Mid Ridge Length of Great Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ l_{Ridge(Mid)} = \frac{1+\sqrt{5}}{2} \times \frac{3\sqrt{3}(5+4\sqrt{5})}{\frac{1}{4}(25+9\sqrt{5}) \times \frac{RA}{V}} \]

Mid Ridge Length of Great Icosahedron:

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1. What is the Mid Ridge Length of Great Icosahedron?

The Mid Ridge Length of Great Icosahedron is the length of any of the edges that starts from the peak vertex and ends on the interior of the pentagon on which each peak of the Great Icosahedron is attached. It is an important geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge(Mid)} = \frac{1+\sqrt{5}}{2} \times \frac{3\sqrt{3}(5+4\sqrt{5})}{\frac{1}{4}(25+9\sqrt{5}) \times \frac{RA}{V}} \]

Where:

\( l_{Ridge(Mid)} \) — Mid Ridge Length of Great Icosahedron (m)
\( \frac{RA}{V} \) — Surface to Volume Ratio of Great Icosahedron (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the mid ridge length based on the surface to volume ratio of the Great Icosahedron, incorporating the golden ratio and various geometric constants.

3. Importance of Mid Ridge Length Calculation

Details: Calculating the mid ridge length is essential for understanding the geometric properties of the Great Icosahedron, including its symmetry, structural integrity, and spatial relationships between different components of this complex polyhedron.

4. Using the Calculator

Tips: Enter the surface to volume ratio of the Great Icosahedron in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is one of the four Kepler-Poinsot polyhedra, consisting of 20 triangular faces that intersect each other.

Q2: How is the surface to volume ratio defined?
A: The surface to volume ratio is the total surface area of the polyhedron divided by its volume, representing how much surface area exists per unit volume.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size and specific geometry of the Great Icosahedron, but it generally decreases as the size increases.

Q4: Why does the formula include the golden ratio?
A: The golden ratio (1+√5)/2 appears frequently in icosahedral geometry due to the mathematical properties and symmetries of these shapes.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Great Icosahedron and its unique geometric properties.