This icosahedron balloon is one of my favorites to make. Kids love playing with this balloon and tossing it around. I’ve posted a smaller version of this balloon before, but this is the large version (more than a foot in diameter.) It is made from six 260 balloons with a large round in the middle. (I like to use a polka-dotted balloon.) Each 260 balloon is used to create one basic unit, and after twisting six of them, you connect the six units with the balloon ends to form the icosahedron (20 faces.)
Check out this paper from Vi Hart, which contains instructions and pictures for making this balloon.