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Passing a Larger Cube Through a Smaller One

Date: 09/25/2003 at 11:47:26
From: Daryl
Subject: spatial reasoning

I've been told that, improbable though it may seem, it is possible to
cut a hole through a solid cube so that a cube, larger than the
original, can be passed in one end and out the other. 

How do you cut the hole?


Date: 09/25/2003 at 14:18:17
From: Doctor Douglas
Subject: Re: spatial reasoning

Hi Daryl,

If you take a cube and look along the body diagonal (straight through
from one corner to the opposite corner), you see the profile of a
hexagon.  You can put a square in this hexagon whose side length is
greater than the original side length of the cube.  This means that
you can pass a larger cube completely through a smaller one.

To verify this, you can draw a regular hexagon and try to maximize the
area of a square that is inscribed within it.

For directions on constructing a demonstration of this very
interesting mathematical fact, check out the following page:

  Gabriel Nivasch -- Square Perforation on a Cube
  http://yucs.org/~gnivasch/cube/ 

Note that while the hexagon construction above does not give the
_largest_ cube that can pass through another one.  That cube is called
Prince Rupert's Cube:

  Eric Weisstein's World of Mathematics -- Prince Rupert's Cube
  http://mathworld.wolfram.com/PrinceRupertsCube.html 

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Higher-Dimensional Geometry
High School Puzzles

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