 TOPICS # Barrel A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.

The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset of a topological linear space is absorbing if for each there is an such that is in if for each such that . A subset of a topological linear space is absolutely convex if for each and in , is in if .)

When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov 1991). For sides consisting of an arc of an ellipse, the equation of the side is given by (1)

with . Solving for gives (2)

so the sides have equation (3)

Using the equation for a solid of revolution then gives   (4)   (5) For sides consisting of a parabolic segment, the equation of the side is given by (6)

with . Solving for gives (7)

so the sides have equation (8)

Using the equation for a solid of revolution then gives   (9)   (10)

Cylinder, Hyperboloid, Solid of Revolution

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## References

Harris, J. W. and Stocker, H. "Barrel." §4.10.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998.Kepler, J. New Solid Geometry of Win Barrels.MacDonnell, J. "The Mathematician's Quest for Superlatives from Geometrical and Calculus Considerations." http://www.faculty.fairfield.edu/jmac/ther/superlatives.htm.Shechter, B.-S. "Kepler's Wine Barrel Problem in a Dynamic Geometry Environment." http://www.math.uoc.gr/~ictm2/Proceedings/pap420.pdf.Taylor, A. E. and Lay, D. C. Introduction to Functional Analysis, 2nd ed. New York: Wiley, 1980.Tikhomirov, V. M. "New Solid Geometry of Wine Barrels." Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991.

Barrel

## Cite this as:

Weisstein, Eric W. "Barrel." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Barrel.html