Comment by peter_d_sherman

>"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."

https://en.wikipedia.org/wiki/CORDIC

lainga 3 days ago

What privileges the square root over any other fractional power?

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xyzzyz 3 days 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.

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xrisk 3 days ago

Presumably the hypotenuse of a right angled triangle.

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- gilleain 3 days 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/Ammann%E2%80%93Beenker_tiling
  
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- hinkley 3 days 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
  
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