How many corners does a semi-circle have?
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- Опубліковано 17 чер 2024
- Most people think it can be 0, 2, or infinite. But I think 3 corners is a perfectly acceptable answer too.
0:00 question
1:30 preliminaries
4:18 answers 0 or 2
5:16 infinite
6:37 answer is 3
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As a former sports car racer I would say there are three corners. One secret of racing fast is geometry; you try to make tight turns into larger radius corners that can be safely navigated at higher speeds. When doing this I would think of the track as an extended plane and my path like the marks left in the ice by a figure skater. So this shape has a long continuous corner we often call a sweeper. Then two abrupt corners for a total of 3. Maybe the answer is influenced by your perspective.
That is absolutely correct, this problem is up to interpretation and your definition of a corner.
Your comment inspired me to research the types of corners in a racetrack. Very interesting perspective! www.reddit.com/r/formula1/comments/vw2v5a/a_little_guide_to_corners_that_i_found_in/?rdt=49678
Not a corner
No. Those are edges not corners. This whole concept is just incorrect. The answer is only 2.
@@apophenic_ not an edge. Corner.
Difficult questions are good but questions that rely on interpretational ambiguity are problematic.
Before someone comes in to say that "ambiguous questions are good because everything's ambiguous in the real world" or whatever.
The difference is that the real world has many solutions, that are all "correct" under the interpretation. A math exam like this one doesn't. If your interpretation is different from the exam writer you fail even though it would pass in the real world.
While they are problematic, they also serve a very good purpose of creating discourse and encouraging people to work through problems together comparing differing views.
Problematic problems! They're the best kind 😁
Yes, this is more like a semantic question than a math question.
@@cberge8 Just as @jffrysith4365 already said:
These kind of questions are great in a learning situation. As en example in a lecture (such as this video) they promote understanding rather than rote learning.
They are also great essay questions on a test, but hey are bad for simple right/wrong testing because they are ambiguous.
The teacher who came up with this question should sit in the corner and think about what they did.
Lol! Nice one! We’d have to do a better job of defining “corner” though first.
but the room should be semi circle
@@markstahl1464 Isn't a corner an action in soccer?
@@LazloNQ Those who can, do; those who can't, teach; those who can't teach become bullies.
Not if the textbook says: " An angle requires two STRAIGHT lines intersecting at one point." and "A circle is a collection of a point equal-distance from a point and therefore has no straight edges". Given these two premises, the answer would be 0. And if that's what it says in the textbook, students should know it's 0. It does not matter what the parents think.
It's all really context-dependent and exams are strictly in the context of what was taught by the teacher or textbook.
From my experience in 3D modelling, I'd say at least 30. But beyond that, you quickly get to the point where it looks so close to perfectly round that you can't tell just from looking at it!
lmao trueeee
From my experience in CAD modeling, I say it's 2. It's a cylinder that has half its area padded by a rectangle drawn from one side to another through the centerpoint, so it inherits 2 of the rectangle's corners.
@@michaelbuckersa semi circle resembles a sphere in my opinion
?
3d modelling has nothing to do with this since that's an approximation to math, not actual.
@@ldgarius 3d modelling is just malipulating a 3d shape into a different model like a jet
This reminds me of a similar question I had in 6th grade. The question asked to identify all the rectangles and had a bunch of shapes. Along with the obviously wrong ones, like triangles, circles, etc, there was a rectangle with long width and short length, one with long length and short width and a square. I said a square was a rectangle but was marked wrong. Teacher wouldn't listen when I tried to argue it.
In high school we had another one, this time asking to identify the pyramids. Similar thing with the obviously wrong answers, but the right answers included a square based pyramid, a tetrahedron and a cone. I was the only one in class to say a cone was a pyramid. My classmates all said it wasn't beslcause it didn't have any "sides", meaning flat faces that meet at the apex. I said it's a circular based pyramid, which means it is a pyramid with either 1 or infinite sides that meet at the apex. Teacher agreed with me this time.
I would say that a square is just a special case of a rectangle. How about a rectangle with sides 10, 9.5, 10, 9.5 - ie a nearly square rectangle. How different do adjacent sides have to be for a square to become a rectangle?
Growing up, I was taught a square is a rectangle
@bladeofSteele Likewise, a circle is a special case of an ellipse, whose major and minor axes are the same length.
Your 6th grade reacher was just wrong. A square is a rectangle, because it fits the definition of a rectangle. However with the pyramid, it was you who was wrong. The more general shape is the cone, a pyramid is a cone whose base is a polygon.
All squares are rectangles, your teacher was wrong.
If two curves meeting at a point counts as a corner, it's weird to think that a curve and a straight line meeting at a point would not count as a corner.
The anchor point seems to follow conventional physics in a way keeping the object from lapsing along the y axis. Why assume it can only do that in one direction. Infinite Anchor points around the whole perimeter of the object would be justifiable as if it’s orientation of the object relative to the applied force changed or if multiple forces were applied, say in a positive static pressure which actively surrounds us, not necessarily air because our natural body exerts greater pressure than ambient air, but say water at a depth of 8,000 feet. With a question that is so easily undefined without limitations to one’s own justifications for the multiple answers, it would seem best to assume infinite corners. After all a corner is just objectively a sharp turn that can be observed. A continuous radius is also a continuation of infinite turns producing infinite corners even tho they are harder to observe and I explicitly point out, that doesn’t demote them.
@@TheNuclearBoltonyou literally make no sense
@@TheNuclearBolton
A point that falls on a defined curve is by definition not a corner, but a point on the curve...
I would argue that something cannot be a corner if the derivative of a function to describe it is not discontinuous (like the derivative of |x| (the absolute value of x). That may not be enough to describe something as being a corner, and it may not actually be enough to fully disqualify something from bein a corner either. This was just a quick thought I had.
@@TNH91i had the same thought. You cant just integrate over corners a cicle is no problem but if the semi cirkle is turned a little, its a problem.
Ran into this same kinda situation recently. Except it was 1 divided by 0. Teacher said the answer was 0, I told her it was undefined. She said I was wrong. I asked her if she had 1 apple and divided it with 0 people would she have 0 apples or still be holding one. I got a nice email from the school superintendent telling me that I shouldn’t question the teachers because it undermines their ability in front of my child.
This happened.
It is well known that you can't divide by 0. The teacher was incredibly undereducated. And also not able to think properly, beacause you don't need any eduaction to understand the example with the apple.
I remember a video by Eddie Woo where he goes explaining further that x/0 is also undefinable
last week i had a teacher that said anything divided by 0 is 0
Do you share it with yourself?
2:51 - "Don't let the cold air inside the home" is actually an example of convection and quite valid.
It's generally an example of advection, not necessarily of convection.
I had a similar question in my 3rd grade history exam. The question was true or false: The pyramids are hollow. I answered true, cause my thinking was that there are some rooms for the tombs inside it which makes it hollow, just with really thick walls. It was marked incorrect and I'm still angry about it.
I find your reasoning entirely justified.
That's funny, I'm 62 and still a little salty about a third grade project I got counted off for 😂
When does an object become hollow?
If I have a solid glass marble with a tiny air bubble in the middle, is it hollow?
Is a birds egg hollow because it has an air sac?
You have a fair point with the pyramids but the room space is very small compared to the total area taken up by stone.
But how could you get inside the pyramid if it is the opposite of hollow?
All answers are correct but not all questions are correct 😊
@@johnnyfearlesszrx Essentially the paradox of the heap. At what point does an internal cavity in a solid cause the solid to become hollow?
