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Prince Rupert's Cube

cube section by plane
In issue 25 of the Magic Edges, we turned our attention to the fact that by cutting a cube with a plane, we get a cross-sectional cut at the cut point.
And this geometric shape will change depending on the angle of inclination of the plane.
 

Taking three cubes, we successively cut them with a plane, and got seven different sections:
- two regular triangles, differing in their size;
- square;
- rectangle;
- diamond;
- pentagon;
- regular hexagon. Here it can be seen on video.

As you probably guess, this is not all possible polygons that can be obtained by cutting a cube. But if the classic cube cuts seemed boring to you, then here is the story told in 1693 by the English mathematician John Wallis.
Prince Rupert of Palatinate argued that in a cube you can cut a hole large enough to be able to drag a cube of the same size through it.

 

Prince Rupert of Palatinate (germ. Ruprecht) (1619–1682).
Rupert is one of the founders of the Hudson's Bay Company and its first manager. In 1646–1648, Rupert led British units as part of the French army. In addition, he was a talented inventor, as well as an excellent engraver who introduced the mezzotint technique in England.

 

So what do we have? In the presence of two identical cube.
two cubes
 
 

In one of the cubes you need to make a through hole.
To fit a cube into this hole, the size of the hole must be a suitable size.
Which one Most likely, this should be the smallest hole area. In this case, the size of the hole will be square and equal to the area of ​​the side of the cube. In such a square hole a cube can pass.
With the shape of the hole, we decided - it is a square equal to the side of the cube. Now let's try to rotate the second cube so that a square hole can be made inside the first (green) one. The complexity of the problem is that it is an internal opening, and the outer walls of the cut cube would be inseparable.

Rupert cube cross section 
 
At first it may seem that the hole obviously cannot be only internal and will go beyond the face of the cut cube.
But this assumption will work until you turn the green cube as follows:
Rupert cube cross section In this case, we can make such an opening inside the green cube that a yellow cube will go right through it. John Wallis proved it mathematically and Prince Rupert won his argument.
Wallis suggested that such a hole would be parallel to the spatial diagonal of the cube.
Rupert cube cross sectionThe projection of the cube on a plane perpendicular to this diagonal is a regular hexagon, and the largest hole parallel to the diagonal can be obtained by drawing the largest square that can be inscribed in this hexagon.
Rupert cube cross section 
 
 
 
The cube model with a hole cut should look like this:
Rupert cube cross section 
 
This is how a situation will look like when one cube passes through another:
Rupert cube cross section 
 
 
After about 100 years, the Dutch mathematician Peter Newland figured out that a better solution could be found. The thing is that the physical model of Prince Rupert’s cube is very difficult to manufacture, since it has very thin walls at four points. The optimal solution proposed by Peter Newland can be obtained by cutting a hole at a different angle than the spatial diagonal. This allows you to increase the thickness of the walls in thin places.
 
P.S.
Rupert cube cross sectionIn the logo of the well-known program from Microsoft, you can see in something similar a picture, with a cube which has a middle cut out.
 
© polyhedr.com  13/03/2018

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