Proving # Equilateral Triangles on Sphere

In summary, two similar problems were discussed, one using N and the other using n. The first problem asked to show that N can only be 4, 8, or 20, while the second problem asked to show that n can only be 3, 4, or 5. The key difference is that N represents the total number of triangles on the sphere, while n represents the number of triangles meeting at one vertex. Using Euler's formula, it is possible to prove both statements.
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Homework Statement


Ok, I have this problem this week. (1)
[tex]
Consider\ a\ tiling\ of\ the\ unit\ sphere\ in\ \mathbb{R}^{3}\ by\ N\ equilateral\\
triangles\ so\ that\ the\ triangles\ meet\ full\ edge\ to\ full\ edge\ (and\ vertex\ to\ vertex).\\
Show\ that\ the\ only\ possibilities\ for\ N\ are\ N = 4,\ N = 8,\ or\ N = 20.[/tex]

Last week, we had the problem that follows (2)
[tex]
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,\ n = 5.[/tex]

Alright so my problem is that it seems to be nearly the exact same statement except (2) uses n while (1) uses N.

If that is true then wouldn't it be impossible to prove N can only be 4, 8, or 20 if I proved it was only 3, 4, or 5 last week.

Homework Equations


Area of each triangle = 3a - ∏

The Attempt at a Solution


Considering my question isn't for the solution rather than help understanding what it is saying/how this would be possible, I don't have any work yet.

If requested, I can show my answer to (2)
 
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Last edited:
  • #3


Alright thanks man, that's all I needed, I got it!
 

FAQ: Proving # Equilateral Triangles on Sphere

What is an equilateral triangle on a sphere?

An equilateral triangle on a sphere is a triangle with three equal sides and three equal angles, formed on the surface of a sphere. It is a special case of a spherical triangle, where all three sides are of equal length.

How can we prove that a triangle on a sphere is equilateral?

There are different ways to prove that a triangle on a sphere is equilateral. One method is to show that all three sides have equal length using the spherical law of cosines. Another method is to show that all three angles are equal using the spherical law of sines. Both methods require knowledge of spherical geometry and trigonometry.

What are the properties of an equilateral triangle on a sphere?

An equilateral triangle on a sphere has three equal sides and three equal angles. It is also inscribed in a single great circle of the sphere, meaning that all three vertices lie on the same circle. Additionally, the angles of an equilateral triangle on a sphere are all less than 180 degrees, unlike a planar equilateral triangle.

How is an equilateral triangle on a sphere different from an equilateral triangle in Euclidean geometry?

In Euclidean geometry, an equilateral triangle has three equal sides and three equal angles, and it lies on a flat plane. In contrast, an equilateral triangle on a sphere lies on the curved surface of the sphere, and its angles are measured differently due to the curvature of the sphere. Additionally, the sum of the angles in a spherical equilateral triangle is greater than 180 degrees, unlike a planar equilateral triangle where the sum is exactly 180 degrees.

What are the applications of proving equilateral triangles on a sphere?

Proving equilateral triangles on a sphere is important in various fields of science and mathematics. It is used in spherical trigonometry, navigation, and cartography. In addition, it has applications in physics, such as in the study of crystal structures and the analysis of electromagnetic fields on a spherical surface.

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