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Edge Length of Great Dodecahedron Formula:
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The Edge Length of Great Dodecahedron is the distance between any pair of adjacent peak vertices of the Great Dodecahedron. It is a fundamental geometric measurement that defines the size and proportions of this complex polyhedron.
The calculator uses the mathematical formula:
Where:
Explanation: This formula establishes the precise mathematical relationship between the ridge length and edge length of a Great Dodecahedron, based on its geometric properties and the golden ratio.
Details: The formula \( l_e = \frac{2 \times l_{Ridge}}{\sqrt{5} - 1} \) derives from the geometric relationships within the Great Dodecahedron, where the denominator \( \sqrt{5} - 1 \) relates to the golden ratio φ (approximately 0.618), a fundamental constant that appears frequently in pentagonal symmetry.
Tips: Enter the Ridge Length of the Great Dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding Edge Length based on the mathematical relationship between these two measurements.
Q1: What is a Great Dodecahedron?
A: The Great Dodecahedron is one of the Kepler-Poinsot polyhedra, consisting of 12 pentagonal faces that intersect each other, creating a complex star-shaped polyhedron.
Q2: What is the relationship between edge length and ridge length?
A: The edge length is approximately 1.618 times the ridge length, which corresponds to the golden ratio φ (1.618...), reflecting the pentagonal symmetry of the polyhedron.
Q3: Can this formula be used for any Great Dodecahedron?
A: Yes, this formula applies to all regular Great Dodecahedra, as it represents a fundamental geometric relationship that remains constant regardless of size.
Q4: What are the practical applications of this calculation?
A: This calculation is essential in geometric modeling, 3D design, architectural applications, and mathematical studies of polyhedral structures.
Q5: How accurate is this calculation?
A: The calculation is mathematically exact, provided the input ridge length is measured accurately. The result is precise to the number of decimal places specified.