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**Finding the average distance between 2 points on a hypercube**

This is the natural extension from this previous post which looked at the average distance of 2 randomly chosen points in a square – this time let’s explore the average distance in n dimensions. I’m going to investigate what dimensional hypercube is required to have an average distance of more than one, and then also what happens to the average distance as n approaches infinity.

**Monte Carlo method**

The Monte Carlo method is a very powerful technique which utilizes computational power. Basically we use the fact that the average of a very large number of trials will serve as an approximation to an exact result. In this case I will run a Python program 10 million times – each time it will select 2 coordinate points and then work out the distance between them. It will then find the average of these 10 million trials. The code above generates 2 coordinates in 3 dimensional space inside a unit cube. We can modify this for n-dimensional space by remembering that Pythagoras still works in higher dimensions.

**Results**

Running this code helps to generate the above results. This answers our first question – we need a 7 dimensional unit hypercube until the average distance between two randomly chosen points is greater than 1. We can also see that the difference between the average distances is reducing – but it’s not clear if this will approach a limit or if it will continue growing to infinity. So let’s do some more trials.

**Further trials**

This takes us up to a 22-dimensional hypercube. At this point it’s probably useful to plot a graph to see the trend.

**Reciprocal model**

This reciprocal model is of the form:

We can see that this is a pretty good fit (R squared 0.9994). If this model is accurate then this would suggest that the average distance approaches a limit as n approaches infinity.

**Polynomial model**

This polynomial model is of the form:

We can see that this is also a very good fit (R squared 0.9997). If this model is accurate then as b is greater than 0, this would suggest that the average distance approaches infinity as n approaches infinity.

**Reflection**

Quite annoyingly we have 2 model which both fit the data very accurately – but predict completely different results! Logically we could probably say that we would expect the average distance to approach infinity as n approaches infinity – and also we could possibly justify this by the fact that the polynomial model is a slightly better fit. Given the similarity between the 2 models it probably time to find out the actual results for this.

**Average n-dimensional distance bounds**

Not surprisingly the mathematics required to work this out is exceptionally difficult – and ends up with non-solvable integrals which require analytic solutions. The Monte Carlo method with very large numbers of trials is a reasonably good approach to approximating this answer. There is however a very useful lower and upper bound for the average distance in n dimensional space given by:

This shows immediately that the average distance will approach infinity as n grows large – as the lower bound will grow to infinity. Quite pleasingly we can see that the polynomial model we derived is similar to the lower bound. We can plot both upper and lower bound along with our polynomial model to see how these all compare. We have lower bound (green), polynomial model (black) and upper bound (green):

We can see that our polynomial model very closely follows the upper bound in our domain. As we extend the domain this polynomial approximation remains above the lower and tracks the upper bounds before gradually growing less accurate. When n is 50 our model predicts a distance of 2.94, whereas the upper bound is 2.88. This is quite a nice result – we have used the Monte Carlo method to derive a polynomial approximation to the average distance in n-dimensional hypercubes and it both closely follows the upper bound over a reasonable domain and also is of a very similar form to the lower bound. We can use this lower bound to see that a 36 dimensional hypercube (and higher) would be guaranteed to have an average distance of more than 2.

**Conclusion**

This was a nice example of the power of the Monte Carlo method in these kind of problems – we were able to use it quite successfully to get a polynomial approximation which turned out to be reasonably accurate. We could have significantly improved this accuracy by running 100 million (or 1 billion etc) trials each time – though this would have probably required a more powerful computer!

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank 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!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

**Essential Resources for IB Teachers**

If you are a **teacher** then please also visit my new site. This has been designed specifically for teachers of mathematics at international schools. The content now includes over **2000 pages of pdf content** for the entire SL and HL Analysis syllabus and also the SL Applications syllabus. Some of the content includes:

**Original pdf worksheets**(with full worked solutions) designed to cover all the syllabus topics. These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.**Original Paper 3 investigations**(with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.- Over 150 pages of
**Coursework Guides**to introduce students to the essentials behind getting an excellent mark on their exploration coursework. - A large number of
**enrichment activities**such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more. I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

**Essential Resources for both IB teachers and IB students**

1) Exploration Guides and Paper 3 Resources

I’ve put together a **168 page** Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission. Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator! I have also made **Paper 3 packs** for HL Analysis and also Applications students to help prepare for their Paper 3 exams. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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