Photo:
THE MIRROR
The following problem is several centuries old.
But his solution is still amazing today.
Two cubes of the same size are given.
You should drill or mill a hole in one cube through which the second cube will fit.
At first glance, this hardly seems possible
.
But think again carefully: maybe it will work after all?
But how?
A cube actually fits through itself. The most obvious solution is the following: you take two spatially diagonally opposite vertices of the cube between your thumb and forefinger.
Then rotate the cube between your two fingers so that one corner is pointing straight ahead towards you and the diagonally opposite corner is pointing backwards.
In the last step, tilt the diagonal axis running through your thumb and forefinger so that the two corners at the front and back are exactly on top of each other.
From this perspective, the cube, whose edge length is 1, has the silhouette of a regular hexagon – see the following drawing on the left.
We can easily calculate that the edge length of the hexagon is sqrt(2) divided by sqrt(3) (tip for anyone who wants to do the math themselves: the diagonal between thumb and forefinger is not parallel to the plane of the figure, otherwise the vertices would be in front and behind not exactly on top of each other).
The edge length seen from this perspective is less than 1 because the edges are tilted.
We can now calculate relatively easily (Pythagorean theorem suffices) how large a square can be at most so that it fits into the hexagon.
The edge length is sqrt(6) minus sqrt(2), which is about 1.035...
The edge length is greater than 1 (albeit just a little), so a cube with edge length 1 fits through the hole.
If we change the tilt angle of the cube that we hold between thumb and forefinger, we'll find an even better solution.
The square opening then has an edge length of 3/4 square root(2), which corresponds to about 1.06...
The die through which a larger die fits is called the Prince Rupert die.
The geometric problem goes back to Ruprecht von der Pfalz, a prince of the House of Wittelsbach with a great interest in art and science.
At the end of the 17th century, the English mathematician John Wallis presented the solution described above with an edge length of 1.035...
Around a hundred years later, the Dutch mathematician Pieter Nieuwland found the solution with an edge length of 1.06...
In case you can't imagine what a cube looks like with an opening that it can fit itself through, the Wikipedia page has the Prince Rupert Cube as a 3D rendered object that you can use at least in desktop browsers can rotate.
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