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

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Models

Models Above:
Some more models are shown above on the left, including perhaps, a religious symbol for 'Trinity in Unity'. Also consider the star for some more symbolism. With the same little tetrahedron and some imagination one can create various attachments. Top right you will note that Pythagoras need not only provide for squares!: Size 9 + size 16 = size 25 (Pythagorean sizes for ladies) are shown. Above on the right: Port Elizabeth has a pyramid and other models.   
As done with the lady and her copies according to Pythagoras above, one could also try to do a model of say the finger of a lady, complete with painted nail, in accordance with Swannies  theorem, using some mathematical manipulation to correct the distortion brought about by projection. See those elliptical cylinders of the model under Equations, turning them into circular cylinders with the same volumes? Such models Swannie shall, for now at least, leave for those  wishing to attempt it. ((April2012: Try Theorem III.)) The variety of shapes that can be turned out seems endless and some could well be pleasing to the eye. Swannie likes to think he has made a contribution to mathematical art!  
  
Contributors and Recommended Reading:
James Stewart's Calculus,  International Student Edition 5e: A problem set therein, no.3 page 858 provided the clue that much later turned out to be De Gua de Malves's Theorem.
Some Adventures in Euclidean Geometry: Michael de Villiers, (trading as Dynamic Mathematics Learning.)
Swannie only learned the names of the following contributors after he found the basics of the theorem:                                                                       
D'Amons Charles de Tinseau: ±1774  General theorem on areas squared of  figures on faces of trirectangular tetrahedrons. A figure on the base projected on the three rectangular faces. (See cylindrical model on next page)
J.P. DeGua de Malves:  ±1783 Special case of de Tinseau on  areas squared of the triangular faces of trirectangular tetrahedrons i.e. base area squared = sum of areas squared of other three faces.(See the prismatic model.)
The Maths Fairy: The one who makes you 'click' and also makes 'things' dissappear for Swannie.
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