Long Edge of Deltoidal Hexecontahedron Given Short Edge Calculator

Formula Used:

\[ \text{Long Edge} = \frac{22 \times \text{Short Edge}}{3 \times (7 - \sqrt{5})} \]

Long Edge of Deltoidal Hexecontahedron:

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1. What is the Long Edge of Deltoidal Hexecontahedron?

The Long Edge of Deltoidal Hexecontahedron is the length of the longest edge of the identical deltoidal faces of a Deltoidal Hexecontahedron. It is a key geometric parameter in understanding the properties of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Long Edge} = \frac{22 \times \text{Short Edge}}{3 \times (7 - \sqrt{5})} \]

Where:

\( \text{Short Edge} \) — The length of the shortest edge of the deltoidal faces (in meters)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes a precise mathematical relationship between the long and short edges of a Deltoidal Hexecontahedron, derived from its geometric properties.

3. Importance of Long Edge Calculation

Details: Calculating the long edge is essential for geometric analysis, 3D modeling, and understanding the spatial relationships within a Deltoidal Hexecontahedron. It helps in determining various other properties of the polyhedron.

4. Using the Calculator

Tips: Enter the short edge value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding long edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: A Deltoidal Hexecontahedron is a Catalan solid with 60 congruent deltoidal faces. It is the dual polyhedron of the rhombicosidodecahedron.

Q2: Why is the formula dependent on √5?
A: The square root of 5 appears naturally in the geometry of the Deltoidal Hexecontahedron due to its relationship with the golden ratio and the polyhedron's symmetry properties.

Q3: What are the typical applications of this calculation?
A: This calculation is used in mathematical geometry, architectural design, crystallography, and in the study of polyhedral structures.

Q4: How accurate is this formula?
A: The formula is mathematically exact for a perfect Deltoidal Hexecontahedron. The accuracy of the result depends on the precision of the input value.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Deltoidal Hexecontahedron. Other polyhedra have different geometric relationships between their edges.