Insphere Radius of Hexakis Icosahedron given Truncated Icosidodecahedron Edge Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \times (275 + 119\sqrt{5})}}{4} \times \frac{2}{5} \times l_e \times \sqrt{15(5 - \sqrt{5})} \]

Insphere Radius of Hexakis Icosahedron:

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1. What is the Insphere Radius of Hexakis Icosahedron?

The Insphere Radius of a Hexakis Icosahedron is defined as the radius of the sphere that is contained by the Hexakis Icosahedron in such a way that all the faces just touch the sphere. It's an important geometric property of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{15}{241} \times (275 + 119\sqrt{5})}}{4} \times \frac{2}{5} \times l_e \times \sqrt{15(5 - \sqrt{5})} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( l_e \) — Truncated Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula calculates the radius of the largest sphere that can fit inside the Hexakis Icosahedron while touching all its faces, based on the truncated edge length.

3. Importance of Insphere Radius Calculation

Details: The insphere radius is crucial in geometry and materials science for understanding the packing properties, volume relationships, and spatial characteristics of complex polyhedra like the Hexakis Icosahedron.

4. Using the Calculator

Tips: Enter the truncated edge length of the Hexakis Icosahedron in meters. The value must be positive and greater than zero.

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 icosahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: How is this different from a regular Icosahedron?
A: A regular icosahedron has 20 triangular faces, while a Hexakis Icosahedron has 120 scalene triangular faces, making it a much more complex polyhedron.

Q3: What practical applications does this calculation have?
A: This calculation is important in crystallography, nanotechnology, and the study of fullerene molecules which often have icosahedral symmetry.

Q4: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Hexakis Icosahedron, which involves the golden ratio (φ) and its mathematical properties.

Q5: Can this formula be simplified?
A: While some terms can be precomputed as constants, the formula fundamentally reflects the complex relationship between the truncated edge and the insphere radius in this specific polyhedron.