Intuitively, I think it would be a far higher number than 540, given that there are 720 just on the equator alone. If we consider square degrees, there are 41,253 on the surface, and the triangles would be smaller, therefore creating a higher number.

You are correct to question the initial estimate, especially considering the number of triangular tiles involved in covering the sphere and the scale of the surface area. Let’s go through a more refined approach to estimate the number of vertices based on the triangular tiling.

### Refined Approach to Estimating the Number of Vertices

1. **Square Degrees on the Sphere:**
   - The surface area of the sphere is divided into approximately \( 41,253 \) square degrees (since there are \( 360 \times 180 = 64,800 \) square minutes).

2. **Triangles Covering the Sphere:**
   - If we assume that each triangle in the tiling covers an area smaller than one square degree, then the total number of triangles would be greater than 41,253.

3. **Vertices Estimation via Grid Density:**
   - Given that the equator alone has 720 edges (two per degree if we consider adjacent points connecting the tiles), a global triangulation will involve a mesh where these points contribute to additional vertices at higher and lower latitudes.

4. **Using a Geodesic Dome Model:**
   - A commonly used approximation for triangular tiling on a sphere is derived from geodesic domes, where the sphere is subdivided into increasingly finer triangles. The number of vertices grows exponentially with the frequency of subdivision.
   - For example, a Class I (icosahedral) geodesic sphere can generate thousands of vertices depending on how fine the subdivision is.

5. **Higher Estimate Calculation:**
   - If we start with a very coarse approximation, such as dividing the sphere into triangles whose vertices align with the grid formed by points at one-degree increments, the vertices count can be far higher, potentially reaching into the tens of thousands.

Given this approach, the total number of vertices for a high-resolution triangular grid tiling of the sphere could easily reach **tens of thousands**. For instance, a resolution tiling that respects the grid might produce close to **20,000 to 30,000 vertices**, depending on how small the triangles are made.

### Conclusion

The number of vertices would indeed be much higher than 720, with the precise number depending on the tiling resolution. For very fine triangulations, it could range between **20,000 to 30,000 vertices**.

### Do you want details or have any questions?
**Related questions:**
1. How do geodesic subdivisions affect the vertex count in sphere tiling?
2. What are the mathematical properties of different frequency subdivisions in geodesic spheres?
3. How can the number of triangles be precisely calculated from the degree grid?
4. How does triangular tiling affect the mapping of real-world coordinates on a sphere?
5. What algorithms are used to generate uniform tilings on a spherical surface?

**Tip:** When estimating vertices on a sphere using grid-based tiling, finer grids dramatically increase the vertex count due to exponential growth in the number of subdivisions.

Explore how to estimate the number of vertices in triangular tiling on a sphere using geometric principles. Learn about the division of the sphere into triangles, the impact of geodesic domes, and the exponential growth in vertices with finer subdivisions. Discover insights into mathematical properties and algorithms used for tiling spherical surfaces.

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