Truncated Icosidodecahedron Edge of Hexakis Icosahedron given Midsphere Radius Calculator

Formula Used:

\[ \text{Truncated Edge of Hexakis Icosahedron} = \frac{5}{2 \times \sqrt{15 \times (5 - \sqrt{5})}} \times \frac{8 \times \text{Midsphere Radius of Hexakis Icosahedron}}{5 + 3 \times \sqrt{5}} \]

Truncated Edge of Hexakis Icosahedron:

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1. What is Truncated Edge of Hexakis Icosahedron?

The Truncated Edge of Hexakis Icosahedron is the length of the edges of a Hexakis Icosahedron that is created by truncating the vertices of an Icosidodecahedron. It represents a specific geometric measurement in this complex polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Truncated Edge} = \frac{5}{2 \times \sqrt{15 \times (5 - \sqrt{5})}} \times \frac{8 \times \text{Midsphere Radius}}{5 + 3 \times \sqrt{5}} \]

Where:

Midsphere Radius - The radius of the sphere tangent to all edges of the Hexakis Icosahedron (in meters)
√5 - Square root of 5 (approximately 2.23607)

Explanation: This formula mathematically relates the truncated edge length to the midsphere radius through geometric relationships derived from the polyhedron's structure.

3. Importance of Truncated Edge Calculation

Details: Calculating the truncated edge is essential for understanding the geometric properties of Hexakis Icosahedrons, which have applications in crystallography, architecture, and mathematical modeling of complex structures.

4. Using the Calculator

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

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, featuring 120 faces, 180 edges, and 62 vertices.

Q2: What does "truncated" mean in this context?
A: Truncation refers to the process of cutting off the vertices of a polyhedron, creating new faces where the vertices used to be.

Q3: How is the midsphere radius defined?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q4: What are typical values for midsphere radius?
A: The midsphere radius depends on the specific dimensions of the Hexakis Icosahedron, but it's typically in the range of centimeters to meters for practical applications.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Icosahedron and its relationship between truncated edge length and midsphere radius.