Midsphere Radius of Snub Cube given Circumsphere Radius Calculator

Formula Used:

\[ r_m = \sqrt{\frac{1}{4(2 - [Tribonacci_C])}} \times \frac{r_c}{\sqrt{\frac{3 - [Tribonacci_C]}{4(2 - [Tribonacci_C])}}} \]

Midsphere Radius of Snub Cube:

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1. What is Midsphere Radius of Snub Cube?

The Midsphere Radius of Snub Cube is the radius of the sphere for which all the edges of the Snub Cube become a tangent line on that sphere. It represents the sphere that touches the midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \sqrt{\frac{1}{4(2 - [Tribonacci_C])}} \times \frac{r_c}{\sqrt{\frac{3 - [Tribonacci_C]}{4(2 - [Tribonacci_C])}}} \]

Where:

\( r_m \) — Midsphere Radius of Snub Cube (m)
\( r_c \) — Circumsphere Radius of Snub Cube (m)
\( [Tribonacci_C] \) — Tribonacci constant ≈ 1.839286755214161

Explanation: This formula relates the midsphere radius to the circumsphere radius using the mathematical constant Tribonacci_C, which is characteristic of the snub cube geometry.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometric modeling, crystallography, and architectural design where snub cube structures are used. It helps in understanding the spatial relationships and proportions within this complex polyhedron.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius using the mathematical relationship specific to snub cube geometry.

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 geometric properties.

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 contexts including snub polyhedra.

Q3: How is the midsphere different from circumsphere?
A: The circumsphere passes through all vertices of the polyhedron, while the midsphere is tangent to all edges at their midpoints.

Q4: What are practical applications of snub cube geometry?
A: Snub cube geometry is used in molecular modeling, architectural design, game development, and mathematical research on polyhedral structures.

Q5: 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.