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**Sphere packing problem: Pyramid design**

Sphere packing problems are a maths problems which have been considered over many centuries – they concern the optimal way of packing spheres so that the wasted space is minimised. You can achieve an average packing density of around 74% when you stack many spheres together, but today I want to explore the packing density of 4 spheres (pictured above) enclosed in a pyramid.

**Considering 2 dimensions**

First I’m going to consider the 2D cross section of the base 3 spheres. Each sphere will have a radius of 1. I will choose A so that it is at the origin. Using some basic Pythagoras this will give the following coordinates:

**Finding the centre**

Next I will stack my single sphere on top of these 3, with the centre of this sphere directly in the middle. Therefore I need to find the coordinate of D. I can use the fact that ABC is an equilateral triangle and so:

**3D coordinates**

Next I can convert my 2D coordinates into 3D coordinates. I define the centre of the 3 base circles to have 0 height, therefore I can add z coordinates of 0. E will be the coordinate point with the same x and y coordinates as D, but with a height, *a*, which I don’t yet know:

In order to find *a *I do a quick sketch, seen below:

Here I can see that I can find the length AD using trig, and then the height DE (which is my *a* value) using Pythagoras:

**Drawing spheres**

The general equation for spheres with centre coordinate (a,b,c) and radius 1 is:

Therefore the equation of my spheres are:

Plotting these on Geogebra gives:

**Drawing a pyramid**

Next I want to try to draw a pyramid such that it encloses the spheres. This is quite difficult to do algebraically – so I’ll use some technology and a bit of trial and error.

First I look at creating a base for my pyramid. I’ll try and construct an equilateral triangle which is a tangent to the spheres:

This gives me an equilateral triangle with lengths 5.54. I can then find the coordinate points of F,G,H and plot them in 3D. I’ll choose point E so that it remains in the middle of the shape, and also has a height of 5.54 from the base. This gives the following:

As we can see, this pyramid does not enclose the spheres fully. So, let’s try again, this time making the base a little bit larger than the 3 spheres:

This gives me an equilateral triangle with lengths 6.6. Taking the height of the pyramid to also be 6.6 gives the following shape:

This time we can see that it fully encloses the spheres. So, let’s find the density of this packing. We have:

Therefore this gives:

and we also have:

Therefore the density of our packaging is:

Given our diagram this looks about right – we are only filling less than half of the available volume with our spheres.

**Comparison with real data**

[Source: Minimizing the object dimensions in circle and sphere packing problems]

We can see that this task has been attempted before using computational power – the table above shows the average density for a variety of 2D and 3D shapes. The pyramid here was found to have a density of 46% – so our result of 44% looks pretty close to what we should be able to achieve. We could tweak our measurements to see if we could improve this density.

So, a nice mixture of geometry, graphical software, and trial and error gives us a nice result. You could explore the densities for other 2D and 3D shapes and see how close you get to the results in the table.

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