Long Edge Of Pentagonal Icositetrahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ Long Edge = \frac{\sqrt{[Tribonacci_C]+1}}{2} \times \frac{3\sqrt{\frac{22(5[Tribonacci_C]-1)}{(4[Tribonacci_C])-3}}}{SA:V \times \sqrt{\frac{11([Tribonacci_C]-4)}{2((20[Tribonacci_C])-37)}}} \]

Long Edge of Pentagonal Icositetrahedron:

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1. What is the Long Edge of Pentagonal Icositetrahedron?

The Long Edge of Pentagonal Icositetrahedron is the length of the longest edge which is the top edge of the axial-symmetric pentagonal faces of Pentagonal Icositetrahedron. It is a key geometric parameter in this complex polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ Long Edge = \frac{\sqrt{[Tribonacci_C]+1}}{2} \times \frac{3\sqrt{\frac{22(5[Tribonacci_C]-1)}{(4[Tribonacci_C])-3}}}{SA:V \times \sqrt{\frac{11([Tribonacci_C]-4)}{2((20[Tribonacci_C])-37)}}} \]

Where:

\( [Tribonacci_C] \) — Tribonacci constant (1.839286755214161)
\( SA:V \) — Surface to Volume Ratio of Pentagonal Icositetrahedron (1/m)

Explanation: This formula relates the long edge length to the surface-to-volume ratio using the mathematical properties of the Tribonacci constant and geometric relationships specific to the pentagonal icositetrahedron.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is crucial for understanding the geometric properties, structural integrity, and mathematical characteristics of pentagonal icositetrahedrons in crystallography, materials science, and mathematical geometry studies.

4. Using the Calculator

Tips: Enter the Surface to Volume Ratio (SA:V) in 1/m. The value must be positive and valid for accurate calculation of the long edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A pentagonal icositetrahedron is a Catalan solid with 24 pentagonal faces, 60 edges, and 38 vertices. It is the dual of the snub cube.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the real root of the equation x³ - x² - x - 1 = 0, approximately equal to 1.839286755214161.

Q3: Why is the Tribonacci constant used in this formula?
A: The Tribonacci constant appears naturally in the mathematical relationships governing the geometry of pentagonal icositetrahedrons and their geometric properties.

Q4: What are typical SA:V values for Pentagonal Icositetrahedrons?
A: The surface-to-volume ratio depends on the specific dimensions and scaling of the polyhedron, but typically ranges based on the geometric configuration.

Q5: Can this calculator be used for other polyhedrons?
A: No, this specific formula is designed exclusively for pentagonal icositetrahedrons due to their unique geometric properties and mathematical relationships.