Long Ridge Length of Great Icosahedron Given Circumsphere Radius Calculator

Formula Used:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \frac{4 \cdot r_c}{\sqrt{50 + 22\sqrt{5}}} \]

Long Ridge Length of Great Icosahedron:

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1. What is Long Ridge Length of Great Icosahedron?

The Long Ridge Length of Great Icosahedron is the length of any of the edges that connects the peak vertex and adjacent vertex of the pentagon on which each peak of Great Icosahedron is attached. It is an important geometric measurement in the study of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \frac{4 \cdot r_c}{\sqrt{50 + 22\sqrt{5}}} \]

Where:

\( l_{Ridge(Long)} \) — Long Ridge Length of Great Icosahedron (m)
\( r_c \) — Circumsphere Radius of Great Icosahedron (m)

Explanation: This formula calculates the long ridge length based on the circumsphere radius, incorporating mathematical constants and geometric relationships specific to the Great Icosahedron.

3. Importance of Long Ridge Length Calculation

Details: Accurate calculation of the long ridge length is crucial for understanding the geometric properties of the Great Icosahedron, including its symmetry, surface area, volume, and other dimensional characteristics.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding long ridge length of the Great Icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is one of the four Kepler-Poinsot polyhedra, featuring 20 triangular faces that intersect each other, creating a complex star-shaped polyhedron.

Q2: How is the circumsphere radius defined?
A: The circumsphere radius is the radius of the sphere that contains the Great Icosahedron such that all the peak vertices lie on the sphere's surface.

Q3: What are typical values for these measurements?
A: The values depend on the specific size of the polyhedron. The circumsphere radius and long ridge length are proportional to each other based on the mathematical formula.

Q4: Are there any limitations to this calculation?
A: This calculation assumes a perfect geometric Great Icosahedron and may not account for manufacturing tolerances or material properties in physical implementations.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Icosahedron. Other polyhedra have different geometric relationships and require different formulas.