Short Edge Of Deltoidal Hexecontahedron Given Midsphere Radius Calculator

Formula Used:

\[ \text{Short Edge} = \frac{3}{22} \times (7 - \sqrt{5}) \times \frac{20 \times \text{Midsphere Radius}}{3 \times (5 + 3\sqrt{5})} \]

Short Edge of Deltoidal Hexecontahedron:

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1. What is the Short Edge of Deltoidal Hexecontahedron?

The Short Edge of Deltoidal Hexecontahedron is the length of the shortest edge of the identical deltoidal faces that make up this polyhedron. It is an important geometric measurement in understanding the structure and properties of this complex shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ \text{Short Edge} = \frac{3}{22} \times (7 - \sqrt{5}) \times \frac{20 \times \text{Midsphere Radius}}{3 \times (5 + 3\sqrt{5})} \]

Where:

\( \text{Midsphere Radius} \) — The radius of the midsphere of the deltoidal hexecontahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the relationship between the midsphere radius and the shortest edge length of the deltoidal hexecontahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for understanding the geometric properties of deltoidal hexecontahedrons, which have applications in crystallography, architecture, and mathematical modeling of complex structures.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding short edge length of the deltoidal hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a deltoidal hexecontahedron?
A: A deltoidal hexecontahedron is a Catalan solid with 60 identical deltoidal faces, 120 edges, and 62 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the deltoidal hexecontahedron.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to the deltoidal hexecontahedron due to its unique geometric properties.

Q5: What are practical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, crystallography, and mathematical research involving polyhedral structures.