Squares are 83.2% circle

[00:00:00]So, I was playing around with this website where you try to draw a perfect circle and it gives you a score based on how close you were. But then I looked online and what the hell is this?

[00:00:09]Possibly the ugliest circle I've ever seen gets 99.2%.

[00:00:13]And I first assumed this was photoshopped or something, but here are three more images from different people.

[00:00:18]Now, there is no objectively correct way to measure how circular a shape is. But this sure as hell does not deserve 99.4%. So, what I'd like to do this video is try and come up with some different ways to measure this number.

[00:00:31]And to make things more interesting for you, here are 14 shapes, and we're going to measure the circularness of each one.

[00:00:38]Take a guess. Which shape will be the most circular? Which shape will have the highest percentage? We have a square, a semicircle, a right triangle, an ellipse, a star shape, four circles, a very long octagon, a ven diagram, a ring, this parabola shape, the expandomino pizza slice, the rest of the pizza, and the mandler set. All right, lock in your guess and let's start.

[00:01:03]My first idea was to take the shape, draw a perfect circle with the same area as the shape, and now we move the circle around to try and overlap as much area as possible. And the logic behind this is shapes that are not very circular, like this rectangle here, will only have a little bit of overlapping area, while shapes that are really similar to a circle will have a lot of overlapping area. So the idea here is pretty simple.

[00:01:27]The more area you can overlap, the higher your percentage will be. All right, here's an example. Let's measure a square. First, draw a circle with equal area. Now, we move the circle around and we try to overlap as much area as possible. This happens when the circle is directly centered on the square, in which case 90.9% of the circle's area is overlapping. So, that is our circularness measurement. Squares are 90.9% circular.

[00:01:54]Now, let's try a semicircle. This calculation is a bit more difficult because we need to figure out where to place the circle in order to get maximum overlap. It turns out that the circle needs to be around here which leads to 78.4% overlap. So that's our measurement for semicircles. And we keep going. The isosles triangle measures 77.5%.

[00:02:14]The ellipse measures 78.4% which strangely enough is exactly the same as the semicircle. And I don't know why. If you manage to figure out why, let me know in the comments because I have no idea. Anyway, the star shape measures 76.0%, the four circles shape measures 81.4%, or at least I'm pretty sure it's 81.4%.

[00:02:33]Maybe there's a more optimal position for the circle that overlaps more area, but I don't think there is. Moving on, the long octagon measures 84.2%.

[00:02:42]The ven diagram measures 86.5%.

[00:02:45]The ring measures 87.5%.

[00:02:48]This parabola shape measures 85.7%.

[00:02:52]The x penttoino measures 78.2%. 2%. The pizza slice measures 85.3% which took me a while to figure out. And the rest of the pizza measures 86.3% which also took me quite a while to figure out. Now we have the Mandelra set and it turns out that the area of the Mandlerra set is not actually known. So I don't know how large this circle is supposed to be exactly. And even if I did know, I have absolutely no clue how to calculate the overlapping area. So the Mandelra set is disqualified from this round. Oh yeah, I should mention that there are going to be several rounds. The first round was overlapping area and the next round will be something different and so on. Each shape will earn points depending on how high they manage to score. And we'll tally up the points at the end of the video to see which shape is the winner.

[00:03:35]The square wins the first round with the only score above 90% and takes home 14 points.

[00:03:43]So that was the overlapping area metric.

[00:03:46]Now what makes this a good metric and what makes this a bad metric? Think about that for a second.

[00:03:53]All right, first let's talk about the good parts. There are three important things I would like you to pay attention to here.

[00:04:00]First, if we have a shape and we make it more rounded, then the overlapping area will be greater. So, our measurement will be greater. This is exactly what we want. If we make a shape rounder, it will have a higher circularness.

[00:04:14]Second, let's try to measure the circularness of a perfect circle. Well, we simply draw a circle of equal area and 100% of the area is overlapping. So, the circle's circularness is 100%. This is not the case for any other shape.

[00:04:29]Only circles are 100% circular, which again makes sense. Third, a big square and a small square both have the same circularness because the circle that's used to measure them scales with the square, if that makes sense, because of the equal area thing.

[00:04:46]All right, we'll come back to these three things soon. Now, remember when I said that there is no objectively correct way to measure circularness?

