Simo K. Kivelä, 1 Nov 2014

Presentation in Nordic GeoGebra Conference, Ylöjärvi, Finland

*Prince Rupert of the Rhine*, *Duke of Cumberland* or
*Ruprecht von der Pfalz* as he is also called, lived 1619--1682
and participated in many fights of the century, but he was also
interested in other things: arts, natural sciences, mathematics
etc.

Rupert in a propaganda picture (Wikimedia Commons).

The following problem is named after Prince Rupert:

Is it possible to cut a hole in a cube in such a way that another cube of the same size can be passed through?

Actually, he believed — correctly — that this is possible, and asked more: How big can the cube be, if it can be passed? He did not solve the problem himself, and it was solved approximately 100 years later by a Dutch mathematician Pieter Nieuwland.

The problem has been standard material for many years in the courses of
decriptive geometry, also in Finland up to sixties. The method in solving
the problem has usually been so called *Monge projection*, named after
French engineer and mathematician *Gaspard Monge*, who lived in the era
of the great revolution and Napoleon. The method has been a military secret
before the revolution.

Monge projection consists of two orthogonal projection, one to the xy-plane, the other to the yz-plane. The former, top view, is usually placed in the lower part of the drawing, the latter, front view, above it in such a way that y-axis is common. GeoGebra sheet below shows the traditional drawing method, except that some interactivity is availlable.

[download]

The first cube is seen in the top view as a square in red and 45 degrees turned in black. The black cube is also seen in the front view. The cube is projected with an orthogonal projection to a slanted plane; this is parallel to x-axis and its intersection lines with xy-plane and yz-plane are shown. The original slope of the plane (angle α) is controlled with the slider. When the plane is turned around the xy intersection line, the picture of the cube will turn to the xy-plane (circular arcs), and the picture of the cube is seen in the top view in hexagonal form. Depending on the original slope of the plane, the red square can be placed inside this hexagon. Hence, there is a direction for cutting the desired hole.

Although the problem is solved, it is not very easy to understand what kind of solid the cube with the hole really is.

GeoGebra's 3D tools make possible to build a threedimensional model of the situation. The sheet below gives the original cube in green; length of the edge is 2. The canal for pushing the cube is yellow; its size and slope can be controlled with the two sliders. In addition there are two points for turning the canal and for controlling its length. The whole system can be rotated threedimensionally.

[download]

The optimal slope of the canal and the maximum size of the cube to be passed may be approximately found with the sliders.

The construction protocol of the model is available as a separate GeoGebra file.

In order to see the green cube with hole the Boolean difference of the yellow and green prisms should be formed, but at the moment GeoGebra does not support solids. Other possibilities may exist, but they are not very easy. The software package Mathematica has appropriate tools (although neither Mathematica supports solids) and the picture below is made in Mathematica.