Sunday, 21 July 2013

A Zometool model for an astrologer's platform

In ancient China, the solid as shown in the picture is called chutong 芻童, which can be roughly translated as "truncated hill". As the reader can see, it has a bottom face and a top face, both of which are oblongs with unequal lengths and widths. On its four side faces are four trapezoids. This solid is often used as an astrologer's platform for observing the heavens, or as an emperor's mausoleum.

If the length and width of the top face are a and b, the length and width of the bottom face are c and d, and the height of this solid is h, then its volume formula is given in the Nine Chapters of Mathematical Art (1st century) as

V = h [(2a + c)b + (2c + a)d]/6.

In the third century, Liu Hui gave a demonstration of the correctness of this formula. He used an example of this solid, in which the top length, top width, bottom length, bottom width and height are 2 units, 1 unit, 4 units, 3 units and 1 unit, respectively. The Zometool model in the picture was made exactly as the dimensions in Liu Hui's example.

After setting up the dimensions, Li Hui then divided the solid into several basic "chessmen": cubes, yangma and qiandu. (See this for explanations). As the reader can see, this model can be divided into 2 cubes, 6 qiandu and 4 yangma. Can you, the smart the reader, demonstrate the volume formula of the chutong according the method of this division and the formulae of those chessmen?