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The problem involves a geometrical concept called a "Reuleaux square," created within a cube of edge length \(2r\), with intersecting spheres forming a new shape. Given the cube and spheres, we need to determine the ratio of volumes \( \frac{V_1}{V_2} \).

Here's the breakdown:

1. **Volume of the Sphere**:
   - The volume \( V_1 \) of a sphere with radius \( r \) is given by:
     \[
     V_1 = \frac{4}{3} \pi r^3
     \]

2. **Volume of the Reuleaux Square (V2)**:
   - The Reuleaux square's volume \( V_2 \) is derived from the intersection of two perpendicular spheres within the cube. However, we need more context or a specific formula typically provided in the problem statement or referenced literature for the exact calculation of \( V_2 \). 

For the sake of solving this, assume that \( V_2 \) represents the volume of a geometrical shape formed by the intersecting spheres. Let's denote \( V_2 \) as the combined volume of these intersections.

3. **Ratio Calculation**:
   - The given problem requires the ratio \( \frac{V_1}{V_2} \).

Given the relationship in a Reuleaux square, one might need specific integrals or geometric constructions not explicitly provided in the problem statement. Typically, if such geometrical constructs are studied, they might have predefined ratios or volumes.

Without additional information or assumptions, it is challenging to provide the exact ratio. However, often in problems like these, the ratio might simplify to a notable constant or fraction derived from integral calculations of intersecting volumes.

Would you like more details on how to approach these types of geometric problems, or do you have another specific question? Here are some sample questions you might find useful:

1. How do you calculate the volume of a Reuleaux square?
2. What is the integral method to find volumes of intersecting spheres?
3. Can you explain the geometric properties of Reuleaux shapes?
4. How do you determine the volume of shapes formed by intersecting solids?
5. What are some real-life applications of these geometric concepts?

**Tip**: When dealing with complex geometric shapes, breaking down the problem into simpler volumes and using symmetry can often simplify the calculations.

Learn how to calculate volumes involving a Reuleaux square formed within a cube, intersecting spheres, and their respective volumes. Explore the relationship between the volume of spheres and the volume of intersecting shapes. Suitable for advanced high school students.

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