Constructibilty

A few years ago, in ancient Greece, some dudes wanted to make geometric pictures. They generally drew these in sand with sticks, which made it difficult to draw pictures accurately. They did have a few tools at their disposal though. They had straight sticks which they could lie in the sand and trace. They also had some equivalent of a compass, so they could draw perfect circles.

Note: While this is more or less accurate, please don’t fact check me on it. There is likely some detail(s) I have falsified for my own purposes.

Regardless, there are three main problems from antiquity That people wanted to solve, but were unable. They are known as

  1. Trisecting an angle
  2. Squaring the circle (Construct a square with the same area as that of a given circle)
  3. Doubling the cube (Construct a cube with twice the volume as that of a given cube)

In order to be precise, I need to tell you the rules.

The Rules:
  1. Between any two points, you can draw a line
  2. Given two points, you can draw the circle centered at one that goes through the other
  3. If two lines, two circles, or a line and a circle intersect, you can put a point down at each point of intersection.
  4. You start out with two points a distance of 1 apart from each other.
The idea behind #4 is that if we want to know what numbers we can construct, we need a base unit to measure in.
For starters, we can construct the number two as follows:
  • Draw a circle C centered at one point p going through a point q a unit distance away.
  • Draw the entire line \ell through p and q.
  • One intersection of C and \ell is already labeled q. Label the other one r
  • Now the segment between q and r have the length 2.
This is how constructions are done. Now, this blog is not supposed to be about geometric constructions, so I don’t want to show you a bunch of solutions. However, they do make fun puzzles. Try to come up with a way to construct the following (I promise, all of these can be done):
  • You are given 1, and we constructed 2. Construct the other positive integers.
  • Given a line \ell, and a point p not on \ell, construct the line through p perpendicular to \ell. (Hint: Start by drawing a big circle centered at p… big enough to intersect \ell at two points)
  • Given a line \ell, and a point p not on \ell, construct the line through p parallel to \ell. (Hint: Use perpendiculars)
  • Given two line segments, of length a and b construct a line segment of length a+b (Hint: translate the b segment using parallels)
  • Given line segments of length a and b, construct a line segment of length ab and one of length \frac ab. (Hint: You can use parallels to make similar triangles)
  • Given an angle, bisect it (Hint: You can first bisect an appropriate line segment)
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2 Responses to Constructibilty

  1. Pingback: The field of constructible lengths, part 3 « Andy Soffer

  2. Pingback: Oops, Square roots! « Andy Soffer

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