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Truncated Cuboctahedron Edge of Hexakis Octahedron Calculator

Formula Used:

\[ l_{e(Truncated\ Cuboctahedron)} = \frac{7}{2} \times \frac{1}{\sqrt{60 + (6 \times \sqrt{2})}} \times l_{e(Long)} \]

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1. What is the Truncated Cuboctahedron Edge of Hexakis Octahedron?

The Truncated Cuboctahedron Edge of Hexakis Octahedron is the length of the edges of a Hexakis Octahedron that is created by truncating the vertices of a Cuboctahedron. This geometric relationship demonstrates the mathematical connection between these two polyhedral forms.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{e(Truncated\ Cuboctahedron)} = \frac{7}{2} \times \frac{1}{\sqrt{60 + (6 \times \sqrt{2})}} \times l_{e(Long)} \]

Where:

Explanation: This formula establishes a precise mathematical relationship between the long edge of a Hexakis Octahedron and the corresponding edge length after truncating the vertices of a Cuboctahedron.

3. Importance of This Calculation

Details: This calculation is important in geometric modeling, crystallography, and architectural design where precise relationships between polyhedral forms are required. Understanding these geometric transformations helps in creating complex 3D structures and analyzing spatial relationships.

4. Using the Calculator

Tips: Enter the Long Edge of Hexakis Octahedron value in meters. The value must be positive and greater than zero. The calculator will automatically compute the corresponding Truncated Cuboctahedron Edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron, featuring 48 faces, 72 edges, and 26 vertices.

Q2: What is a Cuboctahedron?
A: A Cuboctahedron is an Archimedean solid with 8 triangular faces and 6 square faces, featuring 12 identical vertices and 24 edges.

Q3: Why is the constant (7/2)*(1/sqrt(60+(6*sqrt(2)))) used?
A: This constant represents the precise mathematical ratio derived from the geometric relationship between the truncated cuboctahedron and the hexakis octahedron through their dual relationship.

Q4: What are practical applications of this calculation?
A: Applications include molecular modeling, architectural design, computer graphics, and materials science where precise geometric transformations are required.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, providing precise results based on the geometric properties of these polyhedral forms.

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