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Edge Length of Snub Dodecahedron given Volume Calculator

Formula Used:

\[ l_e = \left( \frac{V \cdot 6 \cdot \left(3 - \left( \left( \frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} + \left( \frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} \right)^2 \right)^{\frac{3}{2}}}{\left( \left(12 \cdot (3\phi + 1) \cdot \left( \left( \frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} + \left( \frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} \right)^2 - \left( (36\phi + 7) \cdot \left( \left( \frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} + \left( \frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} \right) \right) - (53\phi + 6) \right)} \right)^{\frac{1}{3}} \]

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1. What is the Edge Length of Snub Dodecahedron given Volume?

The edge length of a snub dodecahedron given its volume is calculated using a specific mathematical formula that relates the volume to the edge length through the golden ratio (φ). The snub dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles).

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \left( \frac{V \cdot 6 \cdot \left(3 - \left( \left( \frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} + \left( \frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} \right)^2 \right)^{\frac{3}{2}}}{\left( \left(12 \cdot (3\phi + 1) \cdot \left( \left( \frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} + \left( \frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} \right)^2 - \left( (36\phi + 7) \cdot \left( \left( \frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} + \left( \frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2} \right)^{\frac{1}{3}} \right) \right) - (53\phi + 6) \right)} \right)^{\frac{1}{3}} \]

Where:

Explanation: This complex formula derives from the geometric properties of the snub dodecahedron and involves the golden ratio, which is fundamental to its structure.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from volume is essential for understanding the scale and proportions of a snub dodecahedron, which has applications in mathematics, architecture, and molecular modeling.

4. Using the Calculator

Tips: Enter the volume of the snub dodecahedron in cubic meters. The volume must be a positive value. The calculator will compute the corresponding edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a snub dodecahedron?
A: A snub dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles), 150 edges, and 60 vertices.

Q2: Why does the formula use the golden ratio?
A: The golden ratio appears naturally in the proportions of the snub dodecahedron, making it fundamental to calculations involving this polyhedron.

Q3: What are typical values for edge length?
A: The edge length depends on the volume. For a unit volume (1 m³), the edge length is approximately 0.4-0.5 meters, but this varies with the specific dimensions.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the snub dodecahedron due to its unique geometric properties.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect snub dodecahedron, though practical measurements may have slight variations.

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