Long Edge Of Pentagonal Icositetrahedron Given Volume Calculator

Formula Used:

\[ Long Edge = \frac{\sqrt{[Tribonacci_C]+1}}{2} \times Volume^{\frac{1}{3}} \times \left( \frac{2 \times (20 \times [Tribonacci_C] - 37)}{11 \times ([Tribonacci_C] - 4)} \right)^{\frac{1}{6}} \]

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 mathematical formula:

\[ Long Edge = \frac{\sqrt{[Tribonacci_C]+1}}{2} \times Volume^{\frac{1}{3}} \times \left( \frac{2 \times (20 \times [Tribonacci_C] - 37)}{11 \times ([Tribonacci_C] - 4)} \right)^{\frac{1}{6}} \]

Where:

\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)
\( Volume \) — Volume of Pentagonal Icositetrahedron in cubic meters

Explanation: This formula relates the long edge length to the volume of the pentagonal icositetrahedron using the Tribonacci constant and various mathematical operations.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is essential for understanding the geometric properties of pentagonal icositetrahedrons, which have applications in crystallography, materials science, and mathematical research on polyhedral structures.

4. Using the Calculator

Tips: Enter the volume of the pentagonal icositetrahedron in cubic meters. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A pentagonal icositetrahedron is a Catalan solid with 24 pentagonal faces, 38 vertices, and 60 edges. 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: What are typical volume values for these structures?
A: Volume values depend on the specific dimensions, but typically range from very small (nanometer scale) to larger geometric models.

Q4: Are there other edges in a pentagonal icositetrahedron?
A: Yes, besides the long edge, pentagonal icositetrahedrons also have medium and short edges, creating a complex polyhedral structure.

Q5: What applications does this calculation have?
A: This calculation is important in crystallography, material science research, and mathematical studies of polyhedral geometry and symmetry.