Introduction

The packing fraction of a closely packed arrangement of circular disks can be defined as the ratio of the total area occupied by the disks to the total area of the space in which they are packed. In this arrangement, the disks are arranged in a hexagonal pattern, where each disk is surrounded by six neighboring disks.

Explanation

Step 1: Determining the area of a single disk
The area of a single disk can be calculated using the formula for the area of a circle: A = πr^2, where r is the radius of the disk.

Step 2: Determining the area of the space occupied by a single disk and its neighbors
In this arrangement, each disk is surrounded by six neighboring disks. The centers of these neighboring disks form an equilateral triangle, with each side length equal to 2r (twice the radius of a disk). The area of this equilateral triangle can be calculated using the formula: A = (√3/4) * (2r)^2.

The total area occupied by a single disk and its neighbors can be calculated by adding the area of the disk and the area of the equilateral triangle: Total Area = πr^2 + (√3/4) * (2r)^2.

Step 3: Determining the packing fraction
The packing fraction can be calculated by dividing the total area occupied by the disks by the total area of the space in which they are packed.

In this arrangement, the distance between the centers of the disks is 2r. Therefore, the total area of the space occupied by a single disk and its neighbors is equal to the area of a square with side length 2r. The area of this square can be calculated using the formula: A = (2r)^2.

The packing fraction is given by the ratio: Packing Fraction = (πr^2 + (√3/4) * (2r)^2) / (2r)^2.

Simplifying the expression, we get: Packing Fraction = (π + √3/2) / 4.

Conclusion
The packing fraction of the closely packed arrangement of circular disks with a distance of 2r between their centers is given by (π + √3/2) / 4. This value represents the ratio of the total area occupied by the disks to the total area of the space in which they are packed.