On polyhedron presses

A traditional press presses things by exerting a pressure from above and below. A 3-dimensional object that is pressed in this way becomes flattened. Is it possible instead to press an object from all sides? Is it possible to make a closed space that can reduce steplessly? Is it possible for this space to preserve its shape - just with smaller and smaller dimensions?

At first it sounds impossible, but when you look into it, it turns out to be quite simple. As examples of closed spaces I have chosen the platonic polyhedrons. By letting the sides slide across each other, the space they bound can reduce in size - the challenge is to do it without the sides cutting through each other.

By trial and error you quickly find solutions for the tetrahedron and the octahedron (the two topmost). And you quickly realize that they have more than one solution. The solutions for the cube and the icosahedron (the two bottommost) can be a bit more tricky to find. They are possibly unique modulo rotation and reflection.

The polyhedron presses has, in a way, a 2-dimensional analogue in optics. The iris diaphragm, that is inside some lenses, "presses" a many-sided polygon so its area becomes smaller.

But (the assiduous reader might think) there are five platonic polyhedrons and I have only shown presses for four of them - what about the dodecahedron? The answer is that there is no such thing as a dodecahedron press.

The sides of the dodecahedron consists of 12 pentagons. When a pentagon slides over the edges of a smaller pentagon, it will slide over at least 3 of them. If pentagons from two different sides tries to slide over the same edge, they will cut through each other. In other words there can be at most 1 pentagon sliding over an edge. The dodecahedron has 30 edges, and that is not enough for the 12*3=36 edges that is needed. Therefore there is no dodecahedron press. QED

# In practice

So polyhedron presses exists in theory, but can you build one in reality? I tried to look at the cube, and found a design that should work in practice.

To build it you make 8 blocks, which looks like this:

The blocks are assembled by pushing the dovetails into the appropriate grooves. It is probably a good idea to either make the cube press with low-friction surfaces, or to apply a lubricant.

When the blocks are pressed together a cube shaped space in the middle will shrink. The pressure must be on the two blocks that there were only one kind of. They must be pressed towards each other, and at the same time you must ensure that they do not change their orientation.

Seen from inside, the room will shrink by the floor going under two of the walls, the ceiling sliding over the two other walls, and the walls themselves sliding behind each other and the floor or the ceiling.

 http://www.h33.dk/polypres_index.en.html last changed 27.mar.2005 © 1997-2005 Bjørn Hee, mailto:webmaster@h33.dk