User:Dalene Oertel/sandbox

SWANNIES THEOREMS
Swannies Theorems feature the trirectangular tetrahedron and in particular the areas of its faces, with shapes drawn thereupon and attachments fixed thereto as per definition.

SWANNIES FIRST THEOREM 2006
The First Theorem is a spatial/volume-type follow-up to the area-type theorem of Pythagoras and further more to that of De Tinseau and its special case that of De Gua De Malves. It culminates in the construction of models made up of five parts of which the central connecting part is a trirectangular tetrahedron and the remaining four has a volume relationship such that the volumes of three of them add up to that of the fourth.

In its basic, special De Gua form, the volumes of prisms fitted onto the three rectangular faces of the tetrahedron add up to that of a prism fitted onto the base of the tetrahedron. The prime example of such a model is one representing Plato's 3^3 + 4^3 + 5^3 = 6^3 cubes and which is shown above on the right.

Definition (As for the special De Gua case): 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, keeping to the same unit type, the sum of the volumes of the prisms meeting at the vertex equals the volume of the prism on its base.

De Tinseau's theorem provides for projections of any enclosed shape drawn on the base onto the rectangular faces. A Swannies De Tinseau model with a right circular cylindrical base is shown above on the left.

Formula: x^3 + y^3 + z^3 =  w^3   (Note x, y, z, and w does not refer to any specific linear dimension of a model) The above description should be adequate for anyone wishing to complete a model with an available tetrahedron. Building one to specified values such as Plato's cubes, requires equations to be set up to find the three right angled sides of the tetrahedron. (See Reference below) Note: The choice of a unit of measurement determines whether a model is slender or stubby (e.g. cm or inch).

Proofs for the theorem can be lined up with both Pythagoras's  d^2 = a^2 + b^2 + c^2 equation for the diagonal of a box (using the dimensions as vectors) and De Gua De Malves's  area squared theorem.

SWANNIES SECOND THEOREM 2010
It is an area theorem with the same built-up as given above for the first theorem, up to finding projected images on the rectangular faces. However is this case the largest shape similar to the shape on the the base is fitted inside each projected image. The areas of the three similar shapes thus found adds up to that of the shape on the base of the tetrahedron. See above diagram and model. Also see the diagrams below. Note that his theorem allows for a number of variations and extensions. (See: Reference below)

Definition: The sum of the areas of shapes on the rectangular surfaces of of a trirectangular tetrahedron that are similar to the one on the base equals its area provided the most shortened corresponding dimension between the shape on the base and its projection onto a face is used as the key dimension for drawing the similar shape on the face in each case. General Formula: x^2 + y^2 + z^2 = w^2 Green to red projection to blue. Proof when all fits into a square. TRIRECTANGULAR TETRAHEDRON        > SAME WITH FACES FLATTENED OUT. WITH EQUILATERAL  BASE                 RED + BLUE + GREEN =  BLACK BASE ABC TO MAKE THE FIT: The diagram on the right shows the rectangular faces ADB, BDC and CDA of tetrahedron ABCD with equilateral base. The red, blue and green triangles are also equilateral as per definition. Flipped over the red triangle covers triangle EFG. Likewise flipped over the blue and green triangles overlap the red one on triangle EFG. The areas between A and E, Band F and C and G remains uncovered. Divided into similar little purple triangles as shown triangle EFG contains 12 excess little triangles due to overlapping while the uncovered areas have the exact same space to accommodate them. Models: As with the First Theorem objects (solids) can be fitted onto the derived shapes, subject to the following requirement: The objects fitted onto the rectangular faces shall be in each case be of the same height as the one on the base, or vice versa, and as near as possible in kind to it. Trirectangular tetrahedron with equilateral*                 Semi sphere on the  base with semi base. Regular tetrahedrons fitted onto the                     ellipsoids of the same height on the rectangular faces with a pyramid of the                         rectangular faces. same height on the base. *Special case with equilateral base tetrahedrons models only here shown, simply because they were easier to build.

REFERENCE:

1. See http://swannie.webplus.net for more definitions, proofs, models, 'Art' and e-mail contact. 2. Pythagoras 3. D'Amons Charles de Tinseau 4. J.P. De Gua de Malves.

Swannie =   Francois P. Swanepoel, Port Elizabeth, South Africa