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

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Golden R


                           An Introduction to Swannies Platinum Trirectangular Tetrahedron and its Prisms

The GOLDEN RIGHT- ANGLED TRIANGLE

The Golden Ratio   1,6180 (to 4 th dec.) is covered  by Google. Presented here, however, is a somewhat different approach to it and also a Golden Right-angled Triangle with its squares to serve as an introduction to the Platinum Ratio 1,8393 (to 4 th dec.) and the Platinum Trirectangular Tetrahedron with its prisms; which constitutes, if assembled, a, to be called,  perfectly proportioned Swannies First Theorem model or rather Swannies Platinum Model,  next to the not so different Stainless Steel Plato's Model that appears on  the Home Page and also below.

`The conventional Fibonachi sequence starts with 1 + 1 = 2, then 1 + 2 = 3 etc. always adding the last two values to give the next one.   Here is shown part of the sequence  and below it the ratios  between a value and the previous one fluctuating up and down and settling down to reach, to the fourth decimal, the Golden Ratio:

1    ;    1      ;     2      ;     3      ;     5     ;     8     ;     13     ;     21   ;    later    144    ;    233    ;     377
 1,000   2,0000   1,5000   1,6666   1,6000  1,6250    1,6154                            1,6081     1,6180

Staying with positive numbers, including fractions, you can however start with any two numbers, including your month and day of birth, any way round. Elsewhere I mentioned the 4, 7, 11 ... Eu du Cologne  sequence! You always end up with a 1,6180 ratio  between one and the previous one. First of all one might think of adding a length unit to the numbers and compare linear properties ( Yes there are such examples) . However with the triangles that now follows the numbers are used  to represent area units and later on  with the tetrahedron they represent volumetric units.

A sequence starting with 9, 16 and then 25 is of course the well known 3^2 + 4^2 = 5^2 used to illustrate the Theorem of Pythagoras. Partly copying the above exercise we get by addition:-
9     ;    16    ;     25    ;      41     ;    66       and later       453    ;     733     ;     1186   
 1,7778    ,5625    1,6400    1,6098                                      1,6181      1,6180

Now 9 +16  =  25 or 3^2 + 4^2 = 5 ^2 gives you a nicely numbered  Pythagorean right angled triangle. However 453 + 733  =  1186 or 21,2838^2 + 27,0740^2 = 34,4384^2, the last three figures above gives you a (very near) Golden right-angled Triangle so called as the Golden Ratio applies to the AREAS of squares of/on its sides ((Note, not to confuse, that the name  Golden Triangle is in use for another (isosceles) triangle  somehow also defined by the Golden Ratio.))   

To see how this Golden right-angled Triangle compares with the triangle drawn to the start of its number sequence  3^2 ; 4^2 ; 5^2,  also start off with 3^2 = 9,  then  multiply it with the Golden Ratio = G = 1,6180  and then again with G as follows:
                                     9  + 9G  =  9GG
                           9  +  14,5623   =  23,5620      
                         3^2 +3,8160^2  =  4,8540^2
Still close to its 3^2   +   4^2      =  5^2  origen but now perfectly proportioned!

Below on the Left: A Golden right angled triangle with above dimensions in inches, its squares shown to be made up of 9, nearly 16 and nearly 25 square inches; just slightly smaller/more compact than its start of the sequence 3"^2, 4"^2 and 5"^2 would give.
Below on the Right: The existing steel model (prisms mounted on tetrahedron) with cubes of the same volume as the prisms shown next to it, the number of cubic inches of the three smaller cubes adding up to the 216 cubic inches of the large cube below. The exercise that follows on the next page gives a very similar model, as it is closely related; just more compact, and lets say, better proportioned .

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