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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 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.

pyramid

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:

pyramid 2You 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 .

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IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB Exploration Modelling and Statistics Guide


IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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