Technology Logo

Bengt's friend Alban has started studying at LTH, the Faculty of Engineering at Lund University. He therefore has a sweater with the logo of the student union, TLTH (http://www.tlth.lth.se/). It consists of a square inside an equilateral triangle inside a circle.
Bengt is thinking about whether this is the optimal way to pack these shapes.

1. If the circle has the area 1, what area does the square have? (It is as big as possible.)
   
2. If the shapes contain each other in a different order, one might get a different ratio between the outermost and the innermost. If the outermost shape always has area 1, in which order should they be placed to make the innermost as big as possible, and how big will it be?
   
3. What if they are in three dimensions? A sphere, a regular tetrahedron, and a cube, with the outermost having volume 1 - how big will the innermost be?
   
4. While you're at it: n dimensions?