[ARTICLE] Universal Gear and the Pythagorean Theorem (Australian Mathematics Education Journal AMEJ / 2019)
A leap in knowledge often is accompanied by a necessary leap in the human thought process. This was true at the time when the Babylonians learnt Pythagoras’ Theorem, and it is true now. The present leap in knowledge challenges one’s ability for three-dimensional visualization. It insights the Pythagorean Theorem as being part of a larger three-dimensional concept, here named the Universal Gear. In a nutshell, the Universal Gear comprises of infinite interlaced Pythagorean Theorems embedded in planes along all three orthogonal directions that are constantly connected and in perfect balance. In so doing, this concept governs information exchange in one- (lines), two- (areas) and three-dimensions (volumes) throughout the grid. From this enhanced perspective, the Universal Gear is bigger than the Pythagorean Theorem in that not only it satisfies the later individually — by interrelating the area of three squares via an intermediate triangle — but it also establishes a volumetric equivalent (to the Pythagorean Theorem) by relating two cubes to a half truncated octahedron via a truncated tetrahedron,and vice versa, which is different from saying that the volumetric sum of two cubes (both geometric integers) relates to a third (also a geometric integer) — see Fermat's Last Theorem. Noting that all in all, it is the Pythagorean Theorem that governs this three-dimensional structure. Thus, the Universal Gear establishes that the truncated tetrahedron is the three-dimensional equivalent to the right-triangle — a truncated tetrahedron is indeed formed by three orthogonally alignment right-triangles — and that the diagonal of the cube, that is an hexagon, is the three-dimensional equivalent to the diagonal of a square (the hypotenuse of a triangle). Ultimately, the Universal Gear is governed by the Pythagorean Theorem’s dynamics expressed synchronously in a three-dimensional network. Like reality, the Universal Gear presents a fractal nature in that its structure repeats itself infinitely both inwardly and outwardly. One of its practical applications has been identified in the study of fluids, more particularly in computation fluid dynamics.
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