[00:04:53]While that is true, we still want to have some rules throughout this video.

[00:04:57]Otherwise, we'll end up with completely ridiculous measurements. So, here are our three rules. Number one, the more circular a shape gets, the higher its circularness should be. Pretty self-explanatory.

[00:05:09]Rule number two, a perfect circle should have circularness 100% and it should be the only shape with circularness 100%.

[00:05:17]And rule number three, circularness should not depend on size. If you have two squares and one is bigger than the other, they should still have the same circularness.

[00:05:27]Now, as we just saw, the overlapping area metric satisfies all three of these rules. So, we can consider it a valid metric. However, there are still some weird side effects. For example, consider this ring. It's very thin, so it doesn't have much area. Meaning the circle we use to measure it will be pretty small. Therefore, the amount of overlapping area will also be small. So, this ring only has like 10% circularness according to this metric. While squares have 91%. Me personally, I think this is ridiculous. The ring feels much more similar to a circle than the square does. But, you know, maybe that's just me. So, that's weird side effect number one. Shapes that are very thin tend to have extremely low measurements. Now, there's another problem with the overlapping area metric, which is that it's quite difficult to calculate accurately. If I give you a shape like this, it's very hard to measure because you don't know exactly where the circle should be positioned in order to achieve maximum overlap. You would need to test a bunch of different positions, record the percentage for each one, repeat this process several more times, then pick the highest percentage to use as the measurement. And even then, there's a chance that your measurement is wrong.

[00:06:36]Maybe there was a slightly better location to place the circle in and you missed it.

[00:06:41]Now, you may have noticed something interesting about circles while watching this video. If a shape and a circle have the same area, and you compare the perimeters of both shapes, it turns out the circle's perimeter is always smaller than the shape's perimeter. Verify this for yourself.

[00:06:58]This makes intuitive sense if you think about it because if you have a shape and you round it out, the perimeter decreases. Again, you can verify this for yourself. And if you keep rounding it out, the perimeter keeps decreasing and eventually you end up with a circle.

[00:07:13]So, in other words, the more circular a shape gets, the smaller its perimeter becomes.

[00:07:20]We can actually use this property to create another circularness metric. We take the shape that we want to measure, draw a circle with the same area as the shape, and now we calculate the perimeters of both shapes, and we divide the circle's perimeter by the shape's perimeter to get the circularness. Once again, let's use a square as an example.

[00:07:39]We draw a circle of equal area. And if we divide the circle's perimeter by the square's perimeter, we get 88.6%.

[00:07:46]According to this new perimeter metric, squares are 88.6% circular.

[00:07:52]All right, before we do anything else, let's check whether this perimeter metric satisfies our three rules or not.

[00:07:58]First rule, the more circular a shape gets, the higher its circularness should be. Now, we already know that the more circular a shape gets, the smaller its perimeter becomes. Now, let's look at our formula for circularness. The shape's perimeter is in the denominator.

[00:08:14]So, as it decreases, the circularness should increase. So, the first rule is satisfied. Second rule, circles have 100% circularness and also no other shape has 100% circularness. The first part is pretty obviously true. Circle perimeter divided by the exact same circle perimeter equals 100%. Now this other part is much trickier to prove.

[00:08:37]You'll have to take my word for it. So the second rule is also satisfied. Third rule, circularness should not depend on size. This rule is also satisfied. Just like before, small squares are measured with small circles and large squares are measured with large circles. The ratio between the circle perimeter and the square perimeter remains constant.

[00:08:58]Now, here are some problems with the perimeter metric. Look at this shape. I think you'll agree that it is somewhat similar to a circle. It's basically a spiky circle after all. Now, if you measure the circularness of this shape, it turns out to be something like 20% because the perimeter of this shape is so massive. Now remember that squares have 88.6% circularness and the spiky circle has 20%. Now maybe you disagree with me. Maybe you think that this is perfectly reasonable. But I don't really think so. I consider this a pretty big flaw with the perimeter metric, the details matter much more than the big picture, if you know what I mean. If I take a circle and I modify a really small part of it, the circularness can drop significantly. Now the benefit of the perimeter metric is that it's really easy to calculate. All you need to do is find the perimeter and area of the shape. And using these two values alone, you can calculate the circularness.

