[ARTICLE] Anatomy of the Pythagoras' Tree (Australian Senior Mathematics Journal / 2016)

The grand architecture of nature can be seen at play in a tree: no two are alike. Nevertheless, there is an inescapable similarity that makes us identify a tree anywhere in the world. Just saying “tree” recalls words like green, root, leaves, still, strong, branches. The tree of primitive Pythagorean triples is no different (Figure 1). It has a root, or a beginning. It is rooted not on earth, but on the soil of our mind. It has branches that spring from that root as it grows with the action of nature and time. In this case, it is not the proverbial Mother Nature, but the human nature — a nature formed by the human interpretation of reality. The Pythagoras’ tree presented by Berggren in 1934 has stood still and strong for almost a century, but probably it is even older. Its leaves are triples, and they grow throughout its branches. Ultimately, when one looks at the Pythagoras’ tree, one looks at a ‘tree’. The root is the triple (3, 4, 5). All branches and leafs emerge from, and are dependent, of this root. Like any tree, all it requires is a seed and soil, and all develops automatically. The architecture that defines the tree is present throughout the tree and is a reflection of the beginning — the root. In other words, the root (3, 4, 5) plus the same movement repeated over and over again creates the tree. In this paper, we will look at how this basic geometrical and mathematical movement governs the birth and growth of the Pythagoras’ tree. Pythagoras is included in secondary education around the world including in Australian Curriculum (ACARA, n.d.), and hence this paper will be of interest to all.

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