Classifying Solids Using Angle Deficiency

This article takes you through the classification of the Platonic (regular) and Archimedean (semi-regular) solids, to find all of them and prove that there are no more.

I found it difficult to make a choice of article for the NRICH Tenth Anniversary Celebration. I chose this one because it has been the basis of so many enjoyable sessions that I have had both with young learners and also with teachers.

With any age group, from 10 upwards, I find the best approach is to explain the Schlafli code and demonstrate it with an actual Archimedean solid,and then to give individuals different codes with the task of making their solid for that code. According to how much time you have, and how much your group already know, you can structure the session so that they discover the Euler Relation and/or the total Angle Deficiency, you can arrive at a proof that there are only 5 Platonic Solids and you can fill in the tables for all the Archimedean Solids deducing the number of each shape of face as described in the article. There are many possibilities.

In the ATE Maths Superweek where the photos were taken we had a great Holiday Director called Ian Johnston and a very good cook called Mrs Higgins and the children never knew that these were one and the same person. Just before lunch on the morning we spent on this topic Ian (an engineer) joined us and the children, having decided to test him, excitedly showed him their models "Ian look at mine, it's a 466" , "Ian mine is a 3434, can you explain that?" ... and so on. It did not take long for Ian (an engineer) to work it out and the children were impressed.

Perhaps your class can make the models out of card and hang them from the ceiling.