1. What is the Midsphere Radius of Truncated Icosahedron?

The Midsphere Radius of Truncated Icosahedron is the radius of the sphere for which all the edges of the Truncated Icosahedron become a tangent line on that sphere. It is a key geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_m \) — Midsphere Radius of Truncated Icosahedron (m)
• \( l_{e(icosahedron)} \) — Icosahedral Edge Length of Truncated Icosahedron (m)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula relates the midsphere radius to the original icosahedral edge length from which the truncated icosahedron is derived.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and materials science for understanding the spatial properties and packing characteristics of truncated icosahedral structures.

4. Using the Calculator

Tips: Enter the icosahedral edge length in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 12 regular pentagonal faces, 20 regular hexagonal faces, 90 edges, and 60 vertices.

Q2: What is the significance of the golden ratio in this formula?
A: The term (1+√5)/4 appears in the formula, which is related to the golden ratio φ = (1+√5)/2, a fundamental mathematical constant.

Q3: How is the truncated icosahedron used in real-world applications?
A: The truncated icosahedron is best known as the shape of a soccer ball and is also found in molecular structures like fullerenes (buckyballs).

Q4: What are the units for this calculation?
A: The calculator uses meters, but the formula works with any consistent unit of length.

Q5: Can this formula be derived from first principles?
A: Yes, the formula can be derived through geometric analysis of the relationships between the various elements of the truncated icosahedron.