Triangle on a Sphere Question Interpretation

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.
  • #1
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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##

Homework Equations


The Attempt at a Solution


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

Thank you
 
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  • #2
A (flat) triangle won't fit on the surface of a sphere. Do you mean spherical triangles?
 
  • #3
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.
 

1. What is the concept of "Triangle on a Sphere Question Interpretation"?

The concept of "Triangle on a Sphere Question Interpretation" refers to a mathematical problem in which a triangle is inscribed on the surface of a sphere, and the angles and sides of the triangle are measured and analyzed using principles of spherical geometry.

2. What are the key principles of spherical geometry used in "Triangle on a Sphere Question Interpretation"?

The key principles of spherical geometry used in "Triangle on a Sphere Question Interpretation" are the fact that the sum of the angles of a triangle on a sphere is greater than 180 degrees, the Law of Sines and Cosines, and the formula for calculating the surface area of a sphere.

3. How is "Triangle on a Sphere Question Interpretation" relevant to real-world applications?

"Triangle on a Sphere Question Interpretation" has many real-world applications, such as in navigation and map-making, astronomy, and geodesy. It is also used in various fields of science and engineering, such as geology, physics, and architecture.

4. Is "Triangle on a Sphere Question Interpretation" limited to equilateral triangles?

No, "Triangle on a Sphere Question Interpretation" can be applied to any type of triangle on a sphere, including equilateral, isosceles, and scalene triangles. The principles of spherical geometry used in this concept apply to all types of triangles.

5. Can "Triangle on a Sphere Question Interpretation" be solved using traditional Euclidean geometry?

No, "Triangle on a Sphere Question Interpretation" requires the use of spherical geometry, which has different principles and formulas than traditional Euclidean geometry. It takes into account the curvature of the sphere's surface, which cannot be represented in a flat, two-dimensional plane.

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