Mid Ridge Length of Great Icosahedron given Volume Calculator

Formula Used:

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

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 Great Icosahedron is attached. It is a key geometric parameter in understanding the structure of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( l_{Ridge(Mid)} \) — Mid Ridge Length of Great Icosahedron (m)
\( V \) — Volume of Great Icosahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives the mid ridge length from the volume of the Great Icosahedron using the golden ratio and cubic root relationships.

3. Importance of Mid Ridge Length Calculation

Details: Calculating the mid ridge length is essential for geometric analysis, architectural design applications, and understanding the spatial properties of the Great Icosahedron in mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the volume of the Great Icosahedron in cubic meters. 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, featuring 20 triangular faces that intersect each other.

Q2: Why is the golden ratio (1+√5)/2 used in the formula?
A: The golden ratio appears naturally in the geometry of icosahedra and their derivatives due to their five-fold symmetry properties.

Q3: What are typical values for Mid Ridge Length?
A: The mid ridge length varies with the volume, but typically ranges from a few centimeters to several meters depending on the scale of the polyhedron.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Great Icosahedron and its mid ridge length calculation.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, though practical accuracy depends on the precision of the input volume measurement.