I'd say, things can be more or less hollow. But only when it reaches a certain level of hollowness, then we actually call it that. So maybe the question should be, is the pyramid hollow *enough* ?
True story. My sister was a grade school teacher. She had to give a standardized intelligence test. One question was “can you jump higher than a house?”. A very smart young girl answered “yes”.
My sis pulled her aside after the test and asked why she answered “yes”.
“Because a house can’t jump” she said.
🧠 🧠🧠🧠🧠🧠
Man this is one of the oldest jokes in the book
Depends on what kind of house, a dog house is easily out jumped.
I read “horse” and started thinking about if jumping is well defined. Horses basically thrust themselves forward and up like in a long jump, but a “proper jump” is from a static position and up.. and then I realized I was thinking about the wrong statement…
*Baba Yaga has left the chat*
But isn't that also just interpretation? To me, the question sounds like its asking if you can jump past the height of a house.
It feels like its asking if its possible for you to jump that high. I mean, I guess its creative how she thought about it differently.
I would argue for two 2 corners because:
1. I don't accept just any vertex as a corner, because that would mean there are unlimited corners for non-point shape (even a single straigt line). In order for a vertex to count as corner, I would argue that the lines meeting/intersecting at the vertex must have an angle between them, aka have different directions at that vertex.
2. If you now approximate the half-circle part of the semi-disk with an n-gon of vertices and let n go towards infinity, the inner angle at a) the two intersection points with the straight line goes towards 90° and b) the inner angle at all other vertices goes toward 0°.
I would argue that, as a consequence, there may be an unlimited amount of vertices (and in Adobe-speech: unlimited amount of control points), but only at two of those there is a measurable angle thus counting as corners.
it does make sense to define a corner as a discontinuity of the "slope" (dy/dx) of a curve. However, does that mean that if you have a polygon drawn with rounded corners, it has none?
Technically, I think in graphics the control points are not those that are points on the curve, but rather the points that control the curve that goes through the main point (these are called Bezier curves). Conceptually, the curve at the defined point is tangential to the line between that point and the control point on each side of the point. If the point and the two control points are colinear, then the curve is smooth. Otherwise, the curve has a distinct vertex at that point. So, the semidisk is defined by two points that have non-colinear control points, and one that does (the one at the "top" of the circle).
If a circle is defined as an infinite set of points which are all at an equal distance away from a central point, then a circle wouldn't have vertices because of its definition which does not mention edges or lines. With this definition a circle can be visualized irrespectively of the medium used to visualize it such as atoms or quantum phenomena
However for a computer a circle defined this way is impossible to manipulate. The computer must quantize the circle into an n-gon with a very high but not infinite number of vertices to be able to manipulate a circle. An n-gon is mathematically different from a circle as it involves edges connected at vertices while the definition of a circle does not involve these things
are the angles actually measurable?
I've always liked the definition of a corner as a point where the slopes (derivatives) of the two lines or curves do not match when the lines or curves intersect/meet.
that's good one. formally it wold be something like when the limit of derivative approaching X from one side is not equal to limit of derivative approaching X from the other side, that point X is corner
yep, thats essentially how I was finally able to differentiate b/w what is and what is not a "corner", the moment i learned calculus and was told, you cannot differentiate a "corner" so to say was the moment I was like ohhhh that's what it was
@@lukasdolezal8245 Formally you would probably define it by parametrising the curve at a constant speed and the corners would be the places where the velocity would be discontinuous
I wish I could have answered "This is not a properly defined question, so there is no answer." on a few of my high school tests. There definitely were a couple times where a clarification of terminology was needed to properly answer the question, and I spent way too much time overthinking it.
Same here, and this applied to math and literature. So I would sometimes just have to guess. And get it wrong.
They would be the only questions I get wrong, it was almost always just cruddy phrasing
I would get mad in physics all the time bc there was so many questions worded so poorly that I could easier argue for multiple different correct answers
As a current Grad Instructor for some courses, I think I would take an answer like that if you did a little work to prove how there are multiple distinct ways to answer the question. You wouldn’t have to solve them all just start them a little bit and say there’s no correct answer
I have done it many times.
I did write this on enough tests. Also the adjudicators were usually there for questions and would make an announcement if an error was discovered.
Nothing like wrecking a young person's confidence and making them hate math at a young age. Way to go, Teach!
I don't think saying a kid had a "wrong" answer is necessarily bad, I would have liked it if the teacher were able to talk to the child and see what he though of to come up with that answer.
@skeletorrises6325 they already know why the student came up with that answers. It's a trick question based on a poor drawing. The actual drawing shows corner, it's not hard for the teacher to see why they'd put that. Maybe if the bottom had no outline representing the end of the semi-circle, I'd agree with the teacher.
Such ambiguity. Not only is it not considered an corner, but they also gave an example not corner with an angle of 90 degrees 🤣
@@tiacool7978 Yes, but a semi-circle is a two-dimensional shape (not just the 1-dimensional outline), so even if you don't draw the bottom, you could shade in the semi-circle (even if just in your mind) and "see" the corners.
@@DogMan077 Having an answer marked as wrong when it's correct definitely is bad and could lead to a deterioration in self-confidence that doesn't arise when being corrected for a for an actually incorrect answer. In the latter case, the child can at least understand where they went wrong, and it becomes a teaching moment. In the former case, a child might start thinking that their own thinking is wrong, because being told it's not a corner makes no sense.
that's why I used have a love-hate relationship with math classes in school. Some answers are simply a question of how you interpret the question and what model you use. Teachers however tend to be stuck in a right/wrong mindset.
my old physics teacher got it right. if we could properly justify our answer in an exam we'd get our points no matter what his correction sheet had to say about it.
Here's what's missing. They don't lay out any axioms. Or even, ask the students to lay out axioms.
This was my entire educational experience in school. Multiple possible answers but teachers refusing to acknowledge or accept.
Most of the time the answers to these tricky questions are specifically stated in the text book or homework and is partly testing if you actually paid attention or are just trying to use common sense to answer, which isn't the point of the class.
Dont you love it when youre 8 and your teachers hit you with a philosophical query disguised as a math question and then you lose grade for it
No. (I get the joke, I was just adding this for comedic effect.)
Questions (and marking, particularly) like this seem perfectly and deliberately designed to make people hate mathematics, who might otherwise actually quite enjoy it. I remember a few questions like this back at school, luckily I was stubborn enough not to be discouraged by it and to just tell the teacher that they were wrong (or that their answers were incomplete and subject to interpretation), and move on.
My degree is in physics and my wife is an elementary teacher, so we’ve both seen a lot of math, just from vastly different perspectives.
We both agree though, that the way we teach math to people in grade school is terrible. Kids aren’t being taught the logic and reasoning, they’re just taught to memorize. Once they’ve memorized enough, they’re just shown a bunch of different pieces of math without really understanding how or why they connect. It’s a system which pushes kids away from math, even those who would otherwise do quite well in the subject.
@@isaiahmumaw The whole point of learning math in high school is to sharpen your brain to be trained in structured reasoning. There was a discussion the other day with teenagers about "Why do I need to learn Pythagoras?" And they were right in the sense that very few people need that in their adult life. But it is the training that is transferable to all other aspects of life.
But if math is taught without training your brain, that is very bad. Like having gymnastics without exercising your body.
Yeah, I think a question such as this lacks intellectual sincerity. Good mathematicians (and scientists and engineers) take care to communicate precisely.
