# Triangle on a Sphere Question Interpretation

• tainted
In summary, the conversation discusses the tiling of a unit sphere in ##\mathbb R^3## by equilateral triangles and the relationship between the number of triangles meeting at a vertex and the only possible values for that number, which are ##n=3##, ##n=4##, or ##n=5##. The solution involves considering the area of a triangle and its relationship to the number of triangles that can fit on a sphere, as well as a correspondence to some platonic solids.

## Homework Statement

Consider a tiling of the unit sphere in ##\mathbb R^3## by equilateral triangles so that the triangles
meet full edge to full edge (and vertex to vertex). Suppose n such triangles meet an one
vertex. Show that the only possibilities for n are
## n=3 ##, ##n = 4##, or ##n=5##

## The Attempt at a Solution

I guess the main thing I need help with is interpretation of the question.

Thank you

A (flat) triangle won't fit on the surface of a sphere. Do you mean spherical triangles?

Hint: there's a relation between the angles of a triangle and it's area. Figure out the area of a triangle if n of them share a vertex. How many will fit on a sphere? If you think about there is also a easy correspondence between these and some of the platonic solids.