3
votes
2answers
121 views

Can one prove existence of incommensurables without the Pythagorean theorem?

Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the ...
2
votes
2answers
107 views

Example of a proof using the axiom of commensurability

I'm teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned "axiom" is the Greek axiom of commensurability, which stated in geometric terms the ...
6
votes
7answers
762 views

A question regarding irrational lengths in reality

I have a square stone slab 1 metre by metre, by the Pythagorean identity the diagonal from one corner to another is given as $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
2
votes
1answer
109 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...
0
votes
3answers
135 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
0
votes
1answer
218 views

Spiral of Theodorus - Discussion

The fact that $\sqrt2$ is not rational goes back to Theodorus of Cyrene from the school of Pythagoras, and is discussed in Plato's dialog "Theaetetus". Of course, $\sqrt n$ is not rational for any ...
4
votes
3answers
139 views

prove that $\sqrt{2}$ is irrational using only geometry

Prove that $\sqrt{2}$ is irrational using only geometric concepts and proofs. The proof should look like a proof in Euclid's elements or standard high school geometry. No algebra is allowed. (I know ...
-1
votes
2answers
2k views

Are “perfect” circles mathematically impossible (and do irrational numbers exist)? [closed]

It occurred to me that while $\pi$ is an irrational number, it is nevertheless the ratio between the circumference and diameter of all circles. This seems like a contradiction. Thinking about it ...
11
votes
1answer
250 views

A hole puncher that hates irrational distances [duplicate]

Possible Duplicate: Irrational painting device I have a special hole puncher that does the following: When applied to any point $ x \in \mathbb{R}^{2} $, it removes all points in $ ...
3
votes
2answers
92 views

How do continuity, distance and irrationals arise from discreteness?

Consider a square as rendered on a computer screen: its width and height are $N$ pixels each, and its area is $N^2$ pixels. Its diagonal, when measured in pixels, is also $N$ pixels long. If you ...
1
vote
1answer
163 views

Does anybody know the formula for a quasicrystal structure?

I am an architecture student researching into quasicrystals with the hope of applying it to form a complex truss system. I was wondering if anyone new of a formula for the structure? thanks in ...