Pythagorean Triangles (Dover Books on Mathematics) by Waclaw Sierpinski (PDF)

Description

The Pythagorean Theorem is one of the fundamental theorems of elementary geometry, and Pythagorean triangles — right triangles whose sides are natural numbers — have been studied by mathematicians since antiquity. In this classic text, a brilliant Polish mathematician explores the intriguing mathematical relationships in such triangles.Starting with “primitive” Pythagorean triangles, the text examines triangles with sides less than 100, triangles with two sides that are successive numbers, divisibility of one of the sides by 3 or by 5, the values of the sides of triangles, triangles with the same arm or the same hypotenuse, triangles with the same perimeter, and triangles with the same area. Additional topics include the radii of circles inscribed in Pythagorean triangles, triangles in which one or more sides are squares, triangles with natural sides and natural areas, triangles in which the hypotenuse and the sum of the arms are squares, representation of triangles with the help of the points of a plane, right triangles whose sides are reciprocals of natural numbers, and cuboids with edges and diagonals expressed by natural numbers.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A pythagorean triangle (PT) is a right triangle (i. e. a triangle which has a right angle) the lengths of the sides of which are all ‘natural numbers’ (i.e. positive integers) The smallest and best known PT is (4, 3, 5).Chapter 1, section 1.2 begins: In a pythagorean triangle (as in any right triangle) the biggest side is obviously the hypoteneuse; the other two sides, called arms, contain the right angle. If these (i. e. their lengths) are x and y and the hypoteneuse is z, then by the theorem of Pythagoras,. . x^2 + y^2 = z^2[Sierpinski used superscripts, but Amazon’s text box doesn’t provide for them]Many (I think most) Americans are used to calling the shorter sides of a right triangle legs, not arms, and we might be more comfortable with ‘called’ instead of ‘obviously.’ Also, we are used to naming the legs a and b and the hypoteneuse c, rather than x, y, and z. While differences such as these make this book require a bit more effort to read, the effort is worth it for the many interesting facts you will find here.For example, on page 16 we learn that if the lengths of the two legs of a PT are consecutive numbers, b = a + 1, then (3a + 2c + 1, 3a + 2c + 2, 4a + 3c + 2) is another PT, On page 17 he lists the first six such triangles: (4, 3, 5), (20, 21, 29), (120, 119, 169), (696, 697, 985), (4060, 4059, 5741), and (23660, 23661, 33461). Perhaps because it is so obvious, Sierpinski doesn’t mention that therefore PTs exist with acute angles arbitrarily close (but never =) to 45 degrees.While extensive, the information about PTs in this slim volume (107 pages) is not exhaustive. In addition to the omission cited above, Sierpinski says nothing about infinite matrices of PTs, of which there are two that I know of. One is based on x=2r-1, y=2k, where r is the row number and k is the column number, anda(r,k) = 2xy,b(r,k) = y^2 – x^2,c(r,k) = y^2 + x^2.This has the advantage of formulaic simplicity, compared to:a(r,k) = 4rk+2k(k-1),b(r,k) = 4r(r+k-1) – 2k + 1,c(r,k) = 4r(r+k-1) + 2k(k-1) + 1.However, the latter gives a matrix in which every row is an infinite family of PTs in each of which c exceeds a by the square of the rth odd number (1, 9, 25, 49, . . .)and every column is an infinite family in each of which c exceeds b by twice the square of k (2, 8, 18, 32, . . .).watziznayme@gmail.com

⭐Elemantary facts about Pythagorean triangles (C-squared = A-squared + B-squared), A,B,C integers,developed by the famous topologist Waclaw Sierpinski, known for the Sierpinski gasket, the Sierpinski carpet,and other wonders. An understanding of high-school algebra is all you need to read this book.Book arrived promptly, clean and intact.

⭐Helps me in my personal study of Math’s various areas!! Enjoy studying Pythagoras Theorem

⭐It’s an essay on differents relationship between lot of parts (lengs, area, perimetre,…) of the rectangle triangle with integer sides, the so called Pythagorean Triangles.Don’t you expect any drawings inside it. The only figures in book they are on cover and sec. 11.7Only for Number Theory lovers.

⭐Waclaw Sierpinski may not be a household name amongst non-mathematicians, but in the world of mathematics he is highly regarded. He writes well about this particular Diophantine problem, of integral Pythagorean triples. I particularly like his derivation of the formula in Euclid: A= 2*m*m, B= m^2-n^2, and C= m^2+n^2. He goes on to consider a number of more advanced problems, including pythagorean 3D-boxes.I use his material in planning text and examination problems for Electrical Engineering. All the voltages and currents are Pythagorean triples, in the phasor-domain. For me, this book is beautiful and practical and affordable.

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