/sci/ - Science & Math

>>6552765
A circle, by definition, has no corners.
>3/10

>>6552770
This.
Except in computers, then they have a ton of corners.

>>6552765
infinite amount

>>6552765
Mail part was the best part.

Alsio, made me think of Dylan's song (how many roads...).

If a corner is defined as the vertex of two lines, a circle has 0 corners.

>>6552765
No physical object can exist as a perfect circle because particles from any relative view would always appear in a "connect-the-dots" fashion; so, it would always have a number of sides.

As many as you want baby

>>6552815
>because particles from any relative view would always appear in a "connect-the-dots" fashion;
you're embarassing yourself here.
Wait for physic classes. Real world is not a lego game made of small bricks.

>>6552765
None.
While the shape of a regular convex polygon does approach a circle when the number of corners approach infinity, it doesn't become a circle since infinity doesn't exist

-1/12

>>6552858
If two lines can be drawn on three points on a circle, you have a corner/non-zero angle of intersection of the two lines.
The corners do exist.

>>6552765
>How many corners does a circle have?
Define "corner."

>>6552791
I would have never guessed that Bobby reference
>>6552974
vertex/intersection of line segments

>>6552972
There's no such thing as a perfect mathematical circle. So there wouldn't be any lines to draw to yield a corner.

perfect circles do exist, they do in trigonometry anyway.
for any value r,
rcisx
0 < x <= 360, or
0 < x <= 2pi.

A circle has no corners, a circle is a representation of a line that goes out an equal distance in all angles.

>>6552987
A circle has no line segments, but it has infinite vertices.

So no corners under that definition.

>>6553076
0 < x <= 360, phone went full retard.
or
0 < x <= 2pi.

>>6553079
chanu is broken when it comes to symbols I guess.

a circle is an ideal object with no corners and perfect symmetries everywhere

>>6552860
You 'avin a giggle m8?

>>6553078
To have vertices you must have line segments. A vertex is present where there is a corner by definition.

>>6553112
>To have vertices you must have line segment

Says fucking who?

If I plot a single point on a plane, is that not a vertex?

>>6553120
No.

It isn't.

it has one corner: the circumference

>>6552858
>it doesn't become a circle since infinity doesn't exist

A circle by definition is an infinite set of points.

However, they can't properly be called "corners" because a corner is the intersection of two lines or curves at an angle other than 180°.

>>6552780
not if you define them right.

>>6553494
The way computers deal with curves is as a series of line segments.

You'd have to have some pretty alien (and therefore inefficient) methods of drawing curves to have anything else.

>>6553500
>a series of line segments

That depends on the application. Vector-based graphical applications may display circles and ellipses as many-sided polygons. However for raster based, the circular form will be sampled pixel by pixel, so there are no "sides", only blocks.

Internally the circle may simply be represented by an abstract formula, and sampled on a point-by point basis. Again there are no segments.

Infinite corners


A point <span class="math">p \in \mathbb{R}^n[/spoiler] is a corner of a convex set <span class="math">A \subset \mathbb{R}^n[/spoiler] iff there exists vectors <span class="math">a,b \in \mathbb{R}^n[/spoiler] such that
<div class="math">\{p \} = A \cap \{x \in \mathbb{R}^n: a^T x = b \}</div>

>>6554196
a circle isn't convex

>>6554206
Do you even know what convex set is?

>>6554219
do you? or do you not know what a circle is? either could be causing this confusion

>>6554221
Are you trolling me?

>>6554224
circle != closed disk

>>6554227
>disk
>not a circle
>>6554224

He might be anon

>>6553079

why not 0< x <= tau?

>>6554229
"A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre."

"In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle."

wikipedia

>>6552765
>implying those aren't the same thing

>>6552858
>because infinity doesn't exist

Before postulating something doesn't exist, I'd advise you to think about a reason why it shouldn't exist. If you can't think of one, at least be modest enough to phrase it like:

>I don't know whether the assumption of infinity is intrinsically incosistent, but since I'm a super hipster I'll just restrict my attention to finite mathematics.

If you take a shape like a pentagon, it has a certain number of corners.

If you then take a hexagon, it has more corners, and is slightly more circular..

If you repeat this ad infinitum you eventually will have a circle, with infinite corners and the space between them being zero.

QED.

>>6555325
>you have a circle
No you don't, you have an infinitely sided polygon.

>>6555369
And what would that look like, pray tell?

>>6555374
"looks like" != is

>>6555378
Actually if there is no discernible difference between two things they are for all intents and purposes the same thing.

ie .999999 = 1

>>6555382
Except that's false, there is a discernible difference.

There's no such thing as a line segment with a length of 0, so you can't have a vertex with both segments as 0.

Vertexes=corners, so a circle has no vertexes.
Because your infinigon has infinite vertexes, it cannot be a circle.

>>6555395
Just as there's no such thing as a polygon with infinite sides. Obviously. Just as there is no true circle.

The point is though that at the limits of side length and side number zero and infinity respectively you essentially get a circle.

>>6552972
What do you mean by this? Are you saying that if you chose three points along a circle and draw lines connecting them, those are now corners? Because if you are saying that, then you could pick any three points on any shape, and the notion of corners becomes meaningless.

>>6554206
>>6554221
Convexity doesn't really make sense without a choice of embedding and a metric on the full space anyway.

>>6552765

A perfect circle should have 0 corners.

However nature is not perfect, so there are no thing such as a "circle", it's just a concept.

Computer circles have a ton of corners, it's funny that in order to simulate something with zero corners, we need tons of corners, even infinite corners.

