Long Ridge Length of Great Icosahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \frac{3\sqrt{3} \cdot (5 + 4\sqrt{5})}{\frac{1}{4} \cdot (25 + 9\sqrt{5}) \cdot \frac{S}{V}} \]

Long Ridge Length:

Unit Converter ▲

Unit Converter ▼

From:	To:

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:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \frac{3\sqrt{3} \cdot (5 + 4\sqrt{5})}{\frac{1}{4} \cdot (25 + 9\sqrt{5}) \cdot \frac{S}{V}} \]

Where:

\( l_{Ridge(Long)} \) — Long Ridge Length (meters)
\( \frac{S}{V} \) — Surface to Volume Ratio (1/meters)
\( \sqrt{2}, \sqrt{3}, \sqrt{5} \) — Square roots of 2, 3, and 5 respectively

Explanation: This formula relates the long ridge length to the surface-to-volume ratio of the Great Icosahedron through a complex mathematical relationship involving square roots and constants derived from the geometry of the shape.

3. Importance of Long Ridge Length Calculation

Details: Calculating the long ridge length is essential for understanding the geometric properties of the Great Icosahedron, which has applications in mathematics, architecture, and molecular modeling. It helps in determining the precise dimensions and proportions of this complex polyhedral structure.

4. Using the Calculator

Tips: Enter the surface-to-volume ratio in 1/meters. The value must be positive and greater than zero. 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, which are regular star polyhedra. It has 20 triangular faces that intersect each other.

Q2: How is the surface-to-volume ratio measured?
A: The surface-to-volume ratio is calculated by dividing the total surface area of the Great Icosahedron by its volume, both measured in appropriate units.

Q3: What are typical values for surface-to-volume ratio?
A: The surface-to-volume ratio depends on the size of the Great Icosahedron. Smaller polyhedra have higher ratios, while larger ones have lower ratios.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Great Icosahedron only. Other polyhedra have different geometric relationships.

Q5: What practical applications does this calculation have?
A: This calculation is primarily used in mathematical research, architectural design, and in understanding molecular structures that exhibit icosahedral symmetry.