If we first mark points at one degree increments on an equatorial plane intersection of a sphere, then we have 360 points. Next, we mark another set of points equidistant from the original points above and below the plane, then equilateral triangles are formed from the set of all points, yielding another 720 points and a subtotal of 720 equilateral triangles around the equator. This process is then repeated across the surface of the sphere to create a triangular tiling. How many total triangles and vertices are there?

Let's break down the problem step by step to find the total number of triangles and vertices on the sphere.

### Step 1: Understanding the Initial Configuration
- We start by marking **360 points** at one-degree increments along the equator of the sphere.
- Next, we mark another set of points equidistant from these original points above and below the equator. This results in another **720 points** (360 above and 360 below the equator).
- The first set of points along the equator, combined with the points above and below, form **equilateral triangles**.

### Step 2: Counting the Equilateral Triangles Formed Around the Equator
- Each point on the equator is connected to the points directly above and below it to form **two equilateral triangles** (one triangle above and one below the equator).
- There are **360 points** on the equator, each contributing to **2 triangles**.
- Thus, the total number of equilateral triangles around the equator is:
  \[
  \text{Triangles around the equator} = 360 \times 2 = 720
  \]

### Step 3: Extending the Triangular Tiling Across the Entire Sphere
- The triangular tiling process is then repeated across the entire surface of the sphere. To find the total number of triangles on the sphere, we consider that the entire sphere is being tiled with equilateral triangles.

### Step 4: Counting the Total Triangles and Vertices
- **Vertices:** Each point corresponds to a vertex. The vertices consist of those along the equator (360), plus those above and below the equator (360 + 360 = 720).
  \[
  \text{Total vertices} = 360 + 720 = 1080
  \]
  
- **Triangles:** For every triangle along the equator, there are similar triangles extending across the rest of the sphere. To find the total number of triangles covering the sphere, we must account for the total surface area of the sphere being evenly covered by these equilateral triangles.

  Considering that there are \(720\) triangles at the equator, and that these cover a narrow band around the equator, the entire sphere should be covered similarly. Thus, the total number of triangles on the sphere would be much larger.

However, the number of triangles tiling the sphere should satisfy the general formula for a triangulation of a spherical surface:
\[
T = 2V - 4
\]
Where \( V \) is the number of vertices. Substituting \( V = 1080 \):
\[
T = 2 \times 1080 - 4 = 2160 - 4 = 2156
\]
Thus, the total number of triangles on the sphere is **2156**.

### Summary
- **Total number of triangles:** 2156
- **Total number of vertices:** 1080

Would you like further details or have any additional questions?

### Relative Questions:
1. How does the number of triangles change if the sphere's radius is doubled?
2. What is the significance of the Euler characteristic in spherical tiling?
3. How can this tiling be generalized to other polyhedra, like the icosahedron?
4. How does the number of edges relate to the number of vertices and triangles?
5. Can you explain the process of constructing similar tilings on hyperbolic surfaces?

### Tip:
For any triangulation of a sphere, the Euler characteristic formula \( V - E + F = 2 \) provides a powerful tool to relate vertices (V), edges (E), and faces (F), helping to confirm the consistency of the tiling.

Explore the mathematical concept of spherical tiling and calculate the total number of triangles and vertices on a sphere. Detailed step-by-step solution and application of Euler's formula for polyhedra. Suitable for advanced high school students and beyond.

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