Long Edge Of Pentagonal Icositetrahedron Given Midsphere Radius Calculator

Formula Used:

\[ Long\ Edge = \sqrt{(C_{tribonacci} + 1) \times (2 - C_{tribonacci})} \times Midsphere\ Radius \]

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 an important geometric measurement in this polyhedral structure.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ Long\ Edge = \sqrt{(C_{tribonacci} + 1) \times (2 - C_{tribonacci})} \times Midsphere\ Radius \]

Where:

\( C_{tribonacci} \) — Tribonacci constant (≈1.839286755214161)
\( Midsphere\ Radius \) — Radius of the midsphere of the Pentagonal Icositetrahedron

Explanation: This formula relates the long edge length to the midsphere radius through the mathematical constant known as the Tribonacci constant, which appears in various geometric relationships of this polyhedron.

3. Importance Of Long Edge Calculation

Details: Calculating the long edge is essential for understanding the geometric properties of the Pentagonal Icositetrahedron, designing physical models, and studying its mathematical characteristics in polyhedral geometry.

4. Using The Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and non-zero. The calculator will compute the corresponding long edge length using the established geometric relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is the Tribonacci constant?
A: The Tribonacci constant is a mathematical constant that appears in the Tribonacci sequence, similar to how the golden ratio appears in the Fibonacci sequence. It has the approximate value of 1.839286755214161.

Q2: 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 polyhedron of the snub cube.

Q3: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron. It's also known as the "edge-tangent sphere" radius.

Q4: Are there other edges in a Pentagonal Icositetrahedron?
A: Yes, the Pentagonal Icositetrahedron has edges of different lengths. The "long edge" refers specifically to the longest edge among them.

Q5: What are practical applications of this calculation?
A: This calculation is used in mathematical geometry, architectural design, crystal structure analysis, and computer graphics modeling of complex polyhedra.