Pentagram Chord of Great Stellated Dodecahedron Given Ridge Length Calculator

Formula Used:

\[ l_c = \frac{(2 + \sqrt{5}) \times (2 \times l_{\text{Ridge}})}{1 + \sqrt{5}} \]

Pentagram Chord of Great Stellated Dodecahedron:

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1. What is the Pentagram Chord of Great Stellated Dodecahedron?

The Pentagram Chord of Great Stellated Dodecahedron is the distance between any pair of non-adjacent peak vertices of the pentagram corresponding to the Great Stellated Dodecahedron. It represents a key geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_c = \frac{(2 + \sqrt{5}) \times (2 \times l_{\text{Ridge}})}{1 + \sqrt{5}} \]

Where:

\( l_c \) — Pentagram Chord of Great Stellated Dodecahedron (m)
\( l_{\text{Ridge}} \) — Ridge Length of Great Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the ridge length and the pentagram chord in a Great Stellated Dodecahedron, incorporating the golden ratio properties inherent in this geometric shape.

3. Importance of Pentagram Chord Calculation

Details: Calculating the pentagram chord is essential for understanding the geometric properties and proportions of the Great Stellated Dodecahedron. It helps in architectural design, mathematical modeling, and studying the symmetry properties of this complex polyhedron.

4. Using the Calculator

Tips: Enter the ridge length in meters. The value must be positive and non-zero. The calculator will compute the corresponding pentagram chord using the mathematical relationship between these two measurements.

5. Frequently Asked Questions (FAQ)

Q1: What is a 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, creating a star-shaped polyhedron.

Q2: How is the ridge length defined?
A: The ridge length is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Stellated Dodecahedron.

Q3: Why does the formula include √5?
A: The square root of 5 appears because the Great Stellated Dodecahedron incorporates golden ratio proportions, and √5 is fundamental to golden ratio mathematics.

Q4: What are typical values for ridge length?
A: The ridge length can vary depending on the specific implementation, but it's typically a positive real number representing the geometric measurement in meters or other length units.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula and calculator are designed specifically for the Great Stellated Dodecahedron and its unique geometric properties.