Long Ridge Length of Great Icosahedron Calculator

Formula Used:

\[ \text{Long Ridge Length} = \frac{\sqrt{2} \times (5 + 3\sqrt{5})}{10} \times \text{Edge Length} \]

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:

\[ \text{Long Ridge Length} = \frac{\sqrt{2} \times (5 + 3\sqrt{5})}{10} \times \text{Edge Length} \]

Where:

\(\sqrt{2}\) — Square root of 2 (approximately 1.4142)
\(\sqrt{5}\) — Square root of 5 (approximately 2.2361)
Edge Length — The length of the edges of the Great Icosahedron

Explanation: This formula calculates the long ridge length based on the fundamental edge length of the Great Icosahedron, incorporating mathematical constants that define its geometric properties.

3. Importance of Long Ridge Length Calculation

Details: Accurate calculation of the long ridge length is crucial for understanding the geometric structure of the Great Icosahedron, for architectural applications, mathematical modeling, and in the study of complex polyhedra in geometry and crystallography.

4. Using the Calculator

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

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 this different from a regular icosahedron?
A: While both have 20 triangular faces, the Great Icosahedron is a star polyhedron where the faces intersect, unlike the regular icosahedron which is convex.

Q3: What are the practical applications of this calculation?
A: This calculation is used in mathematical research, architectural design of complex structures, and in understanding the geometric properties of star polyhedra.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Icosahedron as it's derived from its unique geometric properties.

Q5: What units should I use for the edge length?
A: The calculator uses meters, but you can use any consistent unit as the result will be in the same unit as the input.