A student wants to arrange 4 identical spheres (of radiusR) on a two dimensional floor as close as possible. Finally he could arrange them. Identify the type and dimensions of the figure obtained by joining their centers. Could he occupy the whole available space, if not, then what type of voids were generated. Calculate their number and radius of small ball that can be fitted in them.

Answer

To solve the problem of arranging four identical spheres of radius \( R \) on a two-dimensional floor as close as possible, we can follow these steps:

### Step 1: Arrangement of Spheres
To arrange four identical spheres as close as possible, we can position them in a square formation. The centers of the spheres will form the vertices of a square.

### Step 2: Identify the Figure
When we join the centers of the four spheres, we obtain a square. The dimensions of the square can be determined as follows:
- The distance between the centers of two adjacent spheres is equal to \( 2R \) (since the radius of each sphere is \( R \)).
- Therefore, the side length of the square formed by the centers is \( 2R \).

### Step 3: Space Occupation
Next, we need to determine if the spheres occupy the whole available space. Since the spheres are arranged in a square, they do not occupy the entire area of the square. There will be some unoccupied space.

### Step 4: Identify the Type of Voids
The unoccupied space between the spheres can be identified as voids. In this arrangement, the voids formed are triangular voids. Specifically, there are two triangular voids formed at the corners of the square.

### Step 5: Calculate the Number of Voids
In total, there are **two triangular voids** formed in this arrangement.

### Step 6: Calculate the Radius of Small Balls that can Fit in the Voids
To calculate the radius of the small ball that can fit into the triangular voids, we can use the known ratio for triangular voids. The radius \( r \) of the small ball that can fit in a triangular void is given by:
\[ r \approx 0.155R \]

### Summary of Results
- **Type of Figure**: Square
- **Dimensions of Figure**: Side length = \( 2R \)
- **Type of Voids**: Triangular voids
- **Number of Voids**: 2
- **Radius of Small Ball that can Fit in Voids**: \( r \approx 0.155R \)