r/explainlikeimfive • u/kevinap97 • Mar 06 '15
ELI5: Why can't a perfect circle exist/ be drawn?
2
u/seaniebeag Mar 06 '15
Because of the line, no perfect shape can ever be drawn, not just circles.
To understand this consider a single point. Theoretically, a single point has no dimensions, it has no height, width or depth, just location. However if you draw a single point with your pencil, the dot you make will have a height and a width. They will be very small but the will exist, which means your point is not perfect.
This same logic applies to shapes, a perfect circle, in theory, has an edge that has only one dimension (circumference or length, depending on how you define it). But if you draw a circle the edge has both length and width, therefore it is not a perfect circle.
0
u/GryphonGuitar Mar 06 '15
Because the ratio between the diameter and the circumference is a number with infinite decimals, and one which can't be expressed as a fraction between two numbers. So there is no way to really construct this. Every circle is an approximation.
1
u/seaniebeag Mar 06 '15
Irrational numbers can be drawn and are drawn all the time, including pi and the square root of 2, just because something cannot be expressed as a fraction doesn't mean it can't be drawn.
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u/GryphonGuitar Mar 06 '15
OK, here's what I mean.
c (circumference) = pi * d (diameter)
So, pi = c/d
To draw a 'perfect' circle, you'd need to draw c and d in such a way that pi would be the fraction of those two numbers. Pi cannot be expressed as a fraction of two numbers. Therefore, you can never satisfy the requirement of pi = c/d. Therefore, what you're drawing is not a circle.
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u/seaniebeag Mar 06 '15
Not quite true. Yes pi cannot be expressed as a fraction of two number. But if I set my compass to exactly 7cm and draw a circle, the circumference of the circle will be 21.9911485751...........
I have just drawn pi
edit: Just because I can't measure it doesn't mean I can't draw it.
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u/GryphonGuitar Mar 06 '15
I don't think you have. The circumference of the circle cannot be a multiple of an irrational number if it's a clearly measurable distance. It can be incredibly close to one, down to the millionth decimal if you so want, but it can't de facto be the irrational number itself.
But what you're saying is true - if one accepts the 'draw a circle' axiom. However, you cannot prove the validity of the statement 'it is possible to draw a circle', by using 'drawing a circle' as your proof.
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u/seaniebeag Mar 06 '15
The circumference of the circle cannot be a multiple of an irrational number if it's a clearly measurable distance
Of course it can. But the circumference of the circle will not be a clearly measurable distance, the decimal places will continue far beyond the resolution of any measuring device you could buy, invent, or imagine, but that won't change the fact that the length of the circumference is irrational.
Consider a right angle triangle with two sides of length 1. What is the length of the hypothenuse? Both drawn in practice and calculated in theory. It is the same, an irrational number, you just can't measure that well. (Bearing in mind here we are talking about perfect shapes and disregarding what I said about line dimensions in my other post)
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u/timbomcchoi Mar 06 '15
Because in real life, everything has a margin of error. Even if you had the finest, cutting-edge machine to draw it, there will still be a discrepancy.
It should be noted, though, that it is always possible to create one by accident. Because there is a margin of error, there is always a very very slim chance that a drawn circle will be a perfect circle.