By Eli Maor, Eugen Jost
If you've ever idea that arithmetic and paintings don't combine, this beautiful visible background of geometry will swap your brain. As a lot a piece of paintings as a ebook approximately arithmetic, appealing Geometry provides greater than sixty beautiful colour plates illustrating a variety of geometric styles and theorems, followed through short money owed of the attention-grabbing historical past and folks in the back of every one. With art via Swiss artist Eugen Jost and textual content via acclaimed math historian Eli Maor, this specified get together of geometry covers various topics, from straightedge-and-compass structures to fascinating configurations related to infinity. the result's a pleasant and informative illustrated journey during the 2,500-year-old historical past of 1 of crucial and lovely branches of arithmetic.
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Extra resources for Beautiful Geometry
Example text
And, indeed, the long string of decimals in our illustration is not the square root of 2, just a close approximation of it. N ot e : 1. More precisely, the Pythagoreans discovered that the numbers 2 and 1 are incommensurable—they do not have a common measure. That is, one cannot find two line segments of integer lengths m and n such that n ⋅ 2 = m ⋅ 1. Had such line segments existed, it would mean that 2 = m/n, a rational number. 9 A Repertoire of Means A nother subject of great interest to the Pythagoreans was how to find the average, or mean, of two positive numbers.
2 P'S' meets the extension of P'Q' at R', with Q'R' = H. And since we already know that G ≤ A, it follows from the similarity of the triangles P'Q'S' and S'Q'R' that H ≤ G. Combining the two inequalities, we get H ≤ G ≤ A, the arithmetic-geometric-harmonic mean inequality. It is truly remarkable that the circle— perhaps the simplest of all geometric constructs—holds within it so many hidden features waiting to be discovered by a keen observer. No wonder the Greeks held the circle in such high esteem.
He tried to prove it but failed, so he posed the question to Euler. Euler, whose mind was occupied with more pressing mathematical problems, shelved Goldbach’s letter; it was discovered only after Euler’s death in 1783 among his enormous volume of correspondence. Despite numerous attempts to prove the conjecture or find a counterexample, Goldbach’s conjecture remains unsettled. Until a few decades ago, the primes were considered the ultimate object of pure mathematics, existing in the ethereal universe of number theory and devoid of any practical applications.