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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.
Essential Resources for IB Teachers
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Essential Resources for both IB teachers and IB students
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The Mathematics of Cons – Pyramid Selling
Pyramid schemes are a very old con – but whilst illegal, still exist in various forms. Understanding the maths behind them therefore is a good way to avoid losing your savings!
The most basic version of the fraud starts with an individual making the following proposition, “pay me $1000 to join the club, all you then need to do is recruit 6 more people to the club (paying $1000 each) and you will have made a $5000 profit.”
There are lots of variations – and now that most people are aware of pyramid selling, now normally revolve around multi-level-marketing (MLM). These are often still pyramid schemes, but encourage participants to believe it is a genuine business by actually having a sales product which members have to sell. However the main focus of the business is still the same – taking money off people who then make their money back after having signed up a set number of new recruits.
The following graphic from Consumer Fraud Reporting is a clear mathematical demonstration why these frauds only end up enriching those at the top of the pyramid:
You can see that if the requirement was to recruit 8 new members, that by the 9th level you would need to have 1 billion people already signed up. Even with the need to recruit just 4 new members you still have rapid exponential growth which very quickly means you will run out of new potential members. For pyramid schemes it is only those in the first 3-4 levels (the white cells) that stand any real chance of making money – and these levels are usually filled by those in on the scam.
Ponzi schemes (like that run by Bernie Madoff) use a similar method. A conman takes money from investors promising (say) 10% annual returns. Lots of investors sign up. The conman then is able to use the lump sum investments to pay the 10% annual returns. This scam can last for years, with people thinking that they are getting a good rate of return, only to find out eventually that actually their lump sum investment has gone.
This is a good topic to look at with graphs (plotting exponential growth), interest rates, or exponential sequences – and shows why understanding maths is an important financial skill.
If you like this topic you might also like:
Benford’s Law – Using Maths to Catch Fraudsters – the surprising mathematical law that helps catch criminals.
Amanda Knox and Bad Maths in Courts – when misunderstanding mathematics can have huge consequences .
Essential resources for IB students:
1) Exploration Guides and Paper 3 Resources
I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
IB Maths Resources from Intermathematics 