Hexagonal Geodesic Domes

Goldberg greenhouse conjectures

It occurs to me that there are some other related equality conjectures for Goldberg structures.

In particular it might be possible to deform a Goldberg polyhedron in such a way that their vertices remain on a sphere while all the areas of their faces became equal.

A similar equality conjecture is based on the observation that the required strength of the laminar material required to cover a region is generally a function of the largest distance it has to span:

It may be possible to deform a Goldberg polyhedron in such a way that all their faces retain co-planar vertices, while the distances of the points near the middle of each face which are furthest from any edge all become equal.

Defining "area" of a face which may not have co-planar vertices - and decdiding which point of such a face is furthest from an edge are issues not addressed here - except by noting that these are tricky issues - but that there are solutions to them.

Lastly, I note that it may be possible to create three new conjectures (equal angle, equal area, equal span) by dropping the constraint that the points must all remain on the surface of a sphere - and adding the constraint that the vertices of each face remain co-planar.

The equal-angle conjecture is associated with the problem of minimising the number of cuts during construction.

These conjectures are motivated by considerations of structures which could be covered by panes of glass - where co-planar faces is critical - and where having variations in the maximum unbroken span might result in variations in the strength of the panes, for a given thickness of material.