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Further forays into n dimensions

In an article in the Gazette for November 2005 [1] Martin Griffiths described some ideas which he had explored with a Further Mathematics class based on extensions of the Platonic solids in more than 3 dimensions. This reminded me of an investigation I had myself undertaken with similar groups of students. Whilst giving a glimpse of ways in which mathematics can develop through generalisation, it also led to some unusual applications of the binomial theorem.