Short Edge Of Pentagonal Hexecontahedron Given Long Edge Calculator

Formula Used:

\[ \text{Short Edge} = \frac{31 \times \text{Long Edge}}{((7\phi + 2) + (5\phi - 3) + 2(8 - 3\phi)) \times \sqrt{2 + 2 \times 0.4715756}} \]

Short Edge of Pentagonal Hexecontahedron:

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1. What is the Short Edge of Pentagonal Hexecontahedron?

The Short Edge of Pentagonal Hexecontahedron is the length of the shortest edge which is the base and middle edge of the axial-symmetric pentagonal faces of the Pentagonal Hexecontahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{31 \times \text{Long Edge}}{((7\phi + 2) + (5\phi - 3) + 2(8 - 3\phi)) \times \sqrt{2 + 2 \times 0.4715756}} \]

Where:

\(\phi\) — Golden ratio (approximately 1.618034)
Long Edge — Length of the longest edge of the pentagonal hexecontahedron

Explanation: This formula calculates the short edge length based on the given long edge length using the mathematical properties of the pentagonal hexecontahedron and the golden ratio.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for understanding the geometric properties of pentagonal hexecontahedrons, which are complex polyhedra with 60 pentagonal faces. This calculation is important in crystallography, materials science, and mathematical geometry studies.

4. Using the Calculator

Tips: Enter the long edge length in meters. The value must be positive and greater than zero. The calculator will compute the corresponding short edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A pentagonal hexecontahedron is a polyhedron with 60 pentagonal faces. It's one of the Catalan solids and is the dual of the snub dodecahedron.

Q2: Why is the golden ratio used in this formula?
A: The golden ratio appears naturally in the geometry of pentagonal shapes and is fundamental to the mathematical properties of pentagonal hexecontahedrons.

Q3: What are typical values for long and short edges?
A: The ratio between long and short edges is fixed by the geometry. For a given pentagonal hexecontahedron, the short edge is always shorter than the long edge by a specific proportion.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to pentagonal hexecontahedrons due to their unique geometric properties.

Q5: What practical applications does this calculation have?
A: This calculation is used in crystallography, nanotechnology, architectural design, and mathematical research involving complex polyhedral structures.