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THE TRIRECTANGULAR TETRAHEDRON

PYTHAGORAS's 'Areas' THEOREM (5C BC)

25 Centuries later…

SWANNIE's 'Volumes' THEOREM (2006 AC)

The photographs and diagrams alone should give a good idea of what the theorems are about; being simply the areas or volumes of three shapes that add up to give a fourth, all arranged on the four faces of a *corner of a box*.
Swannie proves his theory with a mathematical model of cylindrical shape
A Spatial, 3D, Volume – type theorem that allows the design and building of an infinite number of models.
In addition to the wish that it will contribute towards making Maths more interesting for students Swannie wishes to reveal the therapeutic value to a senior citizen of a quest for finding a solution to a maths puzzle.
Whereas a problem in a maths text book – a De Gua De Malves type problem (but this was only pointed out to him afterwards and he was also off computers for years) got him going, a fourth power area squared thing put him off track for at least four months.
During this time many little boxes of many shapes were built out of  frustration but that and the theorem models building is another story.
Not related to the theorems, you are rewarded with ‘Bra and Ball‘ on the common origin of  two useful commodities.
A dodecahedron 'cut out' from a cube such as the one it is mouted on
A non-theorem arty shape of a dodecahedron 'cut out' from a cube such as the one it is mounted on. Easy and quick to find the volume of the cube. Not so with the dodecahedron unless you do a 'dunk it' measurement with a large measuring cylinder and water! Could one call this one's centrepiece a twisted prism or perhaps an antiprism?
The Moons of Hippocratus of Chios
A non-theorem arty shape, depicting the Moons of Hippocratus of Chios. The sections have whole number ratios between them: 28:8:7:4.