منابع مشابه
Trisection, Pythagorean Angles, and Gaussian Integers
Pythagorean angles, that is angles with rational sines and cosines, provide an interesting environment for studying the question of characterizing trisectable angles. The main result of this paper shows that a Pythagorean angle is trisectable if and only if it is three times some other Pythagorean angle. Using the Euclidean parametrization of Pythagorean angles, this result allows an effective ...
متن کاملPythagorean triples, rational angles, and space-filling simplices
The ancient Greeks posed and solved the problem of finding all right triangles with rational sidelengths. There are 4 natural nonEuclidean generalizations of this problem. We solve them all. The result is that the only rational-sided nonEuclidean triangle with one right angle is the isoceles spherical triangle with legs of length 45 and hypotenuse 60. We next ask which simplices have rational d...
متن کاملPythagorean Triples
Let n be a number. We say that n is square if and only if: (Def. 3) There exists m such that n = m2. Let us note that every number which is square is also natural. Let n be a natural number. Note that n2 is square. Let us observe that there exists a natural number which is even and square. Let us observe that there exists a natural number which is odd and square. Let us mention that there exist...
متن کاملPythagorean Triples
The name comes from elementary geometry: if a right triangle has leg lengths x and y and hypotenuse length z, then x + y = z. Of course here x, y, z are positive real numbers. For most integer values of x and y, the integer x + y will not be a perfect square, so the positive real number √ x2 + y2 will be irrational: e.g. x = y = 1 =⇒ z = √ 2. However, a few integer solutions to x + y = z are fa...
متن کاملPythagorean Descent
where B(v, w) = 12(Q(v + w) − Q(v) − Q(w)) is the bilinear form associated to Q. The transformation sw is linear, fixes the plane w⊥ = {v : v ⊥ w}, and acts by negation on the line through w. These properties characterize sw. We will use reflections associated to the four vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1), and e1 + e2 + e3 = (1, 1, 1). The vectors e1, e2, and e3 form an ort...
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ژورنال
عنوان ژورنال: Colloquium Mathematicum
سال: 1959
ISSN: 0010-1354,1730-6302
DOI: 10.4064/cm-7-1-103-105