Eaves Void Formula - Calculating Void Spaces Between Four Circles

Introduction

The Eaves Void Formula, developed by Aazim Caln Eaves, provides a method for calculating the void space formed by four tightly arranged circles, such as atoms or other spherical particles. This formula has potential applications in various scientific fields, including physics, chemistry, and materials science. The formula is as follows:

Void Area = 4r² - πr²

Formula Derivation

The derivation of the Eaves Void Formula involves calculating the void area by forming a square with chords in each circle that subtend a 90-degree angle at the center. This square encloses four minor segments and a central void area. The process of derivation is as follows:

Step 1: Defining the Void Area

When four circles are tightly arranged, chords can be drawn in each circle to form a 90-degree angle at their centers. These chords together form a square that includes minor segments from each circle and a central void area. The void area is calculated as:

Void Area = Area of Square - Area of 4 Segments

Step 2: Calculating the Area of the Square

The side of the square, denoted as a, is the length of the chord that subtends a 90-degree angle at the center of each circle. Using the Pythagorean theorem, we can express a in terms of the radius r:

a² = r² + r² = 2r²
Thus, a = √(2r²) = r√2

Therefore, the area of the square is:

Area of Square = a² = (r√2)² = 2r²

Step 3: Calculating the Area of the Four Minor Segments

Each circle segment subtended by the chord has a 90-degree central angle. The area of one segment is:

Area of One Segment = (θ/360)πr² - 1/2 r² sin(θ)

For a 90° angle (θ = 90°), this simplifies to:

Area of One Segment = (1/4)πr² - 1/2 r²

The total area of the four segments is then:

Area of 4 Segments = πr² - 2r²

Step 4: Calculating the Void Area

By subtracting the area of the segments from the area of the square, we get:

Void Area = 2r² - (πr² - 2r²)
= 4r² - πr²

Applications

This formula can be applied to calculate the void space in various arrangements of circles. Here are two examples:

Example 1: Void Area Between Four Iron Atoms

Suppose each iron atom has a radius of 0.126 nm. Using the formula:

4r² = 4 * (0.126 nm)² ≈ 0.063504 nm²
πr² ≈ 0.0498378 nm²

Void Area ≈ 0.063504 nm² - 0.0498378 nm² ≈ 0.0136662 nm²

Result: The void area between four tightly packed iron atoms is approximately 0.0137 nm².

Example 2: Void Area Between Four Circles with Radius 10 cm

For circles with a radius of 10 cm:

4r² = 4 * (10 cm)² = 400 cm²
πr² ≈ 314.16 cm²

Void Area ≈ 400 cm² - 314.16 cm² ≈ 85.84 cm²

Result: The void area between four circles with radius 10 cm is approximately 85.84 cm².

Author

The Eaves Void Formula was proposed by Aazim Caln Eaves, who first published it to provide a method for calculating void areas in tightly packed circular arrangements. This formula has potential applications in physical sciences, material sciences, and engineering, and is open for further empirical testing and validation by the scientific community.