From Venom71 on Mactalk, comes the question:
Within the entire Universe, large or small, where may I find a true circle?
Perplexing indeed.
A couple of posts later, thebookfreak58 says:
Erm. You can find a true circle by take a cross sectional slice of a cone (along the horizontal axis)
In face, many conic surfaces can be derived from a cone and its respective cross-sections
So, dear reader, within the entire universe, large or small, where may I find a true circle?
If we take this question to mean: where can I find something that has neither start nor end, then the answer is in your nearest copy of Harry Potter and the Deathly Hallows. It’s the answer that Professor McGonagall gives as an answer to the Ravenclaw portrait.
Ha, I’m such a nerd…
Anyway, here’s what banjo has to say:
#1. A circle is a special case of an ellipse where the eccentricity is zero — the foci exist in the same location and the equation can be collapsed in this case to explain a circle. An ellipse is not a special case of a circle because it cannot be arrived at using the simplified mathematical formula of a circle ( r^2 = x^2 + y^2 ) and there are infinite ways of expanding that equation, only one of which can explain ellipses.
#2. Statistically, you can’t. Take for example a perfect sphere. To form it would need to occur in an infinite-sized universe, with a finite amount of universal matter, an infinite distance from the rest of all other matter … and only if you overlook the fact that anything that is made of other things (e.g. a metal sphere made of atoms) will have an irregular surface (like a bunch of marbles approximating the look of a soccer-ball). To have a perfectly circular orbit, you would need one perfect (symmetrically balanced) sphere orbiting another perfect sphere at exactly the right angle, speed, rotation, and altitude with no other gravitational, magnetic, electric, or physical forces acting on it.
And, of course, a circle is only a 2-dimensional concept in a 10- or 11-dimension universe.
P.S. I could be wrong, but this is what I remember from high-school maths and physics (and a little Wikipedia research).
Followed by the answer, by Venom71:
1. A circle must be a special case of an ellipse for precisely the reasons you outlined, and not the other way around. Even if one were to use the conic method to derive a circle and an ellipse: then for each point on the vertical axis of the cone there must only be one circle but an approximate infinite number of ellipses – which empirically supports the notion that a circle is just a special case of an ellipse. (I say approximately infinite when one gets down to the Planck length as the distance between two points on the vertical axis)
2. Where would I find a circle? No circles exist, they are only mathematical constructs.
3. Where would I find an ellipse? No ellipses exist either, again they are only mathematical constructs. (The fall of shot from a gun, and a planet’s orbit do not exist of themselves)
4. Could a real and true circle and ellipse exist? No.
a) As you say, any representation of a circle or an ellipse constructed of particles must be irregular. It also cannot be perfect because of Heisenberg’s uncertainty principle as a perfect circle or ellipse requires that each particle’s exact position and velocity must both be known.
b) General Theory of Relativity may also preclude perfect circle’s and ellipses existing because they could only exist in a matterless and energyless environment. (ie: Gravity and frame dragging would impact the shape of any circle or ellipse but also any measurement device. A measurement device or even devices must be located discretely and thus would have their own “gravity well” and also be in a different “frame” to any position of the circle or ellipse they were intended to measure).
Wait – I’m not a nerd. I don’t know that stuff!!
Speaking about ridiculous maths problems:
http://random.irb.hr/signup.php
http://www.freshbytes.com.au/images/skitch/captchamaths-20081114-142332.jpg
Comments below.