Volume increasing⁠

Can you remember how a milk package looked like in Soviet times? Surpris­ingly, few people remember what was drawn on it, though the whole country bought them almost every day for over than 20 years.

Anyway, everyone remem­bers that milk package was tetra­he­dral (in the form of regular trian­gular pyramid). These pack­ages were invented by Tetra Pak in the 1940s. In those days Tetra Pak brought two inno­va­tions. First, it started to pour out edible liquids into paste­board pack­ages. Second, the produc­tion of tetra­he­dral pack­ages was so easy that it became possible to manu­fac­ture them right at the milk plants.

The most common milk package in the Soviet Union looked like this.

Is it possible to fold a piece of paste­board of which a milk package is made into a package of greater volume?

Math­e­mat­i­cally saying, is it possible to fold a devel­op­ment of a tetra­he­dron into a poly­he­dron with a greater volume?

A theorem of A. Alek­san­drov sais that it's not possible to fold it into a convex poly­he­dron with greater volume. However, we may be able to make a nonconvex poly­he­dron with greater volume.

Surpris­ingly as it may seem, it is possible!

Let us follow David D. Bleecker's construc­tion, suggested in 1996. Move faces apart and add vertices and edges on each face. Consider the central regular triangle with side length addi­tional edges.

Repeat the coplanar and the edges between them disap­pear.

Let's calcu­late the volume of this poly­he­dron. We shall divide it into pieces. Our poly­he­dron consists of 4 equal hexag­onal pyra­mids and trun­cated tetra­hedra. To simplify the calcu­la­tion, attach small tetra­hedra to our tetra­he­dron and then subtract their volume away the whole volume.

It turns out that the volume of the constructed poly­he­dron is 37.7% greater than that of the initial one. Hence, a piece of paper from which tetra­he­dral pack­ages were made can be folded into a package over a third more capa­cious!

Surpris­ingly, this tetra­he­dron is not an excep­tion. It turns out that a devel­op­ment of any convex poly­he­dron with trian­gular faces can be folded into a poly­he­dron with greater volume. This theorem was proved by D.Bleeker in 1996. He proposed an algo­rithm to do it.

In his article D. Bleecker consid­ered, besides poly­he­drons with trian­gular faces, two poly­hedra that do not belong to this class: the cube and the dodec­a­he­dron. Their devel­op­ments can also be folded into nonconvex poly­he­drons which enclose more volume than the initial ones.