Multifaceted Cork

The applet below illustrates a design of a cork plug that fits triangular, square, and circular holes. This is a well known conundrum. (Right click on the shape to remove pieces of the triangulation.)


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Following J. H. Butchart and Leo Moser, Martin Gardner has inquired for the volume of such a shape.

Assume that the shape has a circular base with a radius of one unit, a height of two units, and a straight edge of two units that is directly above and parallel to a diameter of the base. The surface is such that all vertical cross sections which are made perpendicular to the top edge and the base are triangles. What is the volume of that figure?

Imagine the cork tightly set in a right cylinder with the same base and height. Every cross section perpendicular to the base and the edge cuts a triangle from the shape and a rectangle from the cylinder such that the area of the rectangle is twice the area of the triangle. With a reference to Cavalieri's Principle, its generalization, or in the spirit of infinitesimals employed by Rabbi Abraham bar Hiyya Hanasi, the volume of the cylinder is twice that of the cork. The volume is a circular right cylinder with height \(H\) and radius of the base \(R\) is given by \(V=\pi R^{2}H\), which, for the given data (\(R=1\), \(H=2\)) gives \(2\pi,\space\) making the volume of the cork the beloved \(\pi\).

References

  1. M. Gardner, The Colossal Book of Short Puzzles and Problems, (Edited by Dana Richards) W. W. Norton, 2006 (6.8)

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