Another type of geometry problem I’m tired of seeing is those in which there is a figure that was deliberately drawn in such a way that the labels on the edges and angles contradict the proportions of the shape as it is drawn. One is supposed to reason about the shape based on the labels while ignoring the contradictory proportions of the shape. These questions are ubiquitous on standardized tests in America. No professional mathematician deliberately draws misleading figures to trick students or the readers of a paper or book.
Young people in school might take math more seriously if it’s presented with sincerity as something to be taken seriously.
yeah if you look at this question from a elementary standpoint then it is so obviously 2. that teacher was on cocaine
@@isaiahmumaw Not being taught the underlying logic of mathematics was also my biggest complaint about it when I was in school. I remember being extremely frustrated that none of my math curriculums included good explanations of the mechanics of the things I was expected to learn. Math being all about the interactions of of rules and abstractions, it makes it more difficult to understand without regular discussions of the nature of those rules.
Having worked in Adobe Illustrator for the last 20 years, I'd say a corner and a corner point (or in other programs, a vector point) are two different things. You can create a corner from a vector. You can create a curve from a vector. I usually think of them as vectors, having also worked in 3D where similar points do the same thing.
The one control point in the middle of the curve is not needed to make the shape. There could be any number of extra control points on that curve between the corners and they would all be extra and not needed to make that shape. IMO, even in the design contrived situation, there is still only 2 corners, or, it can be any number of corners that you want to add...
And why would the definition of corner in graphical design be a valid answer on a math test? There are a lot of math words that have uses in other places.
I was thinking this too. As someone who has worked with vector graphics, I find "3" as a valid answer incredibly weak.
If you count the midpoint of a curved line as a "corner" you can count any point on any line as a "corner". So you're back at infinity.
Just throwing another variation into the mix: inside corners & outside corners. Likewise your semi-circular table example has four corners (lower and upper side as well as each end of the disc). So you could argue that the table has 8 corners (four inside four outside) and the flat 2D disc has four (counting inside vs outside corners).
It’s all semantics and the way you frame the question and define the parameters is hugely important to the correctness of the answer.
Corners are corners, no inside or outside since lines have no width. That wold be like saying a line has a perimeter.
@@ldgarius tables and the shapes in question here do, however. One corner of a square has both inside and outside corners, how else would you describe the different angles?
Question then is whether the semi circle really has corners if it doesnt contain them. You could say the space containing the semicircle has two corners and the semicircle has the other two.
@@deltalima6703I like this answer better.
The argument for three is wrong for a handful of reasons. The most interesting reason is that bezier curves cant actually draw a circle only approximate it.
You are right, but any digital representation of a circle can only be an approximation. However, setting that aside, the argument still isn't valid. You can easily make the same shape using only 2 "corner points".
@@atomicus5000 That solution would also imply that 4 corners is valid, 5, 6, 7 and so on..
@@fluktuition That's a good point. I guess then there would only be one incorrect answer of 1 perhaps? Anyway, I still don't think what some graphics programs define as a corner should have any influence over any of this. They could have called them points of applesauce because it only matters within the context of the software.
It's not wrong, but a different definition.
@@ryanjackson0xif it's in a math exam for a math class, then in math context it's wrong to say 3, because graphic design isn't math.
A better definition of corner is a point on a shape that has no well defined tangent. This also works in any dimensions, is simple and natural. It aligns with what we think of as a corner. Based on this, the shape has 2 corners.
Might need to work shop it a bit. The end of a line segment where these is a discontinuity (such as with a piecewise function) would also not have a well defined tangent. Actually as I write it, including the word "continuous" might be enough to fix it.
*Edited for one grammar mistake*
@@FrogworfKnightagree. I think the most natural definition of "corner" (without any other context that may motivate a different definition) would be a discontinuity in the tangent function along a continuous path.
@@FrogworfKnight Yes, but when talking about corners in a closed shaped, what OP says makes sense.
Except since all corners by definition have an angle, they have a very well defined tangent: just divide the angle in half and form a 90º angle to it. This works in any dimension, big or small, is actually simple and natural, and alligns with what we actually think of as a corner.
@@ldgarius at what point does the tangent function achieve that value? I think you'll find it hard / impossible to create a reasonable version of that with no discontinuity
Three still doesn't seem right to me because the third point is not distinguishable from the others and therefore it could be anywhere or an infinite location. There isn't anything that makes the center of the arc any different than 1/4th of the way through the arc or 1/5th, etc.
The problem of taking the limit of an increasing number of corners is you reach a countably infinite number of corners for your semicircle, which is something I would absolutely consider distinct from a semicircle which would have an "uncountable number of corners" in some sense. Think of it this way:
You can't say the group U(1) under rotations is isomorphic to the natural number line under translations, even though you might like to. You could do that with the real number line, though.
The problem is that a corner is a discontinuity within a curve. The half disk has two discontinuities at the points where the chord line meets the arc segment. The difference between the n-gon and the half disk is that the n-gon is not a continuous curve, so there are n corners, but the question refers to a half disk, not an n-gon. When n is infinite, the curve becomes continuous, so that there are no corners along the arc segment
This
This still boils down to your definition of a corner. But yes, a discontinuity in the slope of the outline loop would be my definition too
Just in case, don't you mean non differentiable?
@@prodbytukoo I did not want to unsettle anyone who might suffer PTSD from Calculus. 8-P
Lol, by this logic infinity-corner figure can't exist. But it can since we accept it as an answer. Which means contradiction, demonstrating that your argument is wrong
I voted 2, and I stand by my vote. To me, a corner is a point at which two lines (straight or curved) join at an angle -- if you zoom in close enough to those corners, you can't tell if the lines are curved or not, so they might as well not be. And, if there's no angle (0 or 180 degrees), then there's no corner, so a circle has 0 corners, and a half-disk has 2. Not 0. Not infinite. Not 3.
So you're saying you can't measure angles between two points of a circle?
The 3 is the number of control points you need to define a curve. Calling them corner points (just because they are that fir piecewise straight curves) is a bad and confusing model. (yes, I agree with you, just helping out with some of the arguments)
The infinitesimal straightness is a very good argument in my opinion.
@@Fexghadi Of course you can, but the curve is continuous so the angle is 0.
@Fexghadi it's a curve, not trillions or quadrillions of lines making a polygon. We make digital models using lines but the definition of a circle is that it's a locus of a point equidistant from a certain point, so circles DO NOT HAVE any corners
I find that having a discontinuity in the tangent line is the most useful and intuitive definition of a corner
I like that this was a video that didn’t end in “and that’s the answer”. Those are fun too, but leaving you something to think about is fun too.
In general, an arbitrary sector of a disc has three corners: two of them at the ends of the arc and another one in the center of the original disc.
A semi-disk is a kind of a sector, with the central angle of 180 degrees.
A corner may be visualised as a “discontinuity”. And in this case, consider that at the “corners” two lines meet. One line is the diameter & the other is the infinitesimal line segment of the circular arc.
The discontinuity you talk about is the non-Frechet-differentiability of the curve, so yes , its 2.
Edit : non-frechet-differentiable and Gateaux-Differentiable
Non-smooth, not discontinuous
Non-differentiable is maybe a better word for what you're describing
It's not a discontinuity, it is just not differentiable at that point
@@andrewneedham9839specifically continuous and non-differentiable
In Illustrator, there are two kinds of anchor points: corner points, and smooth points. Smooth points have handles which are linked, so they're collinear with the point itself, while corner point handles form some kind of angle with the point. The semicircle shape therefore has two corner points and one smooth point.
