Short Edge of Hexakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \frac{4 \times r_i}{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}} \]

Short Edge of Hexakis Icosahedron:

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1. What is the Short Edge of Hexakis Icosahedron?

The Short Edge of Hexakis Icosahedron is the length of the shortest edge that connects two adjacent vertices of the Hexakis Icosahedron. It is a crucial geometric parameter in understanding the structure and properties of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \frac{4 \times r_i}{\sqrt{\frac{15}{241} \times (275 + 119 \times \sqrt{5})}} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the shortest edge length based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the Hexakis Icosahedron.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for geometric analysis, structural design, and understanding the symmetry properties of the Hexakis Icosahedron in various mathematical and engineering applications.

4. Using the Calculator

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

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 insphere radius defined?
A: The insphere radius is the radius of the largest sphere that can fit inside the Hexakis Icosahedron, touching all its faces.

Q3: What are typical values for the short edge?
A: The short edge length varies depending on the size of the polyhedron, but it is always the smallest edge length in the structure.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect Hexakis Icosahedron shape and may not account for manufacturing tolerances or deformations in practical applications.

Q5: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Hexakis Icosahedron and its geometric properties.