Snub Dodecahedron Edge Of Pentagonal Hexecontahedron Given Midsphere Radius Calculator

Formula Used:

\[ Snub\ Dodecahedron\ Edge = \frac{r_m}{\sqrt{\frac{1+0.4715756}{2(1-2\times0.4715756)}}} \times \sqrt{2+2\times0.4715756} \]

Snub Dodecahedron Edge:

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1. What is Snub Dodecahedron Edge of Pentagonal Hexecontahedron?

The Snub Dodecahedron Edge of Pentagonal Hexecontahedron is the length of any edge of the Snub Dodecahedron of which the dual body is the Pentagonal Hexecontahedron. It is an important geometric measurement in polyhedral studies.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Snub\ Dodecahedron\ Edge = \frac{r_m}{\sqrt{\frac{1+0.4715756}{2(1-2\times0.4715756)}}} \times \sqrt{2+2\times0.4715756} \]

Where:

\( r_m \) — Midsphere Radius of Pentagonal Hexecontahedron (m)
0.4715756 — Constant value used in the calculation

Explanation: This formula calculates the edge length of a snub dodecahedron based on the midsphere radius of its dual polyhedron, the pentagonal hexecontahedron.

3. Importance of Snub Dodecahedron Edge Calculation

Details: Calculating the edge length of polyhedra is crucial in geometry, crystallography, and materials science for understanding spatial relationships and structural properties of complex shapes.

4. Using the Calculator

Tips: Enter the midsphere radius of the pentagonal hexecontahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a snub dodecahedron?
A: A snub dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles), 150 edges, and 60 vertices.

Q2: What is a pentagonal hexecontahedron?
A: A pentagonal hexecontahedron is a Catalan solid that is the dual polyhedron of the snub dodecahedron, with 60 irregular pentagonal faces.

Q3: What is the midsphere radius?
A: The midsphere radius is the radius of a sphere that is tangent to all edges of a polyhedron.

Q4: Where is this calculation used in real world?
A: This type of geometric calculation is used in crystallography, molecular modeling, architecture, and computer graphics.

Q5: What are the limitations of this formula?
A: This formula is specific to the relationship between the snub dodecahedron and its dual, the pentagonal hexecontahedron, and uses a fixed constant value.