I can’t speak for illustrator since I’ve only ever barely used it. PowerPoint has similar points with corners and you can make the semi disc with just two corners. You don’t need the smooth point. To get there, however, I’ve always needed to create a triangle, then define the corner points, then delete the smooth point.
Also that shape is not a circle; Bezier curves can only approximate circles.
@@EthanRooke Well, piecewise linear approximations of the circular arc are also just approximations. As far as I'm concerned, if the minimal Bezier approximation using 3 vertices is acceptable, then all other Bezier approximations with more vertices must also be acceptable. That would then mean that the answer to the thought question is "all non-negative real integers except 1".
@@PSUQDPICHQIEIWCI suppose there are situations where even a line segment can be a useful approximation for a semicircle
Points ?
I'm personally of the 2 corners camp. But maybe these kinds of shapes could be described with with a more exotic corner metric. You get 2 whole corners from the discontinuities at the base, and then for example you add the ratio of the curved path length of the total path length. So, you'd end up with approximately 2.6 corners on that shape. A circle would then have exactly 1 corner.
I think 0, 2, and infinite all make sense. I'm glad that you gave the explanation for 3 corners, and it makes sense, but the issue I have with that answer is at that point, you could also have 4, 5, 6, or any integer value as an answer of the anchor points.
So, if you're going to include 3 as an answer, I think you'd also have to say a semi-circle (half disk) can be said to have any number of corners as long as the number is a positive integer and isn't 1.
I get the infinite corner interpretation, however I feel a corner is where the rate of change is discontinuous. The curve has a constant or continuous rate of change. Therefore it should be considered a segment. The rate of change is only different or discontinues when it meets the straight line. If you are willing to consider a curved line as composed of an infinite number of corners then you'd have to accept the same with a straight line and that seems to be less than useful.
Discontinuity in the rate of change of angle seems like a good definition of a 'corner' to me. However, if the coursework that the kid was supposed to remember had defined a 'corner' as an angle ( > 0 < 180 ) between two connected straight lines, then in that context, 0 is the only correct answer.
@@oldmossystoneimo in any case you need a bit more of mathematical maturity to properly define it, if you can't it's just ambiguous.
@@oldmossystone Why would they ever define it as being between two straight lines? Like, what is the point of that definition? It can't be for simplicity's sake if you then ask them about curved lines.
I disagree as a constant rate of change implies the line is straight. It then becomes discontinuous when it starts bending and curving. Though I'd say two, as a corner is usually an area where you cannot differentiate and find an instantaneous rate of change. And two of these spots exist on the semicircle
I mean the rate of change is constant, not that it's a changing value. An arc has a constant rate of change defined by a continuous function. For a line this is simply a constant, for an arc it would be a polynomial of degree 2
@@dominicballinger6536
I’m definitely in the 2 camp. A corner, to me, is a discrete change in the slope/derivative of a line or curve. This happens twice on a semicircle.
0 reduces it to just the intersection of two straight edges, which is too narrow a definition. Exhibit A: the coffee table with two worrisome corners.
Infinity, by contrast, is too broad. It changes the definition of a corner to *any* change in slope, undermining the idea of a corner being abrupt/sharp. You can’t stand in the corner of a circular room. Now, real life actually doesn’t have any continuous changes in slope. Continuity requires an infinitesimal, which does not exist. Everything is just a bunch of tiny edges and corners (often sharper than we realize) between atoms. But in the spirit of models being useful even when they are wrong, the macroscopic structure of an object is good enough, and pure math world can have those precious infinitesimals.
3 is just a graphics thing.
Infinity is an even worse answer than you say. The same limit method used to prove a circle has infinite corners can 'prove' a square or even a line segment also has infinite corners - just pick the right infinite series of increasingly sided polygons that converge to whatever final shape you need.
It's a good example of how infinity often breaks our intuition, and you have to be very careful inferring properties of limits from the properties of their generators.
yeah I would only say 0 if it had not been a closed polygon.
@nodrogj1 No you're wrong, infinite corners can only and strictly only produce a circular arc
@@fahimnabeel606 Maybe a concrete example will prove my point then.
Consider a series of line segments put end to end with length 1/n, with alternating left and right turning angles of 1/n² degrees. As you crank up the value of n, this alternating zigzag pattern will very quickly smooth out and approach closer and closer to a straight line segment of length 1. The limit of this process as n→∞ is thus a line segment in the exact same sense as was true for the circle example, but at each step there are exactly n-1 'corners'. By this construction, the line segment must then have infinite corners. If you insist on a closed path example, you can just connect 3 of these 'line segments' together into an equilateral triangle shape with 3n sides and 3n corners at each step which again quickly converge to an equilateral triangle, and which again have infinite corners.
As I had alluded to, this is actually a well known problem encountered by mathematicians using calculus, and isn't a problem specific to corners: mathpages.com/home/kmath063.htm
Actually a circular room is more accurately defined as a cylinder and therefore there are 2 corners, one at the top of the room where the ceiling meets the wall, and one at the bottom where the floor meets the room, if you want to stand in the corner of a circular room then you just have to stand against one wall, but then a circular room still needs a door and doors for the most part are flat and therefore require a flat wall to be placed into thus introducing more corners to the now near-circular room
Professor: how many corners are in a half circle?
Carpenter: Do you really wanna know?
I was having trouble seeing how three corners was a conceivable answer. After your explanation, I see the midpoint of the semi-circle as an point of inflection, where the Sine function changes direction. Totally understandable and valid. Thanks!
When designing an arc on a computer, you need at least 3 points. In this video, 2 points give you the width, and the 3rd top gives you the height. It is easier to visualize if the apex of the arc isn't in the center, giving you two different shaped arcs. That 3rd point on a technicality is a "corner"
The question on the test didn't match the graphic. The question said semicircle but the graphic was a semidisc. I interpreted the poll the other day as a semicircle without the straight line, so I said 0. If I was taking a test and saw that graphic, I would have certainly said 2, thinking they were trying to clear up any ambiguity with the graphic.
Same for me, i didn’t understood why so many people agreed that the 2 end points of the cuvrved line were angles. Now, it makes sense
What is the difference?
@@BriBear The difference is that a C and a D aren’t the same shape. And so for me a C has 0 angles but a D has 2, but i said 0 for the poll becausd i was thinking of a C
So, it is common to imagine a semicircle as a circle *cut* in half, and cutting something IRL generally yields two bounded (closed) pieces, so it's easy to imply that the straight edge is there.
it was only the boundary of a semidisk
The only answer I don't understand is 0. Because even by the definition of a corner being the intersection of two straight lines, if you were to place a tangent at each "corner" at exactly 90°, that would satisfy that definition. And the 2 tangents are in the shape itself if it were to be looked at closer and closer (zoomed in) to the angles.
I mean, sure, but we're not considering the intersections of the flat side of the circle and the tangents on either side. That's not what the question asks. If we can just insert imaginary tangents wherever we want, every shape with a curve has infinite corners.
presh asked this question a few days ago, but he didn't include the image of the semi-disc as he did in this video. If you define a semi-circle as half of a circle, without the straight line connecting the ends, then 0 corners makes sense
It is zero if you consider a semi-circle as one half of a circle, which is only the curved bit, not the straight diameter closing the shape (which is termed a "half disk" in this video). A semi-circle, using the stricter definition, will only be a single edge/line with no intersections, and so no corners.
