The figure to the right is the root diagram for the 52 dimensional Lie algebra F_{4}. There are 48 non-zero vertices and 48 independently oriented edges, or roots, each corresponding to a different element of the algebra. The remaining 4 elements live at the origin. Every vertex in the diagram can be reached by adding together particular combinations of four positive roots.
Count the number of vertices... Try it! Take the mouse, grab the picture, and turn it around. How many vertices do you see?
You won't get 48 -- only 32 and the origin! That's because F_{4} lives in four dimensions, and to get a diagram we can visualize in three dimensions, we must project along one special direction. For this diagram, we've projected along the first simple root -- That is, we've projected F_{4}. If you choose the right projection, the diagram can be interpretted as corresponding to a subalgebra of F_{4}. In fact, you can visually identify an algebra's subalgebras using its root diagram.
Here's an animation of the root diagram for F_{4} where we've choosen to project along a direction that smoothly changes from the first simple root to the fourth. You can play the animation and simultaneously turn the figure, and in the meantime, test your knowledge of four dimensions!