The figure below is the root diagram for the 52 dimensional Lie algebra F4. 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. As the diagram lives in four dimensions, it must be projected into three dimensions. In the example to the right, the projection direction is chosen to always be in the plane containing the first and the fourth roots. Press Start on the Animation Panel to show the projection of the root diagram as the projection direction changes. The first frame projects along the first root, and the last frame projects along the fourth root. The second and third roots left alone by the projections -- You can see this by turning the root diagram so that the resulting image never changes - even though the projection is continually changing. On the other hand, you can also turn the diagram so that the projections continually morph from an outer diamond shape to that of a hexagon. For additional controls, right click on the diagram.

Figure 24. Animation of $F_4$, projecting along the direction $p_\theta = \cos(\theta)r^1 + \sin(\theta)r^4$ which is confined to a plane in $\mathbf{R}^4$ containing $r^1$ and $r^4$.