Consequences of transcendental numbers in geometry

Question

The definition of transcendental number imposes, the restriction of their construction in geometry, i.e squaring a circle. What are the consequence(any theorem or anything else regarding this) of this in geometry? For example there are some things/operation which are possible algebraically but not geometrically. Is it a valid mathematical question or there is flaw in my understanding of transcendental numbers?

Answer 1

In general it makes sense to link numbers and geometry in the following way (or something similar to this): suppose you have one point which denotes the origin $(0,0)$, and one point which denotes the unit step in $x$ direction, i.e. $(1,0)$. Then having a number represented in this geometric framework means having a point which has that number as one of its coordinates.

When speaking about general straightedge and compass constructions, the relevant distinction is not algebraic versus transcendental, but between constructible and non-constructible. The set of constructible numbers is essentially all you can construct from zero and one using addition, subtraction, multiplication, division and square roots. So $\sqrt{2+\sqrt3}$ is constructible, but $\sqrt[3]5$ is not.

If you want being transcendental as a criterion for the decision, you should not ask to construct a number but to verify it. Suppose that in addition to the $(0,0)$ and $(1,0)$ you are given another point $(x,0)$, and someone claims that the value of $x$ is a certain number for which you can write down an exact formula. If that number is algebraic, then you can create a construction which is equivalent to evaluating the minimal polynomial of the algebraic number, so in the end you have an incidence which tells you whether the claim is true or not. At least assuming perfect accuracy for the construction, and perhaps in combination with some additional checks to ensure you got the correct root of the minimal polynomial, i.e. lies in a given interval. If the number x is transcendental, there can be no such verifying construction (with a finite number of steps), since any such construction can be translated back into an algebaric constraint for the original value $x$.

To illustrate another relationship between number fields and geometry: if you don't have a compass, but only a straightedge and a right angle (or alternatively you are working in a projective framework), then all you can construct is rational numbers. That's because joining two points with rational coordinates will lead to a line with rational coefficients, and intersecting two such lines will lead to a point with rational coordinates. The perpendicular of a rational line through a rational point is again rational, too.

I just saw an article about an axiomatization of origami, and if I remember it correctly they concluded that taking cubic roots would be possible using that. So it very much depends on what kinds of operations you allow as part of your geometry.