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>"Can a compass and straightedge construct a line segment of any length? By Gauss’s time, mathematicians knew the surprising answer to this question. A length

>"Can a compass and straightedge construct a line segment of any length? By Gauss’s time, mathematicians knew the surprising answer to this question. A length is constructible exactly when it can be expressed with the operations of addition, subtraction, multiplication, division or square roots applied to integers. [...] Remarkably, the rudimentary tools that the ancient Greeks used to draw their geometric diagrams perfectly match the natural operations of modern-day algebra: addition (+), subtraction (–), multiplication (x), division (/) and taking square roots (√). The reason stems from the fact that the equations for lines and circles only use these five operations , a perspective that Euclid couldn’t have envisioned in the prealgebra age." Related: https://en.wikipedia.org/wiki/CORDIC https://en.wikipedia.org/wiki/CORDIC

lainga 2y ago
What privileges the square root over any other fractional power?
- xrisk 2y ago
  Presumably the hypotenuse of a right angled triangle.
  - gilleain 2y ago
    Indeed. Related is the Ammann-Beenker tiling and it's connections to the 'Silver Ratio' of sqrt(2) + 1. * https://en.wikipedia.org/wiki/Silver_ratio https://en.wikipedia.org/wiki/Silver_ratio * https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling
  - hinkley 2y ago
    Hypotenuse of a 1x2 unit right triangle, to be precise. By Pythagoras, the square root of any sum of squares can be drawn trivially with a compass and a straight edge. So 2, 5, 7, 10, 13, 17, etc
- xyzzyz 2y ago
  Finding intersection points of a circle with a line is equivalent to solving a system of equations, where one equation is that of a circle, (x-a)^2 + (y-b)^2 = r^2, and the second is that of a line, Ax + By = C. To solve it, you’ll be taking square roots, and not other roots. Similarly, to find intersection of two circles, you’ll be taking square roots, and not other roots.

lainga 2y ago

What privileges the square root over any other fractional power?

xrisk 2y ago
Presumably the hypotenuse of a right angled triangle.
- gilleain 2y ago
  Indeed. Related is the Ammann-Beenker tiling and it's connections to the 'Silver Ratio' of sqrt(2) + 1. * https://en.wikipedia.org/wiki/Silver_ratio https://en.wikipedia.org/wiki/Silver_ratio * https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling
- hinkley 2y ago
  Hypotenuse of a 1x2 unit right triangle, to be precise. By Pythagoras, the square root of any sum of squares can be drawn trivially with a compass and a straight edge. So 2, 5, 7, 10, 13, 17, etc
xyzzyz 2y ago
Finding intersection points of a circle with a line is equivalent to solving a system of equations, where one equation is that of a circle, (x-a)^2 + (y-b)^2 = r^2, and the second is that of a line, Ax + By = C. To solve it, you’ll be taking square roots, and not other roots. Similarly, to find intersection of two circles, you’ll be taking square roots, and not other roots.