By G. H. Hardy
There could be few textbooks of arithmetic as famous as Hardy's natural arithmetic. on account that its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have became first and foremost in their undergraduate classes. In its pages, Hardy combines the keenness of a missionary with the rigor of a purist in his exposition of the elemental principles of the differential and essential calculus, of the homes of endless sequence and of alternative subject matters regarding the thought of restrict.
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5. Two-sided regular tilings ? ? ?? ?? ? ? ? ? ?? ? ? ?? ????? ??? ?? ?? ?? ????? ? ? ?? ?? ???
He regarded this “discovery” as his main scientific achievement, far more important than the three fundamental astronomical laws that bear his name. Fortunately for his self-esteem, he did not live to see the day when more exact measurements of the distances between the Sun and the planets showed that Kepler’s theory was erroneous. The five regular polyhedra were known to the ancient Greeks, in particular to the philosopher Plato, who expressed his admiration for their unique perfection so beautifully that today they are often called “Platonic bodies”.
For example, the Coxeter scheme of the rectangle consists of two components, each of which has two vertices joined by an edge with weight ∞, while the scheme of an equilateral triangle has three vertices joined cyclically by three edges with weights 1. 3. Theorem. 4. We omit the proof (see the book  or, for readers of Russian, the articles in the issue of Matematicheskoye Prosveshchenie cited above). 6. 1. Three planes P1 , P2 , P3 passing through the z-axis of Euclidean space R3 are given. The angles between P1 and P2 , P2 and P3 are α and β, respectively.