Midsphere Radius Of Hexakis Icosahedron Given Medium Edge Calculator

Formula Used:

\[ r_m = \frac{5+3\sqrt{5}}{8} \times \frac{22 \times l_{medium}}{3(4+\sqrt{5})} \]

Midsphere Radius of Hexakis Icosahedron:

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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 is an important geometric property in the study of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{5+3\sqrt{5}}{8} \times \frac{22 \times l_{medium}}{3(4+\sqrt{5})} \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
\( l_{medium} \) — Medium Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the medium edge length of the Hexakis Icosahedron, incorporating the mathematical constant √5 which is characteristic of icosahedral geometry.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is crucial for understanding the geometric properties of the Hexakis Icosahedron, including its symmetry, packing efficiency, and spatial relationships. This measurement is particularly important in crystallography, molecular modeling, and advanced geometric studies.

4. Using the Calculator

Tips: Enter the medium edge length of the Hexakis Icosahedron in meters. The value must be positive and greater than zero. The calculator will automatically 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: How is the medium edge defined?
A: The medium edge of a Hexakis Icosahedron is the length of the edge that connects two non-adjacent and non-opposite vertices of the polyhedron.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the scale of the polyhedron. For a unit medium edge length, the midsphere radius is approximately 1.721 times the medium edge length.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Icosahedron. Other polyhedra have different formulas for calculating their midsphere radii.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 10 decimal places precision, which is sufficient for most geometric and engineering applications involving the Hexakis Icosahedron.