Icosahedral Edge Length of Triakis Icosahedron given Midsphere Radius Calculator

Formula Used:

\[ Icosahedral\ Edge\ Length = \frac{4 \times Midsphere\ Radius}{1 + \sqrt{5}} \]

Icosahedral Edge Length:

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1. What is Icosahedral Edge Length of Triakis Icosahedron?

The Icosahedral Edge Length of Triakis Icosahedron refers to the length of the edges that form the underlying icosahedral structure of a Triakis Icosahedron. A Triakis Icosahedron is a Catalan solid that can be thought of as an icosahedron with a pyramid attached to each face.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ Icosahedral\ Edge\ Length = \frac{4 \times Midsphere\ Radius}{1 + \sqrt{5}} \]

Where:

Midsphere Radius (rₘ) — The radius of the sphere that is tangent to all edges of the Triakis Icosahedron
√5 — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the precise geometric relationship between the midsphere radius and the icosahedral edge length in a Triakis Icosahedron.

3. Importance of Icosahedral Edge Length Calculation

Details: Calculating the icosahedral edge length is essential for understanding the geometric properties of Triakis Icosahedrons, which have applications in crystallography, molecular modeling, and architectural design due to their unique symmetric properties.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and non-zero. The calculator will compute the corresponding icosahedral edge length using the mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid that consists of 60 isosceles triangular faces. It is the dual of the truncated dodecahedron.

Q2: What is the significance of the midsphere radius?
A: The midsphere radius is the radius of the sphere that touches all the edges of the polyhedron, providing important geometric information about the solid's proportions.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Triakis Icosahedron, assuming precise input values.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to the Triakis Icosahedron due to its unique geometric properties.

Q5: What are practical applications of this calculation?
A: This calculation is used in various fields including materials science, crystallography, architectural design, and 3D modeling where precise geometric relationships are required.