The problem is that the image in the example from the school shows a half disk, not a strict semi-circle. So for that case, zero corners can never be correct. Now, does the written version of the question take precedence over the drawn version? Who decides that? In my opinion, the teacher, sorry "an múinteoir", should have marked both zero and two as correct. Three is too specific to certain tools, and not a generally correct solution. Infinite is also wrong, as that breaks the entire concept of corners for many shapes, and is meaningless in a learning situation at that level.
@SirBrandonKing yes I understand that but I'm not adding tangents to the shape I'm only using tangents to illustrate lines that are already in the shape itself. The tangents are only to show the lines that are already there more clearly.
@mytube001 thanks for explaining this makes a little more sense to me now 👍
Thank you for the video. After all your explanation, I further strengthened my conviction to Team *2-Corners*
While some have expanded the scope of this question tangentially (such as what constitutes a 'corner' in auto racing), I'm going to keep this cut-and-dry by staying within the context of basic geometry.
Long story short, a 'corner' in geometry is the point where two perpendicular lines converge (at a 90° angle) - we can unanimously concur that a rectangle has four such corners. Some might take this farther by asserting that any such convergence constitutes a corner, without regard for the angle of incidence, such as a triangle having three corners. Though this assertion has its merits, it quickly breaks down as both a circle and a straight line can be said to have an infinite number of angles (and by extension, an infinite number of corners). Much as all squares are rectangles but not all rectangles are squares, all corners are angles but not all angles are corners.
By definition, the radius of a circle is perpendicular to the edge of its circle. Also by definition, any arc (pie, wedge, slice, etc) of a circle is bound by two radii - a semicircle is simply an arc which specifically encompasses 50% of a circle, with the radii forming a 180° angle.
Any arc (again, including a semicircle) has two points where perpendicular lines converge at 90° angles - hence, two corners. A 90° or 270° arc goes one step beyond as the radii also converge at the center at a 90° angle - hence, three corners.
tl;dr
The kid is right.
The teacher is wrong.
A semicircle has precisely two corners.
"Interpret an image - grade F because your interpretation is invalid"... I think whoever was that kid's teacher is doing a great job at ruining their students' creative vision. Might as well become a literature teacher
There's a difference between reasoning and having fun with a question that isn't well defined, and putting such a question on a test. A test should never contain ambiguities.
Tell that to the DMV
You're assuming the teacher never defined what a corner is for them.
Do you have the entire test and/or school curriculum?
Because all i have is one out of context question.
That is the point there is no general overall definition, so making test question about it is wrong.
Also teaching them one definition when there's more wrong as well.
This gave me flash backs to all the times "colour" or "realise", for example, has been marked incorrectly wrong and I'd have to have that awkward discussion (that often got me yelled at) that these spellings were also correct English and not wrong.
But one positive thing did come of it, one teacher in Highschool told me "Well if it's so "correct," then try it on your writing ACT and see what you get."
I did, and I got a 12 out of 12. Later the same teacher told me she was fairly sure I was the only person in my senior class to get a perfect writing score.
For me its the opposite
Hate it when my teacher tells me "color" and "center" are incorrect
like, how it be "centre", its not pronounced cent urr, it's cent er
I went to high school in GA, and our junior year we had to take a high school graduation exam and some other test because they were transitioning from one to the other. My literature teacher didn’t like me because I had an undiagnosed at the time learning disorder. I was one of a handful of students who 100% the literature part of the test. She basically stopped interacting with me for the rest of the year and didn’t look at me.
It can be said that , as there are two sharp points in a semicircle, the derivative of its equation at those two points is not defined. As corners of any regular polygon has derivative not defined, then it can be concluded that those are corners of a semicircle.
All those answers are justifiable but in reality 2 makes most sense, in real life because of the table example and another example would be if you walked in to a room that was shaped like a semi circle you’d consider it to have two corners. If some one asked you to put lamps in the corners you’d immediately know what to do. No one would be confused thinking where the hell do I put these lamps.
true lower class math is more about "how would you approach if it were real life" and higher grade math is more of the "theoretical" math that most STEM people think about. but its the failure of school to not be able to accommodate children that are able to thinking more than just of what if it were real life. like what! you are telling me that critical thinking ability are supposed to be punished in school, and the teacher did not even ask the student about their reasoning is just worse
a room can't be shaped like a semi circle.
@@mrossknewhy not? Circular rooms exist, I’m sure semicircular rooms do too
@@tacotuttle they don't, since a room is by definition enclosed by walls on all side. a semicircle is not.
@@mrosskne ah, youre saying that it should be called a half disk room instead of a semicircle room
The ambiguity in the problem comes from three main sources: the text, the visual, and the classroom lesson. The text refers to a semi-circle, which is different than what is shown in the picture. The student is being asked to answer the question based on what the classroom lesson was, which follows the text (semi-circle), not on the visual model presented, the semi-disk. I have had a similar learning moment with my children when trying to explain to them that a line is 1D, because they argue they can measure the thickness of the line (the visual representation of the line).
good that theyre thinking critically about it!
a line is 2d, a point is 1d.
@@adrian4276no. I’m curious what your rationale here is.
@@LK-on6rwlines can only be represented / exist in a space containing 2 or more dimensions.
@@LK-on6rw umm, mathematical descriptions of terms. the poster mentioned trying to explain that a line is 1D. That is false. a point is 1d, no width, length, or thickness. it is essentially a concept. a line is 2d, the distance between two points yet has no width or thickness, also mostly conceptual. 3d is our physical world, length, width, and thickness. Does this help clear up your confusion about my rationale?
Mathematically a corner can be taken as a point where the endpoints of 2 straight edges meet. In that case there would be 0 corners to a semi-circle. But a corner can also be where the endpoints of any two edges meet, whether the edges are curved or straight. So the semi-circle would have 2 corners where its diameter endpoints meet the curved arc.
language of "Mind Your Decision"
There is also a different 3 corner model. If you define the semicircle are being defined by an 180deg angle at it center, with the center point being a 180deg corner (which is totally valid because you would consider it a corner at every other angle). Adding the the two corners on the side you get 3 corner.
When working with univariate functions, corners are actually pretty well defined. A corner is a point where the function at that point is continuous, and the one-sided limits as the derivative of the function approaches that point are not equal (disregarding any endpoints to the function). This semi-disc can actually be defined by a single univariate function using polar coordinates if you use a reference point at the center of the shape, and if you do that you will see that there are only two places where these corner conditions are filled.
Could you call it a "curvature singularity"?
@@simontist Yeah I guess, if you define a singularity as a point where a function is undefined or not well behaved. Vertical tangents also cause a "curvature singularity", producing asymptotes in the derivatives, so that's not just exclusive to corners. Only corners have a derivative where the left limit is not equal to the right limit for such singularities.
@@Darth_InsidiousI mean curvature as defined using intrinsic coordinates, not X and Y, so it doesn't depend on direction. It's basically "as I travel along this line, how curved is it?"
Now _this_ is something I can get behind. Absolutely stellar. Yes, there are only 2 corners in a semicircle as commonly referred to instead of the 0 corners in a mathematical semicircle (which does not include the line connecting the ends of the semicircle arc).
