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.