Proof for Close Packing of Congruent Identical Spheres

In summary, the conversation discusses two different algorithms for calculating the density of close packed congruent identical spheres in two different arrangements: a tetrahedron with four equilateral triangular faces and a square pyramid with a square base and four equilateral triangular faces. It is found that as the number of spheres approaches infinity, the density of both arrangements equals approximately 0.74048... or π/√18. The conversation also mentions using Wolfram Alpha to verify the calculation and getting confirmation from Dr. Thomas C. Hales, who proved the Kepler Conjecture. The final question is to provide a proof that the two algorithms are equal as n approaches infinity. The expert provides a summary of the proof and explains that it is sufficient to
  • #1
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I developed two algorithms for calculating the density of close packed congruent identical spheres in two different arrangements:
  • A tetrahedron with four equilateral triangular faces, and
  • A square pyramid with a square base and four equilateral triangular faces, as shown below.

tetrahedral ball stack w- n=3.jpg

Figure 1. Tetrahedral ball stack.
pyramidal ball stack.jpg

Figure 2. Square pyramidal ball stack.

Here are the Excel-friendly algorithms (n = number of spheres along bottom row):

Density of Tetrahedral Stack (Dt):

Dt = (4*(2^0.5)*n*(n+1)*(n+2)*Pi())/(3*(2*n+2*3^0.5-2)^3 or

Dt equation.jpg
(1)

Density of Square Pyramidal Stack (Dp):

Dp = ( n*(1+n)*(1+2*n)*Pi())/(3*((1+(2^0.5*n)))^3) or

Dp equation.jpg
(2)

I found that, as the number of spheres approaches infinity for both arrangements, that Dt = Dp = ≈ 0.74048... or π/√18.

I resorted to Wolfram Alpha with the following query:

Dt limit.jpg
=
Dp limit.jpg
(3)

and got the following result:

Wolfram Proof RTC = SPC.jpg
(4)

True!

I then forwarded my calculations to Dr. Thomas C. Hales, who proved the Kepler Conjecture, asking him if Dt = Dp was correct, and he responded, saying,

"The reason for the equal densities in a tetrahedron and square pyramid is that they can both be viewed as part of the face-centered-cubic packing, each with a different set of exposed facets."

My question to you is this: can you provide a detailed proof that (1) = (2) as n→∞, i.e., that
Dt equals Dp.jpg
is true?


Thanks for any input!

Fizixfan.
 
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  • #2
You get the equality of the limits by calculating the coefficients of the highest power ##3## of ##n##. It is sufficient to see that
$$\frac{2 \pi n^3}{3 \cdot \sqrt{2}^3 n^3} = \frac{4 \pi \sqrt{2} n^3}{3 \cdot 2^3 \cdot n^3}$$
because all other terms don't grow as fast (and division of the complete fractions by ##n^3## (nominator and denominator) gives ##\frac{1}{n}## terms which become zero at infinity).
 
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1. What is close packing of spheres?

Close packing of spheres is the arrangement of identical spheres in a way that maximizes the use of space. This means that there is little to no empty space between the spheres, resulting in a tightly packed structure.

2. How is close packing of spheres achieved?

Close packing of spheres can be achieved by arranging the spheres in a hexagonal or cubic pattern. In a hexagonal pattern, each sphere is surrounded by six other spheres, while in a cubic pattern, each sphere is surrounded by eight other spheres.

3. Why is close packing of spheres important?

Close packing of spheres is important in many fields, including materials science, chemistry, and physics. It can help us understand the properties of materials and how they behave under different conditions. It is also important in the design of structures, such as crystal lattices and foams.

4. What is the proof for close packing of spheres?

The proof for close packing of spheres is based on the Kepler conjecture, which states that the densest possible arrangement of equal spheres in 3-dimensional space is achieved at a ratio of 74.05%. This was proven by Thomas Hales in 1998 using mathematical techniques and computer algorithms.

5. How is close packing of spheres used in real life?

Close packing of spheres has many practical applications in our daily lives. It is used in the design of packaging materials, such as Styrofoam and bubble wrap, to protect fragile items during shipping. It is also used in the production of metals and ceramics, as well as in the construction of buildings and infrastructure.

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