Above: Some Pythagorean type Art; a "4:7:11 Eu du Cologne art for the eye" equivalent of the one for the nose used by your granny. You have to know how to add/mix colours and the areas before you attempt such paintings! The one on the left is not really true, too much blue! The better mix one on the right also provides a clue to perhaps how, without further ado, you can see that the painting is mathematically true. Note: BCA is a right angle and D is halfway 'up' from BA to C. The ratios between the relevant areas appears above. However, can you prove it, i.e. that the ratios between the areas of the inside triangles is the same as that between the outside semicircles. Clue: Re-
By the way -
Museum: The arrangement in the photograph on the left is what Swannie has in mind for his little science museum. It would have a 4,4m square base with the heights and areas in a 1:2:2:3 proportion and total height of 7,26m.
And then there is the quadro-
By the way and to lead you off the track: Actual values were used on a previous page to convey the workings of the First theorem as it is appreciated that the conversions can be confusing. The theorems of the Frenchmen deal with areas squared / fourth powers , fairy stuff to some of us, that Einstein and mathematicians play(ed) with. It required somewhat of a witches wand to turn it into third powers for volume. That is at least how Swannie experienced it and what side tracked him from 'getting there' for some months.
Back to where were: Model on the left: What would you call the pyramid at the centre of the yellow model to which the prisms are attached? Just use x° for those vertex angles.
March 2012 addition of new theorems -
Photographs down below: Views of a twin model. If crystals can have twins, why not models?