Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$? ($Ω$ is from the Big O Notation.)
Equivalently, is it true that $∃ε>0 \, ∀n>20$ every triangulation of a sphere into $n$ triangles must have at least $ε\sqrt{n}$ edges longer than $(1+ε)s$ or shorter than $(1−ε)s$, where $s$ is the side-length of an equilateral triangle that covers $1/n$ of the area of the sphere?
This problem is probably already solved somewhere, but here is what I have:
- A stronger version of the conjecture is that for all small enough $ε$ and all $n>1/ε$, we must have at least $C\sqrt{n/ε}$ edges as above (for some constant $C>0$ independent of $ε$).
- For $n>20$, not all triangles can be equilateral. However, starting with the icosahedron, we can triangulate such that all angles are above 54° and all edge lengths are within a factor of 1.26 of each other (as commented by Matt F.). With some effort, we can even ensure (perhaps with a bit worse lengths/angles) that all triangles have approximately the same area.
- Without the area restriction, all but $O(1)$ triangles can be, within a factor of $1+ε$, equilateral (use a triangulation (of a combinatorial sphere) with $O(1)$ 'defects', and then apply a conformal mapping; note: many triangles will have $o(R^2/n)$ areas there).
- Approximately equilateral triangles have a locally unique triangulation topology (assuming no hinge vertices), allowing a grid-based local coordinate system and a global coordinate chart, with (under the coordinates) isometric connections.
- For a 'sturdy' simply-connected region and similar triangle sizes, a single coordinate system gives correct distances within a factor of $1+ε$ or so (hence "approximately planar"). However, a long thin strip can easily bend.
- A nonsimply connected region can have topological defects. A topological defect is characterized by a dislocation and holonomy (which affects how dislocation propagates); orientation is preserved for us.
- If all defects have zero holonomy, then the dislocations are additive, and due to the intrinsic curvature (for example, circles on spheres have low perimeters for their areas), we need $Ω(\sqrt n)$ coordinate dislocation (assuming similar triangle sizes) to cover a large convex area of the sphere.
- Nonzero defect holonomy, being quantized, ordinarily gives many distorted (or wrong-sized) triangles, but I do not have a simple characterization.
I conjecture a positive answer, and that dealing with the distortions from the ideal grid leads to fractal-like optimal asymptotic solutions for a number of optimization problems on the sphere, including maximum sphericity, minimal length equal area partitions of the sphere, and the Thomson problem (classical electrons on a sphere). I discuss this in detail in the Asymptotic optimal sphericity question. A positive answer here implies $1 - \frac{5 \sqrt{3} π}{27n} - Θ(n^{-3/2})$ maximum sphericity for polyhedra with $n$ faces.
