I have struggled to offer an intuitive explanation (to U.S. college students) why the number of regular polytopes in dimension $d$ is:
- $d=2$, number: $\infty$.
- $d=3$, number: $5$, the five Platonic solids.
- $d=4$, number: $6$, with the $24$-cell the polytope with no clear $\mathbb{R}^3$ analog.
- $d \ge 5$, number: $3$, the simplex, hypercube, and its dual the cross-polytope (or orthoplex).
The derivations are convincing without providing clear intuition. Is there some intuitive explanation, perhaps connected to the maximum volume of a unit-ball achieved in dimension $5$?

Plot: Dave Richeson's blog, 2010.
(But see Bill Thurston's remarks on the unit-ball volume.)