Download e-book for kindle: Foliations in Cauchy-Riemann geometry by Barletta E., Dragomir S., Duggal K.L.

By Barletta E., Dragomir S., Duggal K.L.

ISBN-10: 0821843044

ISBN-13: 9780821843048

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Finally, why did Liu Hui dissect the three squares into exactly fourteen pieces as opposed to twenty? Archimedes (287BCE212BCE), a Greek and one of the three greatest mathematicians of all time—Isaac Newton and Karl Gauss being the other two—may provide some possible answers. 15 on the next page. In the Stomachion, a 12 by 12 square grid is expertly dissected into 14 polygonal playing pieces where each piece has an integral area. Each of the fourteen pieces is labeled with two numbers. The first is the number of the piece and the second is the associated area.

My own intuition tells me that two complimentary observations were made: 1) the area of the lightly-shaded square and rectangle are identical and 2) the area of the non-shaded square and rectangle are identical. Perhaps both observations started out as nothing more than a curious conjecture. However, subsequent measurements for specific cases turned conjecture into conviction and initiated the quest for a general proof. 10. Euclid’s proof follows on the next page. 10: Annotated Windmill 38 First we establish that the two triangles IJD and GJA are congruent.

41 We close this section with a complete restatement of the Pythagorean Theorem as found in Chapter 2, but now with the inclusion of the converse relationship A 2  B 2  C 2    90 0 . Euclid’s subtle proof of the Pythagorean Converse follows (Book 1 of The Elements, Proposition 48). The Pythagorean Theorem and Pythagorean Converse Suppose we have a triangle with side lengths and angles labeled as shown below. 12 on the next page shows Euclid’s original construction used to prove the Pythagorean Converse.

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Foliations in Cauchy-Riemann geometry by Barletta E., Dragomir S., Duggal K.L.


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