Can You Trisect an Angle in a Straight Edge Compass Construction?
Simon Xu
Using a straight edge and a compass we can construct many lengths if we are given a unit length. If we take the unit length as ‘one’, we can copy its length using the compass and add it to itself to construct ‘two’; add another one to construct ‘three’; take one away from ‘four’ to make ‘three’. When we can do addition and subtraction, multiplication and division a length can be made as well to represent the answer. Here we can see how some geometrical operations are equivalent to axiomatic algebra.
So what are the rules for straightedge-compass construction? What is the limit for what number we can construct?
Rules for straight/edge compass construction:
Basic operations for straight- edge/ compass construction: Creating the length through two existing points Creating a circle through one point with centre another point Creating the point which is the intersection of two existing, non-parallel lines Creating the one or two points in the intersection of a line and a circle (if they intersect) Creating the one or two points in the intersection of two circles (if they intersect)
In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. All rational numbers are constructible, and all constructible numbers are algebraic numbers. A field of rational numbers ( ℚ ) is a set containing operations including addition and multiplication. Also, if and are constructible numbers with ≠ 0 , then ± , × , , and √ are all constructible. A complex number is constructible if, and only if, the real and imaginary parts are both constructible. In a straight-edge/compass construction, we can also create numbers such as root 2 by using the method as shown on the right, where we can construct as any rational number, and we can construct a number = + √ , where , , are rational numbers, which forms a new field, F1, based on the field of rational numbers, 𝐹0 . We can construct another number from another field by constructing = + √ where , , are from 𝐹1 , and so on we can construct many numbers and values. angle, and then prove that one third of that angle is not constructible, so we use that as an example to prove that we cannot trisect that angle. To do this, we need to prove that we cannot construct a number such as (40) , which means that an angle of 40˚ is not constructible, but as we can construct a 120˚ angle and therefore the operation of trisecting a 120˚ angle is not possible. Therefore, to prove that a given angle cannot be trisected in straight-edge/compass construction, we need to construct an
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