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Spiral of Theodorus
In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral)[1] is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.
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[edit]Construction
The spiral is started with an isosceles right triangle, with each leg having a length of 1. Another right triangle is formed, an automedian right triangle with one leg being the hypotenuse of the prior triangle (with length √2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is √3. The process then repeats; the ith triangle in the sequence is a right triangle with side lengths √i and 1, and with hypotenuse √(i + 1).
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[edit]History
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells the reader of his achievements. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.[2] Plato quoted Theaetetus speaking to Socrates:
It was about the nature of roots. Theodorus was describing them to us and showing that the third root and the fifth root, represented by the sides of squares, had no common measure. He took them up one by one until he reached the seventeenth, when he stopped. It occurred to us, since the number of roots appeared to be infinite, to try to bring them all under one denomination.
Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.[3]
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[edit]Hypotenuse
Each of the triangles' hypotenuses hi
Spiral of Theodorus
In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral)[1] is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.
-------------------------------------------------
-------------------------------------------------
[edit]Construction
The spiral is started with an isosceles right triangle, with each leg having a length of 1. Another right triangle is formed, an automedian right triangle with one leg being the hypotenuse of the prior triangle (with length √2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is √3. The process then repeats; the ith triangle in the sequence is a right triangle with side lengths √i and 1, and with hypotenuse √(i + 1).
-------------------------------------------------
[edit]History
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells the reader of his achievements. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.[2] Plato quoted Theaetetus speaking to Socrates:
It was about the nature of roots. Theodorus was describing them to us and showing that the third root and the fifth root, represented by the sides of squares, had no common measure. He took them up one by one until he reached the seventeenth, when he stopped. It occurred to us, since the number of roots appeared to be infinite, to try to bring them all under one denomination.
Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.[3]
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[edit]Hypotenuse
Each of the triangles' hypotenuses hi