Swannies Theorems, Art in Maths and a follow-up to Pythagoras, De Tinseau and De Gua De Malves.

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Fairy 2

did not have a clue
as to what something with dimensions 2,83inch  x 2,83inch x 2,83inch x 2,83inch  look like!*    De Gua thus did fine going with his numbers to a next level but starting with triangles and cm^2 landed in a never/nether world  (where for Swannies  maths competency level a Devil rules!) with inch^4 and a none existing shape.

Now, Swannie clearly states in the definition of the First Theorem that use is made of the numerical value of the area of a face, once used for the area of the base of a prism and once, for the height of the prism; such as  an 8 for the area and an 8 for the height giving 64 as its its volume's number. Same done for the other faces. Number  wise this is fully supported by De Guas's Theorem as shown above and validates the First Theorem. The chosen inch units, as inch^3 units (cubed inches = a little block of something, can just be space, measuring an inch by an inch  by an inch) now "engaged" to the number,  gives a volume for the prism of 64 ^1 inch^3, or 4^3 inch^3,  a real property in our world.
Thus for Plato's equation the cube root of the number of  this prisms volume is 4. For the 27, 64, and 216 prisms you have 3, 5 and 6, the other root values for Swannies 3^3 + 4^3 + 5^3 = 6^3 ,  all in inch^3, stainless steel model. By Swannies Theorem  the partially (the sensible part of it) included De Gua's  Theorem can now be visualised! Until now for many centuries De Gua's (and lets not forget De Tinseau the actual originator) Theorem has been pretty useless, hanging around in an equation. But  not anymore - it led, (with skilled? Manipulation - **see later),  to Swannies Theorem which can at least be used for building little models! The spirits of these two Frenchmen can now rejoice, their brainchild has found a haven in this real world,  for  all to see! (But they probably achieved much more than just this.)  

Now to come clear: In no way was it necessary for the Maths Fairy to assist with the above, that was once Swannie knew what to do. But she does exist and so does the (Maths?) Devil! :-
Trying to find a volume equivalent to Pythagoras's "triangle with its attached squares" theorem  the trirectangular tetrahedron seemed to Swannie the obvious choice to start off with, but he also contemplated other shapes, so paged through a number of  text books for a clue. ( He was then, 2005, off computers,  only using 1983 Multimate for real old time letters to a friend or family member, having had enough of watching screens whilst being a member of the working class.)

Then he was most excited to find in James Steward's Calculus, International Students edition 5e a problem set therein, on page 858, no.3 about a  trirectangular tetrahedron:
                                       "Suppose the tetrahedron ****  in the figure has a trirectangular vertex S.  Let A,B and C be the areas of the three faces that meet at S, and let D be the area of the opposite  face PQR . Using the result of problem one, or otherwise show that  D^2 = A^2 + B^2 + C^2  (This is a three***-dimensional version of  the Pythagorean Theorem)"          (Bold lettering by Swannie)
**** A top view of such a tetrahedron, trirectangular, as mentioned appears on the Home Page of  the Swannies Website, showing faces A, B, C; and D as a reflection.

Excitement turned into four months of extreme frustration -  The trirectangular tetrahedron was tied up/seemed married to a unit to the fourth power, some advanced multidimensional maths way beyond  Swannies comprehension and   a shape for which you could try to invoke the addition of the parallel world some folks believe exist as a twin to our one.
Thus convinced by the (Maths?) Devil that numbers get married to units Swannie spent four frustrating months making all sorts of little maths shapes out of cardboard, toasty boxes and bought sheets, glue and Artist  paint, imagining himself  to be an Artist in order to deal with despondency. Then the Maths Fairy  paid him a visited, waved her wand, broke the Devils spell and showed him the way out! The Theorem with its shapes and their volumes  could be defined – a true three dimensional follow up to Pythagoras!
                --- The Maths Fairy

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