Short Edge Of Hexakis Icosahedron Given Truncated Icosidodecahedron Edge Calculator

Formula Used:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \frac{2}{5} \times \sqrt{15 \times (5 - \sqrt{5})} \times \text{Truncated Edge} \]

Short Edge of Hexakis Icosahedron:

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1. What is the Short Edge of Hexakis Icosahedron?

The Short Edge of Hexakis Icosahedron is the length of the shortest edge that connects two adjacent vertices of the Hexakis Icosahedron. It is derived from the truncated edge of an Icosidodecahedron through a specific geometric transformation.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \frac{2}{5} \times \sqrt{15 \times (5 - \sqrt{5})} \times \text{Truncated Edge} \]

Where:

Truncated Edge — Length of the truncated edge of the Icosidodecahedron (m)
√5 — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the shortest edge length of a Hexakis Icosahedron based on the truncated edge length of an Icosidodecahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Short Edge Calculation

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

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: How is this formula derived?
A: The formula is derived from geometric relationships between the Hexakis Icosahedron and its parent polyhedron, the Icosidodecahedron, using mathematical constants and proportions.

Q3: What are typical values for the truncated edge?
A: The truncated edge length can vary depending on the specific polyhedron, but it's typically a positive real number representing a physical or theoretical length.

Q4: Can this calculator handle very large or small values?
A: Yes, the calculator can handle a wide range of positive values, though extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Are there any limitations to this calculation?
A: The calculation assumes perfect geometric proportions and may not account for real-world imperfections or variations in polyhedral structures.