The Hat is made of 3 gnomon triangles. Several joined together make a simple but useful 3D object. It tiles the plane in cookie-cutter fashion and gives rise to all the patterns based on hex tiles.
In 3 dimensions, two hats can join on two short sides to make a half tetrahedron, or join on all three long sides to make a flying saucer. Five Hats joined on two edges for the face of the Great Dodecahedron.
Technically the shape is formed from the intersection of three blades to create an equilateral triangle on a fourth blade.