Plane and Solid Geometry (Universitext)
J. M. Aarts
Language: English
Pages: 357
ISBN: 0387782400
Format: PDF / Kindle (mobi) / ePub
This is a book on Euclidean geometry that covers the standard material in a completely new way, while also introducing a number of new topics that would be suitable as a junior-senior level undergraduate textbook. The author does not begin in the traditional manner with abstract geometric axioms. Instead, he assumes the real numbers, and begins his treatment by introducing such modern concepts as a metric space, vector space notation, and groups, and thus lays a rigorous basis for geometry while at the same time giving the student tools that will be useful in other courses.
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properties of figures in plane geometry in a way that is brief and to the point. We do this in Chapter 1, which also contains a survey of the properties required to read the subsequent chapters. In Chapter 2, we study distance-preserving maps of the plane, also called isometries. We show that every isometry is a reflection in a line, a translation, a rotation, or a glide reflection. We also introduce the notion of congruence, and give the classic congruence criteria. Next, we give a detailed
Sl (X) 69 m B Sl (Y ) ◦ • C • Sl (X) ◦ l A X Y Sm ◦ Sl (Y ) Fig. 2.28. A rotation is the product of two reflections the map Sl ◦ Sm , first Sm and then Sl , it is the other way around; the sign of the rotation angle is determined by the orientation of angle BCA, which is the opposite of that of angle ACB. If ψ = π/2, we saw in the last section that the map Sm ◦ Sl is a reflection in the point C. If l = m, the map Sm ◦ Sl is the identity map. We have now shown that the product of two
apple crosswise. Definition 3.2. We say that a rotation R has order n if n is the smallest strictly positive integer such that Rn = id. Thus rotations over angles π/12, 5π/12, 7π/12, and 11π/12 all have order 12. Let R be a rotation of order n. We consider the set H= id, R, R2 , . . . , Rn−1 . The elements of this set are all distinct; indeed, if Rk = Rl for some l ≤ k, then Rk−l is the identity, whence k = l, because n is the smallest positive integer for which Rn = id. If k and l are positive
Pythagorean theorem. Later, Diophantus (ca. 246–ca. 330) studied equations of the type x2 + y 2 = x3 . As was usual at that time, he looked for rational solutions. In time, the term Diophantine equation has come to mean an equation where one looks for integer solutions. Fermat (1601–1665) studied Diophantine equations of the type x2 + y 2 = p, where p is a prime number. Later, Euler and Lagrange (1736–1813) would extend this study to Diophantine equations of the form Ax2 + 2Bxy + Cy 2 = n, where
It is immediately clear that it satisfies the first two properties. The third property follows from the fact that every real number a satisfies |a| = | − a|. To verify the last property, we first note that all real numbers a and b satisfy −|a| ≤ a ≤ |a| and − |b| ≤ b ≤ |b|. Adding these equations together gives −(|a| + |b|) ≤ a + b ≤ |a| + |b| . By the definition of the absolute value, this last formula implies that |a + b| ≤ |a| + |b| . Setting a = X − Y and b = Y − Z, we obtain property 4. The
