Midsphere Radius of Snub Cube Given Volume Calculator

Midsphere Radius of Snub Cube Formula:

\[ r_m = \sqrt{\frac{1}{4(2-[Tribonacci_C])}} \times \left( \frac{3\sqrt{2-[Tribonacci_C]} \times V}{(3\sqrt{[Tribonacci_C]-1}) + (4\sqrt{[Tribonacci_C]+1})} \right)^{\frac{1}{3}} \]

Midsphere Radius of Snub Cube (rₘ):

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1. What is the Midsphere Radius of Snub Cube?

The midsphere radius of a snub cube is the radius of the sphere that is tangent to all the edges of the snub cube. It represents the sphere that fits perfectly within the polyhedron, touching each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \sqrt{\frac{1}{4(2-[Tribonacci_C])}} \times \left( \frac{3\sqrt{2-[Tribonacci_C]} \times V}{(3\sqrt{[Tribonacci_C]-1}) + (4\sqrt{[Tribonacci_C]+1})} \right)^{\frac{1}{3}} \]

Where:

\( r_m \) — Midsphere radius of snub cube
\( V \) — Volume of snub cube
\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)

Explanation: This formula relates the midsphere radius to the volume of the snub cube using the mathematical constant Tribonacci_C, which is derived from the Tribonacci sequence.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and crystallography for understanding the spatial properties of the snub cube, which is one of the Archimedean solids with interesting mathematical properties.

4. Using the Calculator

Tips: Enter the volume of the snub cube in cubic meters. The volume must be a positive value greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a snub cube?
A: A snub cube is an Archimedean solid with 38 faces - 6 squares and 32 equilateral triangles. It has chiral symmetry and is known for its complex geometry.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the real root of the equation x³ - x² - x - 1 = 0, approximately equal to 1.839286755214161. It appears in various geometric calculations involving the snub cube.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the derived formula, though practical accuracy depends on the precision of the input volume value.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the snub cube due to its unique geometric properties and the involvement of the Tribonacci constant.

Q5: What are typical values for midsphere radius?
A: The midsphere radius varies with the volume of the snub cube. For a unit snub cube (volume = 1 m³), the midsphere radius is approximately 0.7-0.8 meters.