Long Edge Of Pentagonal Hexecontahedron Given Snub Dodecahedron Edge Calculator

Formula Used:

\[ Long Edge = \left( \frac{Snub Dodecahedron Edge}{\sqrt{2+2 \times 0.4715756}} \times \sqrt{2+2 \times 0.4715756} \right) \times \frac{(7 \times \phi + 2) + (5 \times \phi - 3) + 2 \times (8 - 3 \times \phi)}{31} \]

Long Edge of Pentagonal Hexecontahedron:

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1. What is the Long Edge of Pentagonal Hexecontahedron?

The Long Edge of Pentagonal Hexecontahedron is the length of the longest edge which is the top edge of the axial-symmetric pentagonal faces of Pentagonal Hexecontahedron. It is an important geometric measurement in this complex polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

\( Snub Dodecahedron Edge \) — Length of any edge of the Snub Dodecahedron
\( \phi \) — Golden ratio constant (approximately 1.618034)
0.4715756 — Specific coefficient used in the calculation

Explanation: This formula establishes the mathematical relationship between the Snub Dodecahedron edge length and the long edge of its dual polyhedron, the Pentagonal Hexecontahedron.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is crucial for understanding the geometric properties of Pentagonal Hexecontahedron, including its symmetry, surface area, volume, and other dimensional characteristics.

4. Using the Calculator

Tips: Enter the Snub Dodecahedron edge length in meters. The value must be positive and valid. The calculator will compute the corresponding long edge of the Pentagonal Hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between Snub Dodecahedron and Pentagonal Hexecontahedron?
A: The Pentagonal Hexecontahedron is the dual polyhedron of the Snub Dodecahedron, meaning they have a reciprocal relationship where vertices correspond to faces and vice versa.

Q2: Why is the golden ratio (φ) used in this formula?
A: The golden ratio appears frequently in the geometry of regular and semi-regular polyhedra due to its unique mathematical properties and aesthetic proportions.

Q3: What are the typical values for the long edge?
A: The long edge length depends on the size of the Snub Dodecahedron. For a unit Snub Dodecahedron (edge = 1), the long edge is approximately 0.797 units.

Q4: Can this calculator be used for any size of Snub Dodecahedron?
A: Yes, the formula is scalable and works for any positive edge length of the Snub Dodecahedron.

Q5: What are the practical applications of this calculation?
A: This calculation is primarily used in mathematical geometry, 3D modeling, architectural design, and the study of polyhedral structures.