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

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Definitions


Getting on to a definition (Cubing De Tinseau/De Malves):
The diagrams above show a tetrahedron, much like the yellow one in the photograph, lying on its non rectangular base and also opened-up to show calculated values as required for the  construction of  a 'Plato' model (3^3   + 4^3  + 5^3    =  6^3  )

Using these values the areas of the faces are found to be as follows (units not shown):
Areas of the faces:                     5,196       8,000       11,180       14,696  (= the vectors in the box)
Squaring of which gives:         27            64            125            216
Of which the cube roots give     3              4                5                6
Of which the cubes give           27            64            125     =     216   or  3^3   + 4^ 3  + 5^3    =  6^3

Accordingly, going upwards through the above, Alternative Definition of A Theorem on Cubes underlying Swannies Theorem: Regarding a tetrahedron with a trirectangular vertex the sum of the cubes of the cube roots of the squares of the areas of  the faces intersecting at the vertex equals the cube of  the cube root of the square of the area of its base. (Rather see the easier definition below in bold lettering)       

 Definition of Swannies Theorem: The tetrahedron in above figure show the areas of its faces (first decimal only).
 The following  table use these values:
Area of face:             5,2 cm2         8,0 cm2         11,2    cm2             14,7   cm2  
Base of prism:           5,2 cm2        8,0 cm2         11,2    cm2             14,7   cm2
Height of Prism:        5,2 cm           8,0 cm            11,2    cm                14,7   cm  
Volume of Prism:       27 cm 3        64 cm 3          125 cm 3        =     216 cm 3
  
Definition: Regarding a tetrahedron with a trirectangular vertex and with prisms erected on its faces so that the height of a prism equals the numerical value* of the area  of the face, the sum of the volumes of the prisms meeting at the vertex equals the volume of the prism on its base.
*1.keeping to the same unit type. 2. - or ratio thereof.
To provide for De Tinseau's projections it obviously needs some extra wording.

Hint for the construction of a trirectangular tetrahedron: Starting off with an acute angled triangle of a none special  shape and wishing to add on rectangular triangles to its sides to allow the construction of a trirectangular tetrahedron may present a problem. So try the following: Draw semicircles on the sides of the triangle. Drop a perpendicularfrom a corner/vertex  of the triangle onto the opposite side and extend it to intersect the opposing semicircle. This gives you the position of the right angle on that side. Do likewise starting from the other corners of the triangle. Check that you now have three sets of  equal adjacent sides on the perimeter of the total figure.
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