We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In $§$7 we apply our results to the even unimodular lattice $E_8$ and show how the 600-cell transforms $E_8$/2$E_8$, an 8-space over the field $\bf{F}$$_2$, into a 4-space over $\bf{F}$$_4$ whose points, lines and planes are labeled by the geometric objects of the 600-cell.
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