Proportion: Science, Philosophy, Architecture by Richard Padovan

By Richard Padovan

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It remains to show that the triangle ACG inscribed in the semicircle is the same as the triangle ACG contained in the cube. 16 Proof that the square on the diameter of a sphere is triple the face of the inscribed cube. , vol. III, p. 478. < previous page < previous page Page 151 page_150 page_151 next page > next page > But AG/GO=3/l, so AG2 is three times CG2, the square on the side of the cube. Similarly, AG2/AC2=AG/AO But AG/AO=3/2, so AG2=3/2 AC2, and AG2:AC2:CG2=3:2:1 as in the cube. Therefore the triangle ACG inscribed in the semicircle is the same as the triangle ACG contained in the cube, and AG is the diameter of the circumscribing sphere of the cube.

4). 2 Demonstration of the theorem of Pythagoras. 3 Square root of 2. Â Â Euclid, Elements, Dover Publications, 1956, vol. I, p. 349. 5). Now consider this double square as one half of a complete square ABCD, bisected by EF. Construct a semicircle with the diagonal DE of this half square as radius and E, the midpoint of AB, as centre. Let the semicircle cut AB extended at GH. 6). Then the ratio :1 occurs as DE:AE and as GH:GK. Thus GHLK is a rectangle. 6, and the star-pentagon. Let us start with the first source.

Heath, Euclid: The Elements, vol. III, Dover Publications, 1926 p. 438. â 386, a mathematician or group of mathematicians not directly attached to the Academy, embarked on a â programmeâ to rigorously construct the regular polygonsâ ¦The inscription of the pentagon ledâ ¦to the area definition of DEMRâ ¦7 â that is, to a precise mathematical knowledge of the golden section. Only if the architects were well ahead of the mathematicians in this field is it likely that knowledge of the golden section could have developed in time for it to be the basis of the proportions of the Parthenon, as Jay Hambidge and others have speculated.

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