This however means that certain "corners" in common sense won't be considered corners in this one
how about defining a corner as a point on a line where continuity of direction of tangential line breaks? this will rule out "imaginary corners" that depend on how the figure was built (those leading to answer "3"), and would be closeest to common perception
yes that's a great definition of a corner
"Continuity" you say. But the semi-circle is continuous, no matter how you look at it. You might be thinking of "smoothness". But you need to define what "smoothness" is before basing your answer on it.
You could define a corner as a point where the line/curve is not differentiable.
@@BeyondKawaii they said continuity of the tangent line, not continuity of the semicircle.
@@JoeBorrello that's the same thing as op said, innit?
For answer 3 if you consider it's bezier curve, then it's not a half of disk, because you can't represent arc of circle using bezier curves. If you use other splines, for most of them there is a way to subdivide without change of shape, so the number of control points is determined by designer
Before watching the vid, I'd like to post a comment of my own perspective on this problem. The way we define a corner is a place or angle where two sides or edges / lines meet. And yes the definition does specify it to be either a plane or straight line, If u look at the semi circle carefully you'll find it only has one straight line and a circle intersecting the line at two points along it's diameter, and we're only viewing the yellow shaded portion on the 2d plane. You can argue that the line "actually" intersects with the "tangents" drawn on the circle at two point of contacts, hence it has 2 corners of exactly 90 degrees, but there are infinite number of such tangents u can draw on the semi circle and they'll also intersect eachother at an angle tending towards 180+, so infinite corners.
The problem was that they asked for the number of corners in a semi circle and then provided a semi-disk as a reference.
Since the semi circle is just 1 curve, it does not have any corners.
No, the video is wrong. A circle and a disk are the same, a circumference is just the borders of a circle/disk.
A circle would be a filled circumference.
4:22 That's a semi circle/disk
3:22 The "circle" there is actually a circumference, the "disk" there is a circle/disk
A circle and disk are not the same. One is a one-dimensional line the other is a two-dimension shape. Same with the sphere. A hollow sphere is just a surface while a filled sphere is three-dimensional
It is yet another of those poorly defined things I'm afraid.
Yes, a semicircle can just refer to the curve; in which case it will have 2 ends and no area.
Unfortunately, the 'half disk' is also routinely called a semicircle... which is why you find lots of references to "the area of a semicircle" and such.
The "right" answer entirely depends on what the definition of corner the kid was taught in class. The most sensible answer when not given any more context is 2 of course.
BTW: I would totally argue the point (pun intended) with the teacher.
bruh, draw a closed loop around your room following its perimeter and tell me there is no corners
@@Johnny-tw5pr a circle is 2 dimensional just as the disk. A disk and a circle are the same
What is 1 dimensional is a circumference. The video was wrong at saying that a circle is not filled, a circle is filled. What isn't filled is a circumference.
I have a few issues. The infinite answer uses the idea that the property of a limit shape is the same as the limit of the property of each shape in the sequence. Seems like something that may not be true for corners. We could create some shapes that had increasing numbers of corners that tend to say a triangle and therefore claim a triangle had infinite corners? Or make it so each shape in the sequence had 6 corners but slowly converged on a triangle. E.g. move the midpoint of each side a fraction to the side to make a hexagon and then slowly move this closer and straighter. So does this mean that a triangle has 6 corners? Think we need a better definition of corner else it isn't a worthwhile property at all?
When you move a midpoint just enough to make a corner disappear and just as it appears to be a triangle, 3 pairs of 2 sides would fuse to form a single stright side where the slope along the line is the same. Hence the corner ceses to exist. For a corner to exist, change of slope is necessary, which has been eliminated. That triangle now has only 3 points where lines change slope.
@@srinidhikarthikbs981That breaks the "graphic design" use of a corner point though.
Yea, the "infinity" answer proof fake, just like the proof that pi=4
It's just the basic definition of calculus. You can define a general curve as an infinite number of points, and a circle is a curve, thus, it's generalizable as points.
That answer is akin to a bogus argument used by teachers and brain teaser writers to "prove" that sqrt(2) doesn't exist.
A unit square is drawn, and a series of horizontal and vertical segments, a staircase, is drawn joining the two corners. The lengths of all the horizontal steps and all the vertical steps is 2. If the steps are made smaller and more numerous, hundreds, thousands, billions, then it appears to be the same as a simple diagonal line, but their total length stays the same, 2.
So the diagonal is length 2, not sqrt(2) like Pythagorus said? Of course the argument is flawed. The staircase never has tangent lines at any point that aren't exact horizontal or vertical, even as N→∞, while the true diagonal is its own tangent line - these two things are different types of beasts, so what is true of one may not be assumed true of the other.
I think I read in my differential geometry course that a smooth closed curve has at least 4 "corners" and a circle has an infinity of "corners". A corner is defined as the point where the derivative of the curviture function reaches an extreme point. Since the circle has constant curviture everywhere (1/radius), then its derivative is 0 in every point
I love this breakdown.
The way I thought you were going to get to three corners was to consider the continuous set of sectors of a disk - like, a pizza slice has 3 corners, a quarter circle has 3 corners, a 179° sector has 3 corners, a 181° sector has 3 corners, so within the model of sectors of a disk it can make sense to consider the center point to be a sort of 180° corner
I think the 3 is a bit of a stretch because Adobe graphic design is well outside the context of the question being on a math test. While it doesn't completely remove the ambiguity, I think the worst offender is the text of a half circle accompanied by a drawing of a half disk.
If a friend came to me and said a semi-circle has 3 corners, I think they might stop being my friend...
Yeah that is really far fetched
Abandoning your friend, just like that? When what he says is clearly a cry for help?
I definitely think 2 makes the most sense. People typically think of corners as discontinuities in bearing when tracing a path - places where the direction instantly changes without being eased in.
The answer of 0 restricts the definition of a corner to a discontinuity where the bearings are constant in either direction, while the answer of infinitely many uses a definition of any point where the bearing changes at all, even continuously.
It's like the Bézier curve, what you said about the graphic design case. And yeah there could be 3 or 4 corners (or even more actually) in such a Bézier curve
Another argument for the two edges case is that if you zoom in, you have straight edges in the limit.
yeah i think of a corner as a nondifferentiable point, so two
The problem with this reasoning is that you can do the same for every single point of the circle, therefore making it infinity instead
@@leo-um3pj perfectly said honestly.
@@youssefchihab1613 If you zoom into any other point, you have a single line (or 2 straight edges meeting collinearly if you prefer), so the 2 corners are different.
@@youssefchihab1613 Are you always tripping at the corners in the curvature of the Earth's surface? LOL
If three is a valid answer using your argument, we could extend it and say that any whole number that isn't negative is a valid answer, as we could just add that amount of control points or whatever you called them.
yep. 3 is definitely not a valid answer.
Which is why 3 is not a valid answer. Any reasoning for 3 is just a reasoning for infinity.
Yeah, the control points argument didnt make sense to me. Why would there be exactly one of them on the curve, at the "top" of the semi-circle?
@@Cowtymsmiesznego Because to make any shape you only need 3. You cannot just spontaneously add more control points.
@@williamsplays8528 That's completely wrong though? What do you mean "to make any shape you only need 3"? Why is a shape created with 3 control points a semi-circle and not a triangle? (or any other valid shape) And why did moving that control point around weird out the semi-circle instead of squashing it to a semi-elipse?
