A Circle-Stacking Theorem

Bottle stacking in a wine rack leads to a delightful and unexpected result. However mathematics does not end there. A. Brown has observed that if the bottles form a pyramid then the center of the top bottle always projects onto the midpoint of the base.

Theorem

Let N > 1 and place N bottles of equal radius into a rack capable of holding N, but not N + 1 bottles. Form a pyramid by placing more bottles on top until there is only room for a single bottle. The center of the latter is equidistant to the centers of the leftmost and rightmost bottles in the bottom layer.


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A Circle-Stacking Theorem


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In fact even more is true. The heavy broken lines (make sure the Hint box is checked) formed by connecting the first and the last bottles of successive layers can be obtained from each other by reflection and translation. A. Brown's Circle-Stacking Theorem is then obtained as a simple corollary. We'll do that elsewhere with a proof without words.

References

  1. A. Brown, A Circle-Stacking Theorem, Math Magazine, v. 76, no 4 (Oct., 2003), pp. 301-302

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