In 5D space, shapes can get so complicated that you'd need an infinite number of colors just to keep the sides different.
arXiv · March 18, 2026 · 2603.16298
The Takeaway
While a standard 2D map only ever needs four colors to ensure no two neighbors share a color, this proof shows that in 5D, you can build a shape so interconnected that no matter how many colors you use, you can always find a version that needs even more.
From the abstract
The transversal ratio of a polytope $P$ is the minimum proportion of vertices of $P$ required to intersect each facet of $P$. The weak chromatic number of $P$ is the minimum number of colors required to color the vertices of $P$ so that no facet is monochromatic. We will construct an infinite family of $d$-polytopes for each $d\geq 5$ whose transversal ratio approaches 1 as the number of vertices grows. In particular, this implies that the weak chromatic number for $d$-polytopes is unbounded for