Ridge Length of Great Stellated Dodecahedron Calculator

Formula Used:

\[ l_{Ridge} = \frac{1 + \sqrt{5}}{2} \times l_{e} \]

Ridge Length of Great Stellated Dodecahedron:

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1. What is the Ridge Length of Great Stellated Dodecahedron?

The Ridge Length of Great Stellated Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Stellated Dodecahedron. It's an important geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{1 + \sqrt{5}}{2} \times l_{e} \]

Where:

\( l_{Ridge} \) — Ridge Length of Great Stellated Dodecahedron (m)
\( l_{e} \) — Edge Length of Great Stellated Dodecahedron (m)
\( \frac{1 + \sqrt{5}}{2} \) — The golden ratio (approximately 1.618)

Explanation: The ridge length is directly proportional to the edge length through the golden ratio, which appears frequently in the geometry of regular polyhedra.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for understanding the complete geometry of the great stellated dodecahedron, for construction purposes, and for mathematical analysis of this complex polyhedron.

4. Using the Calculator

Tips: Enter the edge length of the great stellated dodecahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the great stellated dodecahedron?
A: The great stellated dodecahedron is one of the Kepler-Poinsot polyhedra, formed by extending the faces of a regular dodecahedron until they intersect.

Q2: Why does the formula use the golden ratio?
A: The golden ratio appears naturally in the geometry of regular dodecahedra and their stellated forms, making it a fundamental constant in these calculations.

Q3: What are typical values for edge length?
A: Edge length can vary significantly depending on the scale of the polyhedron, from mathematical models to architectural structures.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the great stellated dodecahedron. Other polyhedra have different geometric relationships.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, as it's derived from the geometric properties of the perfect great stellated dodecahedron.