There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$.

There are eight convex deltahedra (all faces equilateral triangles), five in $\mathbb{R}^4$ (see, "Convex deltahedra in higher dimensions"), and then three deltatopes in every dimension $d > 4$.

Q. Is there some intuitive reason why freedom is removed in higher dimensions?

I ask because, if I didn't know better, I would think the opposite. There is vastly much more "room" in higher dimensions, and one might think forms proliferate, even under constraints. Concerning "much more room," think of the severe contraints on planar graphs vs. the fact that every graph can be realized as embedded in $\mathbb{R}^3$.

I seek a corrective to my faulty intuition. Thanks!

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$.

There are eight convex deltahedra (all faces equilateral triangles), five in $\mathbb{R}^4$ (see, "Convex deltahedra in higher dimensions"), and then three deltatopes in every dimension $d > 4$.

Q. Is there some intuitive reason why freedom is removed in higher dimensions?

I ask because, if I didn't know better, I would think the opposite. There is vastly much more "room" in higher dimensions, and one might think forms proliferate, even under constraints. Concerning "much more room," think of the severe contraints on planar graphs vs. the fact that every graph can be realized as embedded in $\mathbb{R}^3$.

I seek a corrective to my faulty intuition. Thanks!

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$.

There are eight convex deltahedra (all faces equilateral triangles), five in $\mathbb{R}^4$ (see, "Convex deltahedra in higher dimensions"), and then three deltatopes in every dimension $d > 4$.

Q. Is there some intuitive reason why freedom is removed in higher dimensions?

I ask because, if I didn't know better, I would think the opposite. There is vastly much more "room" in higher dimensions, and one might think forms proliferate. Concerning "much more room," think of the severe contraints on planar graphs vs. the fact that every graph can be realized as embedded in $\mathbb{R}^3$.

I seek a corrective to my faulty intuition. Thanks!

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$.

There are eight convex deltahedra (all faces equilateral triangles), five in $\mathbb{R}^4$ (see, "Convex deltahedra in higher dimensions"), and then three deltatopes in every dimension $d > 4$.

Q. Is there some intuitive reason why freedom is removed in higher dimensions?

I ask because, if I didn't know better, I would think the opposite. There is vastly much more "room" in higher dimensions, and one might think forms proliferate, even under constraints. Concerning "much more room," think of the severe contraints on planar graphs vs. the fact that every graph can be realized as embedded in $\mathbb{R}^3$.

I seek a corrective to my faulty intuition. Thanks!