[00:09:52]All right. Now, let's run the calculations for each of these other shapes. The semicircle measures 86.4%, the right triangle measures 73.4%, the ellipse measures 91.7%.

[00:10:04]The star measures 51.7%.

[00:10:07]The four circles measure exactly 50%.

[00:10:10]The long octagon measures 90.2%, 2%. The ven diagram measures 95.1%.

[00:10:15]The ring measures 70.7%.

[00:10:18]The parabola shape measures 87.1%.

[00:10:21]The x pentomino measures 66.1%.

[00:10:24]The pizza slice measures 84.2%.

[00:10:27]The rest of the pizza measures 79.3%.

[00:10:30]And the mandelro set is 0%. So here is everyone's point earnings for this round. Oh, and also I made a mistake in round one. Here are all the updated and corrected point values.

[00:10:44]The square is in second place with 25 points. The ellipse is in third place with 20 points. The ven diagram takes first place with 26 points. And the parabola shape is tied for third place.

[00:10:56]And I'm sorry to whoever picked mandler set.

[00:11:01]Metrics three and four are going to be kind of similar. Here's metric 3. You take the shape and you draw the circumscribed circle which is the smallest possible circle that contains the whole shape. Now we divide the shape's area by the circle's area and that is the circularness. Let's try it on the square. We draw the circumscribed circle and let's say the square has side length 2. It doesn't matter any numbers.

[00:11:25]By the diameter of the circumscribed circle would be 2 <unk>2. So the radius is <unk>2. Now we divide the square's area by the circle's area and we get 63.7%.

[00:11:38]The idea behind this metric is the rounder a shape is the more area it will take up relative to the circumscribed circle and therefore its circularness will be higher. Now I'm not going to bother going through the three rules again. You can verify for yourself that this metric does indeed satisfy all three rules. So let's go ahead and finish the 14 shapes. The semicircle measures exactly 50%, the right triangle measures only 31.8%, the ellipse measures 50%. Also, the star measures 35.7%.

[00:12:08]Four circles measure 68.6%.

[00:12:11]The long octagon measures 55.8%.

[00:12:14]The ven diagram measures 71.5%, the ring measures 88.9%, the parabola measures 54.3%, the x pentomino is exactly the same as the square. Coincidentally, the pizza measures exactly 50%, the rest of the pizza measures 83.3%, and yet again, I'm not really sure how to calculate the Mandlerra set. But just looking at it, it looks like it's going to come in last place again, so it doesn't really matter. All right, let's add up the points.

[00:12:45]Square is still in second place. Ellipse has fallen out of the top five entirely.

[00:12:50]Van Diagram is still in first. The Ring made a big comeback and is now in third.

[00:12:54]and so did the rest of the pizza.

[00:12:58]So, this is a pretty simple metric, but it has a few problems. If you look at this shape, which is just a circle with a couple spikes sticking out of it, you'll notice that the circumscribed circle is pretty big because of the spikes. And this causes the circularness to be much lower than it probably should be. Also, you can modify the shape in some crazy ways, and the circularness will not change at all because the circumscribed circle is still the exact same way it was before. In other words, what goes on near the center of the shape doesn't really matter at all. You can mess around with it, and as long as the area of the shape and the circumscribed circle don't change, the circularness won't either.

[00:13:36]Our fourth metric is kind of the opposite of our third metric. Instead of drawing the circumscribed circle, we will draw the inscribed circle, which is the largest possible circle that can fit in the shape. And the circularness equals the circle's area divided by the shape's area. In this case, the circle's area is pi and the squar's area is 4. So the circularness is approximately 78.5%.

[00:13:59]Once again, the idea behind this metric is if you make a shape rounder, you'll be able to fit a larger circle inside, which will cause the circularness to increase. Now, if we have a shape such as the star, the inscribed circle sits in the middle. Now, what this means is I can take one of these spikes and make it super pointy, do whatever I want with it, and the inscribed circle will remain exactly the same way it is now. And therefore, the circularness will not change. I can even do wacky stuff like this, and as long as I don't change the total area of the star, the circularness will remain at 15%. Which is definitely a problem with this metric. The stuff near the center of the shape matters a lot, but the rest of the shape doesn't really matter at all. All right, now let's calculate the rest of the 14 shapes. The semicircle measures exactly 50%, the right triangle measures 53.9%.

