Symmetry Diagonal of Deltoidal Icositetrahedron given Midsphere Radius Calculator

Formula Used:

\[ d_{Symmetry} = \frac{\sqrt{46 + 15\sqrt{2}}}{7} \times \frac{2r_m}{1 + \sqrt{2}} \]

Symmetry Diagonal of Deltoidal Icositetrahedron:

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1. What is Symmetry Diagonal of Deltoidal Icositetrahedron?

The Symmetry Diagonal of Deltoidal Icositetrahedron is the diagonal which cuts the deltoid faces of Deltoidal Icositetrahedron into two equal halves. It is an important geometric property of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d_{Symmetry} = \frac{\sqrt{46 + 15\sqrt{2}}}{7} \times \frac{2r_m}{1 + \sqrt{2}} \]

Where:

\( d_{Symmetry} \) — Symmetry Diagonal of Deltoidal Icositetrahedron (m)
\( r_m \) — Midsphere Radius of Deltoidal Icositetrahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the symmetry diagonal based on the midsphere radius of the deltoidal icositetrahedron, using specific mathematical constants derived from the geometry of this polyhedron.

3. Importance of Symmetry Diagonal Calculation

Details: Calculating the symmetry diagonal is crucial for understanding the geometric properties of deltoidal icositetrahedron, including its symmetry characteristics and spatial relationships between different elements of the polyhedron.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding symmetry diagonal of the deltoidal icositetrahedron.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the Midsphere Radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the deltoidal icositetrahedron, provided the input values are accurate.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal icositetrahedron. Other polyhedra have different geometric relationships.

Q5: What are the practical applications of this calculation?
A: This calculation is primarily used in geometry, crystallography, and architectural design where precise understanding of polyhedral properties is required.