Me and my friend were having a chatting and he jokingly told me to sit in the corner of a round room…being the nerd I am I actually thought about this and we got into an argument on wether or not a circle had corners.
My argument was you can break up a circle into an infinite amount of points. Every point has another point immediately adjacent to it. The point where those 2 points meet is a third point inbetween and connecting those 2 points. Since a these points will never be at an angle of 180 degrees… I say those 3 points form an angle, and therefore a corner.
Now, i’m not sure a “corner” can be defined by 3 points an infintesimal distance apart. Or if a corner can be defined by points at all…
does anyone know the mathematical definition of a corner/angle?
If the round room has a flat floor, your question is moot.
I see your theory as similar to the arrow shot at a target that can never get there, because it always has to go half the distance first. Interesting, but not applicable in real life. For a circle to have a single radius all around, it has no corners.
I did a very quick google, and couldn’t find a definition for corner. I don’t think there is a precise definition for the word to be honest, which means you could define corner as you have done, and prove a circle has corners.
My personal feeling though is that a circle doesn’t have a corner. To be “in a corner” implies you are somehow isolated at least on some sides, more so than other positions in the room. This definition doesn’t fit in with any of the positions within a circle.
I was about to agree with this, but no, that’s not quite how it works. Angles and corners are not the same thing. You can have a straight angle which is just a 180° angle. It looks just like a straight line. Squares have millions of them. But squares only have four corners.
And that’s how it is with circles. The only reasonable measure of an angle as part of a circle would be 180°, and we don’t count straight angles as corners.
Agree with the arrow analogy. You can take a shape that has corners, make it more and more circle like, until we say it’s a circle in the limit. But an actual circle does not have corners. It has curves.
You’re playing a semantic game more than a geometry one. If you define a corner as where edges intersect, i.e. a cube has eight corners and 12 edges, then your round room has no corners.
As the distance between the three points approaches zero, the angle described approaches 180[sup]o[/sup]; like this
So in fact the closer you look, the less you see ‘corners’. When the spacing of the points is actually zero, the question as to the angle described by lines connecting them becomes meaningless, but just before that, the line connecting the three points is nearly a straight one.
Either this is wrong, or you mean something different by “immediately adjacent” than what I would mean. I don’t think it makes sense to talk about points being immediately adjacent. For any two distinct points on a line, circle, or other continuous curve, there are infinitely many other points between them.
In the real world there are no corners. Or not ideal ones. Just like the knife cannot be exactly sharp. In the real world there is no such thing as a point or line either.
Mathematically, the way a circle is usually treated is a regular polygon with n sides where n tends to infinity. So a circle mathematically has infinite sides or corners.
Again in the real world there is no such thing as a circle.
Heck, in the real world, most finite areas are usually enclosed by perimeters which can be treated as infinite.
The classic answer to the question - How long is the border of the US with Canada - the answer is depends on what scale you use to measure it
I think the most convenient mathematical definition of a “corner” would be a discontinuity in the first derivative of the curve’s function, the first derivative providing you with the slope of the function. So if there is a sudden jump in the graph of the first derivative of a function, you have a corner. If there is no such jump - basically, if the difference in value between two points on the graph can be made arbitrarily small by bringing the points close enough together - then you don’t have a corner. This nicely avoids such nonsense as “adjacent points on the circle”.
By the above definition of “corner” (which seems to me pretty intuitive, YMMV), a circle would have none.
Jeff
I suppose if you were sitting in a round room you could call the intersection of the wall and the floor a corner. So you could take a chair, lay it on its side and against the wall, then you would be sitting in a corner
Because (planck lengths notwithstanding) if the points are separated by a distance, then a point can be placed at the halfway mark; if the points are not separated by a distance, then there is only one point, not two.
Remember that the definition of ‘point’ here is a zero-sized theoretical set of co-ordinates.
