Short Edge of Deltoidal Icositetrahedron given Symmetry Diagonal Calculator

Formula Used:

\[ \text{Short Edge} = \frac{4 + \sqrt{2}}{7} \times \frac{7 \times d_{\text{Symmetry}}}{\sqrt{46 + 15\sqrt{2}}} \]

Short Edge of Deltoidal Icositetrahedron:

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1. What is the Short Edge of Deltoidal Icositetrahedron?

The Short Edge of Deltoidal Icositetrahedron is the length of the shortest edge of the identical deltoidal faces that make up this polyhedron. It is a key geometric parameter in understanding the structure and properties of this specific shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{4 + \sqrt{2}}{7} \times \frac{7 \times d_{\text{Symmetry}}}{\sqrt{46 + 15\sqrt{2}}} \]

Where:

\( d_{\text{Symmetry}} \) — Symmetry Diagonal of the Deltoidal Icositetrahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula establishes the mathematical relationship between the symmetry diagonal and the short edge length of the deltoidal icositetrahedron, incorporating specific constants derived from its geometric properties.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for various geometric analyses, including surface area and volume computations, symmetry studies, and understanding the overall spatial configuration of the deltoidal icositetrahedron.

4. Using the Calculator

Tips: Enter the symmetry diagonal value in meters. The value must be positive and non-zero. The calculator will compute the corresponding short edge length based on the geometric relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Icositetrahedron?
A: A Deltoidal Icositetrahedron is a Catalan solid with 24 deltoidal (kite-shaped) faces. It is the dual polyhedron of the rhombicuboctahedron.

Q2: How is the symmetry diagonal defined?
A: The symmetry diagonal is the diagonal that cuts the deltoidal faces of the Deltoidal Icositetrahedron into two equal halves, representing a key symmetry axis of the polyhedron.

Q3: What are the practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, mathematical modeling, and any field requiring precise geometric analysis of complex polyhedral structures.

Q4: Are there limitations to this formula?
A: The formula is specifically derived for the Deltoidal Icositetrahedron and assumes ideal geometric conditions. It may not apply to distorted or irregular variations of the shape.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with high precision (up to 10 decimal places), but practical applications may require rounding based on measurement accuracy and specific use case requirements.