If we are going off the logic that a corner is synonymous with a vertex (like shown at 3:47), I feel like yes the answer could be 3. On any graphing calculator, the point where in any direction a function terminates and can't increase its value is typically defined as a vertex point. So in this context, a parabola would have 1 corner. A circle is simply, in a way, 4 parabolas, one for +x, -x, +y, and -y. A circle would have 4 vertexes in this scenario and by the logic of a corner meaning the same as a vertex, a circle would technically have 4 sides. In a semi-circle or semi-disk in this scenario, there would be 3 sides. One for the top (the parabola making up the shape) and 2 for the intersection points where the function terminates and does not increase or decrease (given which side of the bottom line you are referring to) in value in a given direction making a total of 3 points.
This looks like a native language test of math tests. Where you have to interpret text and the score depends on how a specific teacher understood the text themselves.
Questions like these are great for critical thinking or problem solving classes, but are horrible for young students in math classes. A student could mark it with their right answer and still be deemed wrong by the teacher's bias, and can negatively impact the child's learning, especially if there is no explanation given as to why it would be wrong.
Sounds like they accomplish the objective, then.
I'd like to see the question setter run into one of the corners of that semi circle table.
You always get to understand new perspectives here.
As someone that has spent their fair share of time in Photoshop, I disagree that the answer of 3 makes sense from a graphical design perspective. A curved path in the shape of a half-disk could be made up of multiple control handles, not just 1 at the apex. It just so happens that you found an example on Adobe's website that has 3 control handles on it. In actuality, the amount of control handles possible on that curve (and also on the straight edge) also approaches infinity.
I also don't think that restricting the definition of a corner to only be intersecting straight lines is useful in any way, though I posit this as more of an opinion than my previous point. From a mathematical perspective, there is no distinction between a line being straight or curved, as it is just a series of points connected together using an equation. Therefore a vertex of a shape is 2 different equations that make up the shape's edge sharing a point, regardless of whether the equation's line is curved or straight.
So, I think the proper answer to the question is "2 or infinite", neither 0 or 3.
I do not agree with the latter point.
First, I don't think "corner" should be used in a mathematical setting, since it is not definited. But know that in several languages the words for "angle" (which is between two straight lines, or rather two straight half-lines) and "corner" are the same. From that perspective, it would make sense to define "corner" as an intersection between two straight lines.
Furthermore, the fact that, mathematically speaking, there is no distinction between a straight and a curved line is completely false. Of course, since they are both "lines", they share certain characteristics (which are not the ones you described, since there are lines that cannot be described by a single equation and equations that don't describe lines), but that doesn't mean there is no difference between the two: two straight lines, for example, cannot enclose a surface, while two curved lines can.
Nice video!
I immediately went to differentiability, where a corner essentially represents two defined one-sided tangent lines. The two "corners" (as I see them) have both a horizontal and vertical tangent line, depending on whether you're approaching along the arc or along the diameter, whereas every other point has a unique tangent line. So my initial instinct would be 2.
My main problem with the definition of the circle having infinite corners is that you could then start stretching the circle into an ellipse (or squashing it) until it becomes infinitely thin. Basically one has then achieved a straight line with infinite edges. Starting from this infinite edged line one can construct any number of polygons with would, by the rules one has set, now all have infinite edges.
Exactly
This is were math gets interresting.
Your thinking is faultless.
Currently space/time is not considered to be quantised.
When... they find it to be quantised, quantum gravity, this does mean infinite edges from a mathematical perspective. But maths has been proved, incomplete and an incessant pain to the rest of us that have to listen to endless useless mathematical hypothesis that must be true, because they want them to be true, eg string hypothesis.
Matthew
@@mattgroom1 what is true? The lie most people agree on, despite thier individuell belives.
If i (belive to) see a string, is there a string? Becomes the string reality?
If you don't (belive to) see the string, does it disappear? Does it change? Was it ever there?
Yes, the infinite argument has the exact same flaws that allows the π=4 proof to still exist : Uniform convergence of a family of curves doesn't imply convergence of all their properties. Including corners.
When I worked as a land surveyor, we had the concept of property corners where we could set physical markers. This would definitely be 2. I'm going with 2. I can see a case for infinite. But let's be practical when possible.
I would define a corner as a point in a closed curve where you cannot trace a tangent line. With this, the derivative from the arc is gonna be different compared to the derivative of the segment, therefore that point does not have a tangent line, since it depends where you're trying to approximate that point from.
As you said, two edges of any kind meet. That's, after all, our common definition of it, so I'd say it's the most proper one.
The graphics design definition was very interesting, especially because it might come from math, so there's probably some theory behind it.
It totally depends on how you are defining “corner”. It could be some hyper precise mathematical definition or a more colloquial understanding or something in between. Also, since definitions matter, this is actually a semi disc, not a semicircle.
Absolutely I agree with everything you said well I agree with the it depends on how you define a corner
Cuz that's the point at which I stopped reading
But yes first we have to establish how one is defining a corner
Are there any robust definitions of a corner where the number of corners in a semi disc isn't 2? All the definitions I've heard that can be used to say the number of corners in a curve is infinite can also be used to say that coincident lines have infinite corners.
@@Darth_Insidious not in grade school
Corner: Point where two different segments meet in a non-smoot way. I'm seeing two in here. Yeah, a circle are infinite segments, but all met smoothly and if you go infinity the corner lose their corner properties.
What he defined were the vertex, which are indeed infinite here since vertex doesn't care about the "smoothness" of the intersection.
I voted 0 and I stand by it. That’s because it’s literally the answer. A semicircle is just this arc where nothing meets. 2 is a reasonable answer, if you allow a semi-disk (as pictured here, but not in the poll). The others are nonsense. A polygonal approximation is its own idea that may approach the semi-disk but is not the semi-disk. The same goes for cubic Bézier curves, which approximate a circle reasonably well with primary _control points_ spaced by 90°. They do it better with more control points, or worse with fewer. The control point on top is not a corner. You can substitute anything like this. I imagine the semicircle is made of arcs of 30°, 45°, 15°, 50° and 40°. Look, it has 4 corners now, or maybe 6 if you include the endpoints or close the curve.
On the poll, someone commented that a good definition of a corner is a point that isn’t differentiable on a curve. I agree with that.
What I would love to know is what the tutor was using for his/her marking of this exam question, When it came up in the exam it must have been covered during the maths course they done.
Drawing the half-circle as enclosed demonstrates to the reader that the concept being discussed is an enclosed half-circle or half-disk. The instructor is essentially presenting the model to the students and asking them to evaluate it, rather than to produce the model themselves.
I would say to solve the problem mathematically. We can use the concept of calculus to define the corners. A corner can be defined as a point of the edge that is continuous but not diffentiable. From this definition, a semi disk has 2 corners, but a 100 sided inscribed polygon still has 100 corners, but as the number sides tends to infinity, the curve becomes differentiable and the number of corners gets back to 2.
This is why definitions are incredibly important in mathematics. Also, are we really talking about the boundary of the half disk?
4:38 id say that even saying it’s only two strait edges, you could argue for 2 corners, because if you zoom into the point where the curve meets the line infinitely, you could say there is an infinitely small edge there.
I’d argue 2 corners. Why? Because: the bottom line on either side suddenly turns 90 degrees. Whenever a line is changing direction smoothly, with finite derivatives (or relative derivatives, in the case of verticality) it has no corners. Imagine a wet spaghetti noodle, it doesn’t have corners when it is not broken. In the same way, the top of the semicircle is smooth, and has finite derivatives along the non-vertical areas. Now rotate the semicircle such that the straight side is at a 45 degree angle to fix the derivatives. Now the slope at the point is -1 but this instantly switches 1 as you move from the curve to the straight side. Now, because the figure is continuous, and it has instantaneous change in derivatives (implying undefined second derivatives) the point where this change occurs, is indeed a corner.
