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Straightedge and compass constructions

This is a quick note about straightedge and compass constructions, in relation to my set.

A quick definition: a straightedge is an arbitrarily long straight edge, and a compass is a circle drawing tool. Specifically, with a straight edge you can:

  • Pick any two lines and draw the line through them

  • Pick a point and draw a line through it

  • Draw an arbitrary line

And with a compass you can:

  • Pick two points, \(A,B\) and draw a circle centered at \(A\) with radius \(|AB|\)

  • Pick a point and draw a circle centered at that point

  • Draw an arbitrary circle going through a point

  • Draw an arbitrary circle

Just a quick note about a compass, assume in the questions that you are using a non-collapsing compass, that is, if you draw a circle then you can make your next circle have the same radius as the circle you constructed free of charge.

I'll call a geometric construction constructible if it can be constructed with a straightedge and compass. For example

which shows that an equilateral triangle is constructible.

A constructible number, say \(x\), is defined as follows:

If given a unit length, it is possible to construct a segment of length \(x\).

Also a shape is constructible given another shape if it is possible to construct another shape given the first one. So although a \(22\)-gon is not constructible, it is constructible given an \(11\)-gon.

Lastly a use is a single use of a compass or straightedge, as defined above. So the equilateral triangle takes 5 moves.

Again this is in relation to my set.

Note by Wen Z
4 weeks, 1 day ago

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