Is zero = infinite in this case? No one knows....

>>6555403
What about apeirogons?

>>6555451
>apeirogons
Whoops. I meant to say regular polygon.

But they don't exist either because infinity =/= real

2

>>6555446
>yfw there are plenty of perfect circles in physics

>>6552860

>>6555932
yeah maybe in high school physics.

i dare you to name ONE actual PERFECT circle.

>>6555976
>i dare you to name ONE actual PERFECT circle.

Stop talking.

>>6552765
Define "corner."

>>6552974
Deja Vu
>>6556019

If a corner is a point of nondifferentiability where the two sides form a right angle then none.

>>6556009
Tomatoes are fruits.

We have no way of knowing if Amazons really did burn their tits off, or even if they really existed beyond shreds of tenuous evidence. What matters is the Greeks thought they did.

Your picture is dumb.

>>6556009
Dont bash stupidity, being stupid is part of the learning process

>>6556411
tomatoes are fruit based on the biological definition but because of Nix v. Hedden in the US,
>Tomatoes are "vegetables" and not "fruit" within the meaning of the Tariff Act of 1883 based on the common meaning of those words.

>>6556419
>biological definition

And if we look back to the comic
>Biologically, tomato is a fruit

>Biologically

>>6556019

Any linear space distortion on the 2nd dimention.

>>6556421
While there is a Botanical definition of a fruit (the sexual organs of an angiosperm) there is no real agreed-upon definition for vegetable. When we think of the distinction between vegetable and fruit, we are thinking of a non scientific regulatory distinction

>>6556742
fuck off you idiot, tomatoes are fruit even if edgy pic says it's stupid. Learn to science.

>>6555445
A METRIC
LOOK AT THIS FAGGOT AND LAUGH

>>6556742
So we have fruits and non-fruits from a biological standpoint.

Who cares what vegetables are, they're not even a part of the equation.

What, then, is the problem with saying that a tomato is BIOLOGICALLY a fruit?

Not culturally, culinarily, or by any other metric. I mean biologically and botanically.

>>6552765
Infinite.
Curves couldn't have vertexes if they weren't made of corners.

>>6556994
And who says curves have vertexes?

They don't have any line segments, so what's there to join two segments together?

How many corners does this shape have?
How many sides?

isn't a circle the limit of a polygon with n sides?

>>6557225
But your shape would ALSO be the limit of a polygon with n sides.

those rounded edges contain an infinite number of vertices according to that line of thinking, so what's the difference between this and a circle in that regard?

>>6552765
The question has no answer, since vertices are only defined for polytopes, which a circle is not.

>>6556019
also acceptable

>>6556994
>>6552786
>>6552815
nope

>>6556009

infinite
every imaginable number which describes the distance between the starting point and a point on the line of a circle, e.g. 0.00000001 , 0.00000002 ,... has an impulse defined to it which describes the ongoing in the next step

>>6557703
smbc is hillarious. and he isnt smug. if you want smug, go suck randal's dick.

>>6552972
1) you would have to connect three points next to eachother
2) there is no "next" point
3) fail

also, the angle would be 180°

I cant find it right now, but there is a picture that tries to approximate a circle with a polygon. It starts with a square with sides L, and a circle inside it with r=L/2. Then it "bends inwards" the cornes of the square making a dodecagon. The perimeter of the dodecagon is the same as the original square. It repeats this process an infinite amount of time.
The result is an infinite sided polygon, but the perimeter is 4*2r not pi*2r.

Is there a flaw in this logic?

>>6557790
Forgot to say that when it bends the corners inwards, the new "inner" corners lie on the circle.

>>6557419
Nothing, just four corners have extremely disproportionate sides.

>>6557790
>Is there a flaw in this logic?

Yes, obviously.

>>6552765
Infinite because it can be defined as the limit of n-gons as n->inf

>>6557783
also, the angle would be 180°
The angle would be approaching 180 as the points became closer.
180 is the limit

It heavily depend on the definition of a corner. But because any reasonable definition must be invariant under rotations, either circle have no corners or all of it points are corners.

>>6554472
this

>>6554191
You can display them however you want if you code it from scratch. there is no circle primitive in opengl but you can still make one, it's quite simple, just slow cause it doesnt run on the graphics card

>>6555325
repeating ad infinitum does not give a mathematical structure per se and its nature can be different from the entities used in the iteration steps. Therefor, you need to carefully construct such an iteration and show what the result would be.

In any such construction resembling your informal one, the result will consist of countable many vertexes, while the circle has an uncountable amount of points. If you wanted to use the fact that the lines between your vertexes have uncountable cardinality, this would result in a 'circle' with straight edges.

Done right, you would probably get something like all rational points on the circle, which is indiscernible from a real circle when drawn, but different non the less.

>>6559954
0.99999... is also indiscernible from 1, but it's still 1.

>>6561117
The rational points on the circle are discernible from the whole circle dip shit. For example by cardinality. Only, the difference does not show when drawn.

>>6552765
Ok so i think everyone would agree that the real question here is "what is a corner".

But it can be hard to give a reasonable definition for degenerate cases like a circle.

What i would say is close to this annon >>6554196
Take A a convex set of a vectoriel space, a point p is called an extremal point is A\p is still convex. And before someone say " a circle isn't convex" yes i know, but a closed disk is and remember that the corners of a triangle and the corners of a "full" triangle are exactly the same and it goes for every polygon/full polygon.
Now the extremal points of a convex polygon are exacty its corners. And every point on the circle is an extremal point of the disk.

So i would go for the infinite corners.