Find a two dimensional shape so that four congruent copies can be arranged on a plane so that each copy touches all three other copies along some segment of positive length.

Show that there is a three dimensional shape so that any finite number of copies can be arranged in space so that each copy touches all the other copies along some region of positive area.

Note 1: You may use the reflection of a shape.
Note 2: Touching at only a corner is NOT sufficient.

3D question proposed by Leming.

Although I tried to find that would tessellate I am curently at a loss.

I have a sense that this might relate to  4 Colour Mapping in the 2D scenario and 7 in the 3D.

A simple 4 colour map could 2 concentric circles  with the annulus divided into 3.  The minimum colours, so no region borders on of the same colour, are 4.  I expect that here has to be a corollary of this thought.

Similarly, as a 7 colour map is the minimum for a torus, I expect something similar to apply.

Sorry, can't help any further at this point.

Posted by brianjn on 2008-01-22 23:39:27