A geodesic dome, that can easily be built as a 1/4, a 1/2, a 3/4 or a full dome, because it is divided into 4 equal parts. You will have to insert two extra struts, if you divide it, but you can easily see where.
Take a look from the top, and you will see what I mean.
If you want to build a real one, you should visit http://www.domerama.com/calculators/octahedral-5v-mexican-method/ , which I am inspired of. Here you can get the angles and length of the struts, and an assembly diagram