[00:14:51]Ellipses are yet again 50%, the star shape is 13.0%. The four circles is 25%, the long octagon is 43%. The ven diagram is 62.2%.

[00:15:03]The ring is only 12.5%, which is another problem with this metric. By the way, shapes that are thin score extremely low. This is a pretty terrible metric if I'm being honest. There are so many problems. Anyway, the parabola shape is 66.3%.

[00:15:17]The X pentomino measures 31.4%. The pizza slice measures 66.7%.

[00:15:23]And the rest of the pizza measures 30%.

[00:15:25]And the Mandler set actually does decent this time. All right, let's tally up the points.

[00:15:31]Oh, also I messed up the star calculations. Now, let's tally up the points.

[00:15:37]Square is now in first place. The ellipse is in fifth place. The ven diagram is tied for first. The parabola shape is in third. The pizza slice is in fourth. And the rest of the pizza is tied for fifth.

[00:15:51]Now, for our last metric, we're actually going to go back to the website at the start of the video. What metric does this website use? Well, you'll notice that there is a dot in the center of the screen, and we're supposed to draw our circle around the dot. And you'll notice that our drawing starts off green, but as we start moving away from the dot, it becomes red, indicating that we're drawing a bad circle. The same thing happens when we get too close to the dot.

[00:16:18]To get a good score, you must maintain constant distance from the dot, which makes sense because that's what a circle is. Now the problems with this metric are if you draw a perfect circle but it's not centered on the dot the score will not be 100% because you are not maintaining a constant distance from the dot. So immediately we already know that this metric does not satisfy the three rules from earlier. Perfect circles should always have 100% circularness which is not the case here. And it gets worse. The same shape can have multiple different circularnesses depending on how you draw the shape. For example, look at this shape and also here's a dot. If you started drawing from here, then most of the shape would be green and the only red part will be when you get closer to the dot. So, your score will be pretty good. However, if you started drawing from here, then you immediately move farther away from the dot, which makes the rest of the shape red. So, your score will be much worse.

[00:17:15]We'd like to modify this metric a little bit to fix up these problems. The first thing we're going to do is always move the dot to the center of the shape. This way, even if we draw a perfect circle that's not centered on the dot, we will still get the score we deserve. But what if we draw a semicircle? What is the center of a semicircle? We need to be more specific about what we mean by center. So, we will use the centrid, which is the center of mass of the shape. This way, there is no confusion.

[00:17:44]Now our circularness will equal this length divided by this length. The distance to the closest point on the boundary divided by the distance to the farthest point on the boundary. Once again, let's use a square as an example.

[00:17:57]The centrid of the square is right in the middle and the circularness equals this distance divided by this distance.

[00:18:04]So the circularness equals 70.7%.

[00:18:08]Now, if I'm being honest, this metric is pretty terrible. It's probably the worst out of the five. For example, this shape has its centrid here. So, its circularness is this inner radius divided by the outer radius. So, a very small number, which shouldn't be the case for a shape that is basically a circle. Another problem with this metric is that it only takes into account the closest and farthest points from the centrid. And all the other points on the boundary are neglected. Which means this ellipse has the same circularness as this mess because the closest and farthest points from the centrid remain unchanged. There are a lot more problems, but let's just go ahead and calculate the rest of the 14 shapes.

[00:18:46]This will be the final round. The semicircle measures 39.1%, the right triangle measures 31.6%, the ellipse measures exactly 50% for the third time in a row. The star shape measures 38.2%.

[00:18:59]The four circles measures 17.2%, the long octagon measures 49.0%. The ven diagram measures 57.7%.

[00:19:07]The ring measures 33.3%. The parabola shape measures 57.3%.

[00:19:13]The x pentomino measures 44.7%.

[00:19:16]The pizza slice measures 50%. The rest of the pizza measures only 11.4%. And I have no idea where the centrid of the mandler set is. So I'm going to give it 0% again. All right. After tallying up all the points, the square wins first place with 63 points. The ven diagram comes in second and the parabola shape comes in third. And here are the rest of the rankings. And all of the rankings seem pretty reasonable if I'm being honest. So yeah, that's all I have for you today. I really hope you enjoyed.

[00:19:46]This was a really fun video to make and I'll see you next week. Peace.