"Squaring the circle"

FraggleRocker: From your Post #19:

That is at best a forlorn hope.

I am certain that such a tool will not be found. Transcendentals are very special numbers, not amenable to analytic methods.

 

Log in or Sign up to hide all adverts.

All that is needed is a manufactured tool with an angle of about \(71 \, \frac{1392}{3169}\) degrees such that the cosine is exactly 1/pi. (The approximation is already better than 1 part in a billion, but \(71 \, \frac{47}{107}\) is good enough if you need only 1 part in a million precision.) I don't know if it is possible to build a physical tool that allows replication of this angle to tolerances higher that known geometric approximations to pi, but the big point isn't that it would never be physically exact, but that it goes outside the definition of straightedge and compass construction so that the mathematically modelling of the ideal use of an idealized tool would allow exact squaring of any circle which the deliberate limitations of pure straightedge and compass construction cannot achieve.

It is analogous to allowing \(\sqrt{2}\) to be a number even though it is not represented as a ratio of integers which was at one time the most developed idea of number.

 

Log in or Sign up to hide all adverts.

Squaring the circle with ruler and compass was shown to be a mathematical impossibility by Ferdinand von Lindemann in 1882. This making it like 2 + 2 = 5.

Ruler-and-compass constructions are equivalent to doing a finite number of additions, subtractions, multiplications, divisions, and square roots, starting with 1. It's easy to show that all rational numbers are r&c constructible. Also, all r&c constructible numbers also satisfy integer-coefficient polynomial equations, thus making them algebraic. There are many such equations that no r&c constructible number can satisfy, however, and that demonstrates the impossibility of r&c constructions that solve two other traditional problems: duplicating the cube and trisecting a general angle.

 

Log in or Sign up to hide all adverts.

As to what kinds of numbers there are, here's a list:

• Natural numbers: 0, 1, 2, ... -- from Peano's axioms or set theory
• Integers -- inverses for addition
• Rational numbers -- inverses for multiplication
• Real algebraic numbers -- solutions of polynomial equations with integer coefficients
• Real computable numbers -- numbers that can be approximated to arbitrary accuracy by running some Turing some number of steps
• Real definable numbers -- numbers that can be defined in a certain way
• Real numbers -- numbers that can be represented with Cauchy sequences, like infinite decimal expressions or other-base counterparts

• -1 is an integer that is not a natural number
• 1/2 is a rational number that is not an integer
• sqrt(2) is an algebraic number that is not a rational number
• pi is a computable number that is not an algebraic number
• Chaitin's constant is a definable number that is not a computable number
• I can't think of any real number that is not a definable number

An interesting quirk of infinite sets is that there are as many of all these numbers but real numbers as there are natural numbers -- aleph-0. Georg Cantor's famous diagonal argument shows that there are more real numbers than that. However, there are as many real numbers as there are subsets of the natural numbers.

Chaitin's constant is the probability that an arbitrary Turing machine will stop in a finite number of steps. Since there is no way to find that in general with a Turing machine, it is thus uncomputable.

 

Exellent prediction Rpenner!!

 

There are non-definitions like "smallest uninteresting natural number" and paradoxical concepts that don't exist "smallest positive rational number".

However the best documented examples of a non-computable real number between 0 and 1 is even more mind-bendy:

http://en.wikipedia.org/wiki/Specker_sequence

(That's non-computable, not non-definable, btw.)

 

You guys are inspiring. I'm determined now not to create an ellipsoid with the same area as a unit squircle.

But seriously, that was awesome. Is this simply a result of something simple like pi + 1 = 1 + pi, or is something more fundamental going on here?