3:44 Hey vsauce, Micheal here. What is a corner?
I think 2 is the only right answer.
0 is incorrect because lines can be straight or curved by definition.
Infinity is incorrect because a polygon that has so many segments it only LOOKS like 1 curved line. The question already stated it is a semi-circle.
3 is incorrect if we're talking about MATH. I'm going to assume this was not a test on what Adobe calls corners.
I would say 2. To rigorously define the concept of a corner:
Define S: R -> R^2 as a function mapping a parameter t in [0,1) to a position along the curve. A corner is then defined an argument t such that both dy/dx and dx/dy are undefined and S(t) is continuous.
Intuitively, if the curve suddenly changes direction, it must be a corner.
Great vid! I was feeling so smug because I said 0, but I love how you laid out how each answer could be acceptable.
As a graphic designer, absolutely nobody in my experience has called an anchor point a "corner point".
AI will take your job
I was able to understand zero and two, but I had no idea why three could be correct until you explained it. Very cool
(2:42) Why does he have 6 fingers on his left hand?
3. Zoom in on the ends of the straight line and they will be corners, to within any degree of precision you wish to specify, once you zoom in far enough. Extend any 2 tangents of the curve , and zoom out far enough, and you will see that this is also a corner, to within any degree of precision you wish to name, once you get zoomed out far enough. It's all just a question of your perspective, and the degrees of precision you need to work to :D :D :D
Thank you sir. I like how your channel has only gotten better over time.
5:07 Anybody else notice that the coffee table has 4 corners? 2 on top and 2 on bottom.
Very much a question to think outside the box (or around the corner, as germans might say). Nice Video as always!
Interesting perspectives from all the examples. I think the teacher was looking for zero because a circle has no corners. However, you showed the proofs that depending on the perspective, the answer changes.
I would argue that in the above task ("question 5" in the video) should be solved in the given context, instead of imposing a different definition.
Though i see that you could define a semicircle to be half of an arc or as an infinitely complex equilateral polygon, but in my experience that is pretty unusual. The same is true for the term corner.
Most of the time i encountered a semicircle, it was defined to be the boundary of a halfdisk, which would include the baseline (:= that diameter connecting the endpoints of the arc) and because the child's homework clearly shows a yellow inner disc with a black boundary including the baseline, there is no good reason to assume that the semicircle is defined otherwise (which excludes '0' here).
I also doubt that the child's teacher might have introduced infinitesimal numbers to extend the real numbers, so instead of handling intfinitesimal valued angles, we most probably only have to deal with real valued angles. That means that the value of an angle between two neighboring points in an infinitely complex equilateral polygon is 0, which excludes those points beeing candidates for vertices (and also excludes '+inf' here).
Though i can't know that, i also would highly doubt that the child's teacher might have defined corners in a way to justify the answer '3'.
Therefore, in the implied context of that homework, i would see the answer '2' as the only acceptable answer here.
In case the teacher for example explicitely defined a half circle as the arc only (without the baseline), then i would like to know, why the teacher used a misleading image - in my eyes even giving no image would have been better in such a case.
The question is, what did they teach this student? In primary school (because I guess that's a primary school textbook), particularly in earlier grades, the questions can be answered by finding the correct phrase in the material given (i.e. if the question is "what color was Lucy's dress?" Then the text might have a phrase like "Lucy's dress was green"). In later years, the answer is in the text, but it has to be interpreted (ranging from "That green dress looks fantastic!" to "That dress matches your eyes", while earlier in the text there could be a phrase like "Your eyes are like emeralds").
So, depending on what grade are we talking about, the text must say how to count corners (or maybe it was taught by the teacher). Other answers are too technical or complex for primary school.
This crap only teaches that "the teacher is always right and just comply with their whims".
This sort of thing really annoys me.
@@thomasrussell4674yeah, they teach math in the exact wrong way. It is not just torturous for students, but also completely dissonant with hows and whys math was created.
(I have dodged this bullet fortunately)
@@adamrak7560Had a teacher like this in elementary School. Did math for fun from an early age on, so was at least a grade ahead usually. Adding to 100 isn't that fun when you already know the 10x10 chart. Teacher could not deal with this and always gave me worse grades, despite near perfect scoring (in Germany, got 1 on every exam, did homework, engaged appropriately in class. Final grade was always a 2. She straight up just hated me for liking her subject.)
But also had awesome teachers in highschool and college, working with my experience and adding to it, rather than trying to conform it to the standard.
I don’t really think its a matter of “the teacher is always right” or “complying with whims”. I often feel that people forget that mathematics is not set in stone, definitions vary between countries and institutions. When teaching matematics, we must put forward definitions to avoid ambiguity. If a corner is defined by two straight edges meeting, the answer is clearly 0. If corner is defined to be any two edges meeting at a point, the answer is 2. My guess is that this teacher put forward the former definition (which is most videly used in my epxerience) and wanted to check if students understood what a corner is in that mathematical sense, which may or may not correspond to our everyday usage of the word.
Asking how many corners an object has is some very early education stuff, possibly kindergarten or first grade - at least, I can't imagine it being past that. Presenting squares and triangles as objects with "corners" is easy and I can just imagine a teacher circling those places where obvious angles are formed as examples of "corners". I don't think its beyond the logical capacity of even small children to be presented with a concept such as "circles have no corners, so how many corners does half of a circle have?" and expect the answer to be none, but when you present a picture that includes what are most likely the example you gave of "corners", expecting the answer to be anything other than two is ridiculous unless you have specifically illustrated the question with the answer like that previously.
The "2" written there is quite nice, however - much better than any two I ever managed to scribble when I was the age I would expect to be asked this question at (and frankly, better than my 2s now), so I have my suppositions that this is one of those questions that never actually existed and was posted as a rent due tweet.
This kind of math problem made me recall a time when I was a kid, and we had this joke, to mock math problems at school:
Q: Two irons are flying, one blue and the other heading north, how much will a kilogram of apples cost?
A: Why do I need a refrigerator if I have a bicycle
180, half of a circle in degrees -2 degrees, plus two 90 degree corners on the flat section, all internal angles must add up to 180 degrees
The way I like to define a CORNER is that it's a point in a shape's perimeter where we cannot draw a single tangent ie where slope of tangent cannot be defined.
Summary:
0 - two straight edges of a half disk meet (geometrical)
2 - two edges of any kind of a half disk meet (practical)
infinite - a regular polygon reaches an infinite amount of corners (theoretical)
3 - includes the anchor point (graphical)
all of the above - all perspectives work (alloftheaboveal)
The introduction of the term "alloftheaboveal" puts the icing on this cake.
I thought either 2 or infinite, but wondered if it could be 0. I didn't think of 3 - thanks for explaining about the graphic design aspect.
To make 0 the only possible correct answer, the homework sheet should have provided a definition of "corner," such as:
"For purposes of answering the following question, a 'corner' is defined as the point on a closed shape where two line segments meet, and those two line segments are not part of the same line. Using this information, answer the following question."
To be fair to the teacher, we weren't shown the entire homework sheet, so maybe something like that was on the sheet, and the reporter (or his source) just chose not show it.
As I recall from school the “corner” is geometrical figure constructed of two rays with common origin, excluding straight line. So even triangle have 0 corners, it has vertices. (it was in Russia so that’s maybe just a matter of terminology)