Circumsphere Radius Of Snub Cube Given Midsphere Radius Calculator

Formula Used:

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

Circumsphere Radius of Snub Cube (r_c):

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1. What is Circumsphere Radius of Snub Cube?

The Circumsphere Radius of a Snub Cube is the radius of the sphere that contains the Snub Cube in such a way that all the vertices are lying on the sphere. It represents the smallest sphere that can completely enclose the Snub Cube.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula establishes the mathematical relationship between the circumsphere radius and midsphere radius of a Snub Cube using the Tribonacci constant.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry and 3D modeling for determining the bounding sphere of a Snub Cube, which is useful in collision detection, spatial analysis, and understanding the spatial properties of this Archimedean solid.

4. Using the Calculator

Tips: Enter the midsphere radius of the Snub Cube in meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumsphere radius.

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 is a semi-regular polyhedron with chiral symmetry.

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 Cube calculations.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Snub Cube and the defined formula using the Tribonacci constant.

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 practical applications of this calculation?
A: This calculation is used in computational geometry, 3D modeling, crystal structure analysis, and mathematical research involving polyhedral geometry.