I was very pleased to discover recently that the shape formed from a net I like using for more advanced constructions has a proper name, a “Yangma”. I would normally require pupils to create the net using mathematical instruments then fold up to form a square-based pyramid with the apex above a right-angled corner. I also found out that this was first described in a Chinese book of mathematics called The Nine Chapters on the Mathematical Art or Jiuzhang Suanshu, an anonymous collection of writing from between two and three thousand years ago.
“Now that’s all very interesting,” you might say, “but what use is it?”
Well, if we were to use a set of nets, such as the one on my website, it is possible to make three congruent (identical in all aspects) yangma. They can then be arranged such that they form a cube.
“Okay you’ve made a cube,” you now say, “So what?”
Again, a fair response; time for an anecdote.
In my last job I had the pleasure and challenge of teaching children from various countries across Europe and beyond. Many of the children were a little ahead of their British counterparts at Arithmetic and Algebra but they had done little or no work on several topics, most relevant to this tale being constructions and shapes. A few years ago I had a lower ability set of 12/13 year old children which contained a Spaniard and a Russian. Their computational skills were much stronger than their peers but they struggled to apply their skills in context and had never seen a net before. Whilst the rest of the class were working on some algebraic tasks, I worked with the two boys to build some nets. We started with a cube and built up the difficulty until we each made a yangma. Dima, the Russian boy, was by now getting frustrated with his inability to make the shapes, as he prided himself on his mathematical prowess, with some justification. An ever growing mass of paper and glue was developing in the bin; it is a good job that my knowledge of Russian is limited to hello, goodbye and a few other random words as the crumpled shapes were being accompanied into the bin by a torrent of abuse that I suspect was not entirely appropriate! Finally we created the three yangma and put them together as in the picture above. Immediately his eyes lit up and through a growing smile of comprehension he said, “Ah, is why one third!”
“Very touching story,” you comment, “But what is one third?”
As any fule kno [younger readers who have not come across Molesworth need to enrich their lives immediately!], and Dima instantly recalled, the formula for the volume of a pyramid is, If we call the length of one side of our cube a then this gives us,
“Ah now that is interesting,” you concede, “But most pyramids are not like that, so how is the formula still the same?”
This is an excellent question and allows us to employ a valuable mathematical tool. Firstly, we consider our shape to be made of a series of increasing squares so as such a stepped yangma.
Next we start at the top and shift each layer diagonally across one unit, so it will end up like a stepped Egyptian pyramid.
Finally we repeat the exercise with an increasing number of squares with a decreasing difference in their size so that as the number of squares tends towards infinity, the difference between each one tends to zero and we get a pair of smooth-sided square-based pyramids of equal volume, one with the apex over a base corner the other with the apex over the centre of the base. Thus any square-based pyramid can be shown to have the same volume as the yangma and therefore a third of the volume of the appropriate cube.
“Actually, that was rather good but what if the base isn’t square, say it’s a rectangle. Now your three shapes thing won’t work.”
Ooh, good question but actually you are wrong! Here is a cuboid made from three different yangma. [again, the net is available on my website!]
Assume for a moment that the sides have length 3, 4 and 5. Our yangma will be as follows:
|Base||Height||Base x Height||Volume|
|3 x 4||5||3 x 4 x 5||20|
|3 x 5||4||3 x 5 x 4||20|
|4 x 5||3||4 x 5 x 3||20|
Although the shapes are now not congruent, they will still have the same volume which is one third of the whole cuboid so the formula holds true.
“Hmm”, you think for a moment.” What about triangular-based pyramids. They won’t make a cube.”
No indeed, but three of them will make a triangular prism which the Greek mathematician Euclid described in his Elements written about 300BC. Perhaps you might want to go away and work that one out for yourself?
[The Teachers’ Page on my website has links to all the nets and lesson notes as well as other interesting investigations to try]