Circumsphere Radius of Great Icosahedron given Long Ridge Length Calculator

Formula Used:

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

Circumsphere Radius of Great Icosahedron:

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1. What is the Circumsphere Radius of Great Icosahedron?

The Circumsphere Radius of a Great Icosahedron is the radius of the sphere that contains the polyhedron in such a way that all the vertices lie on the sphere's surface. It is a fundamental geometric property used in polyhedral geometry and 3D modeling.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_c \) — Circumsphere Radius of Great Icosahedron (m)
\( l_{Ridge(Long)} \) — Long Ridge Length of Great Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{2} \) — Square root of 2 (approximately 1.41421)

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

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the Great Icosahedron, which is important in fields such as crystallography, architectural design, and mathematical modeling of complex geometric structures.

4. Using the Calculator

Tips: Enter the Long Ridge Length of the Great Icosahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is one of the four Kepler-Poinsot polyhedra. It is a non-convex polyhedron with 20 triangular faces and 12 vertices.

Q2: How is the Long Ridge Length defined?
A: The Long Ridge Length is the length of the edges that connect the peak vertex to the adjacent vertices of the pentagon on which each peak of the Great Icosahedron is attached.

Q3: What are typical values for the Circumsphere Radius?
A: The circumsphere radius depends on the scale of the polyhedron. For a unit Great Icosahedron, the circumsphere radius is approximately 0.935, but this varies with different long ridge lengths.

Q4: Are there limitations to this calculation?
A: This formula is specific to the Great Icosahedron geometry. It assumes perfect mathematical construction and may not account for manufacturing tolerances in physical models.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Great Icosahedron. Other polyhedra have different formulas for calculating their circumsphere radii.