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^^^** ****INDEX**^^^ **SWANNIES ART / SWANNIES THEOREMS**

all about ****THE TRIRECTANGULAR TETRAHEDRON **

**PYTHAGORAS > DE GUA > TINSEAU > PLATO > SWANNIE**

PYTHAGORAS'S 'Areas' THEOREM (5C BC) -**25 Centuries later** -

2006:** First Theorem: **Pages 1 -** Theorems II to X:** Pages 8 -** ** **Art: **Pages 6-** ****Squaring the circle: **Page 13,** ****Ball Story: **Pages 14-** **2013:** Golden & Platinum Ratios & Fairy: ** Pages 17-** **

It takes about 80 minutes to read this.

Swannie from Port Elizabeth, SA (E-

Copyright Notice: The contents of this site is the property of Francois Swanepoel. It may be freely used provided a theorem used/mentioned is referred to; as a Swannies Theorem, First or Second etc.as applicable; and reference is made to this website (Swannies Theorems or http://swannie.webplus.net) whenever any of the contents is used.

**First Theorem: A follow- up to Pythagoras, De Tinseau**

that allows the design and building of an infinite number of models.

In addition to the wish that it will contribute towards making

the therapeutic value to a senior citizen of a quest for finding

a solution to a maths puzzle.

Whereas a problem in a maths text book -

type problem (but this was only pointed out to him

afterwards and he was also off computers for years) got him

for at least four months. During this time many little boxes of

many shapes were built out of frustration but that and the

theorem models building is another story.

Pythagoras + De Tinseau + De Gua De Malves + De Trick = Swannie**What got the theorem going:** The trirectangular tetrahedron, as per example shown above, with

with faces A,B,C seen from above and D (and the sky!) reflected in a mirror, and the three lines below! ***

Area A2 + Area B2 + Area C2 = Area D2 (De Gua De Malves)

{(A2)^1/3}^3 + {(B2 )^1/3}^ 3 + {(C2 )^1/3}^3 = {(D2 )^1/3}^3 (De Trick)

** x3 + y3 + z3 = w3** (Swannie)

*** Please note that in this article ^ denotes a power/index of the previous number or bracket eg. 5^2 = 5². a² or A cubed is shown as a2 and A3.**Below:** Non theorem shapes, just to give some colour to the page.

On the left: A dodecahedron 'cut out' from a cube such as the one it is mouted on. Easy and quick to find the volume of the cube. Not so with the dodecahedron unless you do a 'dunk it' measurement with a large measuring cylinder and water! Could one call this one's centrepiece a twisted prism or perhaps an antiprism?

**O**n the right is a creation with the Moons of Hippocratus of Chios. The sections have whole number ratios between them: 28:8:7:4. The moons get explained under Potpauri** so please for now just see it as 'Art'.** ^^^^^^^^^ *Return to horizontal Index at top of the page !!!*

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