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  2. Spiral of Theodorus - Wikipedia

    en.wikipedia.org/wiki/Spiral_of_Theodorus

    The spiral is started with an isosceles right triangle, with each leg having unit length.Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3.

  3. Straightedge and compass construction - Wikipedia

    en.wikipedia.org/wiki/Straightedge_and_compass...

    Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.

  4. Constructible number - Wikipedia

    en.wikipedia.org/wiki/Constructible_number

    The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps.

  5. Archimedean spiral - Wikipedia

    en.wikipedia.org/wiki/Archimedean_spiral

    The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. The term Archimedean spiral is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to Archimedes' spiral (the specific arithmetic spiral of ...

  6. Automedian triangle - Wikipedia

    en.wikipedia.org/wiki/Automedian_triangle

    There is only one automedian right triangle, the triangle with side lengths proportional to 1, the square root of 2, and the square root of 3. [2] This triangle is the second triangle in the spiral of Theodorus. It is the only right triangle in which two of the medians are perpendicular to each other. [2]

  7. Theodorus of Cyrene - Wikipedia

    en.wikipedia.org/wiki/Theodorus_of_Cyrene

    Irrational square roots have periodic expansions. The period of the square root of 19 has length 6, which is greater than the period of the square root of any smaller number. The period of √17 has length one (so does √18; but the irrationality of √18 follows from that of √2).

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  9. Squaring the circle - Wikipedia

    en.wikipedia.org/wiki/Squaring_the_circle

    The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number , the length of the side of a square whose area equals that of a unit circle. If π {\displaystyle {\sqrt {\pi }}} were a constructible number , it would follow from standard compass and straightedge constructions that π ...