of the ways of marketing ice cream, we might call it the "frozen cone constant".
For a general definition of surface index, see A232808.
FORMULA
Equals (72*Pi)^(1/3) = 6*(Pi/3)^(1/3).
EXAMPLE
6.0929477853795556034363160089487569369107615150472...
MATHEMATICA
RealDigits[Surd[72*Pi, 3], 10, 120][[1]] (* Harvey P. Dale, Jul 13 2015 *)
Decimal expansion of the minimum surface index of an open cone.
+10
5
4, 1, 8, 8, 0, 7, 7, 9, 4, 8, 6, 2, 3, 1, 3, 8, 1, 2, 8, 7, 2, 5, 5, 9, 7, 2, 0, 5, 9, 3, 6, 6, 8, 8, 1, 1, 9, 4, 4, 0, 5, 2, 7, 5, 8, 8, 7, 0, 4, 3, 4, 9, 6, 8, 6, 2, 1, 2, 8, 3, 0, 5, 7, 4, 3, 1, 7, 8, 2, 3, 9, 8, 3, 2, 7, 7, 8, 7, 2, 3, 9, 4, 2, 9, 5, 1, 8, 6, 8, 5, 8, 9, 7, 9, 0, 0, 8, 7, 3, 2, 1, 4, 2, 1, 6
COMMENTS
Equivalently, minimum external surface area among all open cones (not counting the base) enclosing a unit volume. It is attained when the cone is right circular and its height equals the radius of its base multiplied by sqrt(2). We might call it the 'teepee constant', related to the need to minimize the material needed to construct it.
For a general definition of surface index, see A232808.
EXAMPLE
4.18807794862313812872559720593668811944052758870434968621283...
MATHEMATICA
RealDigits[3 Sqrt[3]Surd[Pi/6, 3], 10, 120][[1]] (* Harvey P. Dale, Nov 23 2022 *)
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