Euler's theorem and Wire sculpting

How may strings do you need to make this face? The answer is one. By Euler's Theorem you can make a wire sculpture with one string if the graph of the sculpture has at most two vertices which have odd degree. We get the graph of the sculpture by drawing the figure on a piece of paper. 

All the twenty vertices of a dodecahedral graph has the odd degree three. Therefore by Euler's theorem it cannot be made with one string. We note that if the number of odd vertices of a graph is n, then it takes at least n/2 strings to construct the graph.
Therefore it takes at least ten strings to make a dodecahedron.

It takes at least two strings to make the left flower whereas the right flower can be made with one piece of wire. This is because the right flower is a general graph, that is, there are multiple edges between the same vertices.

This bunch of flowers is made with one string of wire.

Again , by considering general graphs, this bird can be made with one string of wire.

Different materials make different birds. The material maybe different but Euler's theorem stays true and we can make this bird with one piece of wire.

Guess how many strings you need to make the complete graph K_6? The answer hello is 3.

Ofcourse wire sculpture is not only for counting strings. Have a ball and make one too. Cheers.