1. What is the Midsphere Radius of Hexakis Icosahedron?

The Midsphere Radius of Hexakis Icosahedron is defined as the radius of the sphere for which all the edges of the Hexakis Icosahedron become a tangent line on that sphere. It represents the sphere that touches all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
• \( l_e(Short) \) — Short Edge of Hexakis Icosahedron (m)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the length of the shortest edge of the Hexakis Icosahedron, using mathematical constants derived from the geometric properties of this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of the Hexakis Icosahedron. It helps in determining the sphere that is tangent to all edges of the polyhedron, which has applications in various fields including crystallography, molecular modeling, and computer graphics.

4. Using the Calculator

Tips: Enter the length of the short edge of the Hexakis Icosahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: What are the typical applications of this calculation?
A: This calculation is used in geometric modeling, crystallography, architectural design, and in the study of polyhedral structures in mathematics and physics.

Q3: How accurate is this formula?
A: The formula is mathematically exact for a perfect Hexakis Icosahedron and provides precise results when correct input values are used.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, or convert the result from meters to your desired unit.

Q5: What is the significance of the midsphere in polyhedral geometry?
A: The midsphere (or intersphere) is significant because it touches all edges of the polyhedron, providing important information about the polyhedron's symmetry and spatial relationships.