Short Edge of Deltoidal Hexecontahedron given Insphere Radius Calculator

Formula Used:

\[ \text{Short Edge} = \frac{3}{22} \times (7 - \sqrt{5}) \times \frac{2 \times r_i}{3 \times \sqrt{\frac{135 + 59\sqrt{5}}{205}}} \]

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 complex polyhedron. It is an important geometric measurement in understanding the structure and properties of this shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{3}{22} \times (7 - \sqrt{5}) \times \frac{2 \times r_i}{3 \times \sqrt{\frac{135 + 59\sqrt{5}}{205}}} \]

Where:

\( r_i \) — Insphere Radius of Deltoidal Hexecontahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the shortest edge length based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the deltoidal hexecontahedron.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is crucial for understanding the geometric properties, symmetry, and spatial relationships within the deltoidal hexecontahedron. It helps in various applications including crystallography, architectural design, and mathematical modeling.

4. Using the Calculator

Tips: Enter the insphere 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 deltoidal (kite-shaped) faces, 62 vertices, and 120 edges.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, touching all its faces.

Q3: Are there different edge lengths in a deltoidal hexecontahedron?
A: Yes, the deltoidal hexecontahedron has both short and long edges, with the short edges being the ones connecting the vertices of the deltoidal faces.

Q4: What are typical applications of this calculation?
A: This calculation is used in geometry research, architectural design, and in understanding the properties of complex polyhedra.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, though practical measurements may have precision limitations based on input accuracy.