[ARTICLE] Special Case of the Three-dimensional Pythagorean Gear (Australian Senior Mathematics Journal / 2018)

In mathematics, three integer numbers or triples have been shown to govern a specific geometrical balance between triangles and squares. The first to study triples were probably the Babylonians, followed by Pythagoras some 1500 years later (Friberg, 1981). This geometrical balance relates parent triples to child triples via the central square method (Teia, 2015). The great family of triples forms a tree ― the Pythagorean tree ― that grows its branches from a fundamental seed (3, 4, 5) making use of one single very specific motion. The diversity of its branches rises from different starting points (i.e., triples) along the tree (Teia, 2016). All this organic geometric growth sprouts from the dynamics of the Pythagorean geometric gear (Figure 1). This gear interrelates geometrically two fundamental alterations in the Pythagorean theorem back to its originas ― the fundamental process of summation x+y=z (Teia, 2018). But reality has a three dimensional nature to it, and hence the next important question is: what does the Pythagorean gear look like in three dimensions? The AAMT journal copyright statement allows online sharing as follows: " The authors may publish a copy of the work online on their institutional website or repository, or on third-party research repositories such as Research Gate. " (see link below for further details)