Man, I have never heard of this cross ratio. What a powerful propriety.

reading the title i thought it might be about the probability a cross would result in a goal

@deniztuncer296

6 жыл бұрын

i had the same thought as well

@Triantalex

8 ай бұрын

??

Never apologize for using SI units. However you are allowed to use any units you like if you can arguing for it.

@99joery

6 жыл бұрын

Null Blank the imperial system is defined using the metric system

@quintrankid8045

6 жыл бұрын

FFF System.

@JorgetePanete

6 жыл бұрын

Null Blank It's*

@mal2ksc

6 жыл бұрын

Should have done the whole thing in smoots.

@rillloudmother

6 жыл бұрын

only the small minded care about units when we are talking about proportions.

This guy definitely has one of the better handwritings of this channel 😌

I really like this. I especially like the quote he reads at the end, I find it humanizes math (with respect to geometry) somehow. Furthermore, everything in this video is completely testable using simple pictures of landscapes that can be measured after the calculations have been made. I just might try this sometime...

@numberphile

6 жыл бұрын

Thanks. Glad you liked it.

@Triantalex

8 ай бұрын

??

I counted the stripes on the field. The ball is almost exactly between two stripes. There are 12 stripes between ball and goal line, 7 stripes between ball and centre line. That's 38 stripes in total. The game was played at the Stade de Gerland which has a length of 105m. So each stripe is 105/38=2.76m and hence the ball is 2.76x12=33.16m away from the goal line. Did not expect to get an estimation so close as his!

this video is a great introduction to photogrammetry

I really like his humility in stating that he learned of it whilst teaching a class.

Clever way to teach cross ratios.

@General12th

6 жыл бұрын

Agreed. This approach is brilliant!

@xintongbian

3 жыл бұрын

I was really having a hard time understanding cross ratios reading wikipedia and watching other youtube videos until I stumbled upon this one, duh!

"What we need to figure out is how a change in perspective changes a measurement." That's exactly what Einstein was trying to do when he developed Relativity.

Is it just me or is it cool and refreshing to see real-world measurements incorporated into one of these

@JorgetePanete

6 жыл бұрын

PinochleIsALie you forgot a cool and refreshing question mark

@PinochleIsALie

6 жыл бұрын

Jorge C. M. I prefer ending all sentences with semi-colons, since it always leaves people wanting more;

@GeneralPet

6 жыл бұрын

that's a very egotistical thing to say.

@ObjectsInMotion

6 жыл бұрын

No its disgusting. Math doesn't need "units".

@JorgetePanete

6 жыл бұрын

Objects in Motion No, it's*

This is amazing. I just want to say thank you for making these

And now this is in wikipedia. So it is officially an internet fact.

@MattMcConaha

6 жыл бұрын

Someone needs to do the calculation more precisely (e.g. pixel distances instead of ruler measurements) and be aware of significant figures.

@xooperz

6 жыл бұрын

+Matt McConaha the outcome would hardly change

@subschallenge-nh4xp

6 жыл бұрын

Sam g

@subschallenge-nh4xp

6 жыл бұрын

Johnny Lee y

@letsseepaulallenscard.6604

6 жыл бұрын

idk what you mean by this but the Cross-ratio has been on Wikipedia since 2016

Belgium won because they watched this.

Do not apologize for using metric system, apologize for not determining error bars.

people should apologize for using the imperial system, not the metric one

@miro007ist

6 жыл бұрын

correct

@ZoioGame

6 жыл бұрын

Only americans use It right?

@kristianwilliams441

6 жыл бұрын

I'm an American and I agree with you. The imperial system makes no sense.

@00BillyTorontoBill

6 жыл бұрын

one other do too... in africa... i think liberia maybe.

@blindleader42

6 жыл бұрын

Maybe the Imperial system makes no sense to people that believe either history makes no sense or the Imperial system suddenly sprang into existence with no cause or history. On the other hand, US customary units make all the sense in the world, and people who go on the internet to complain about it have some sort of personal problem that posting on the internet will not alleviate. Such people might benefit from professional help not available in a youtube comment thread.

Great video!! Would love to see some more sports-related videos on numberphile. This one was really fun to watch.

@numberphile

6 жыл бұрын

Cheers. Be sure to click on our soccer/football playlist.

PROFESSOR FEDERICO!!! I THINK HE'S JUST SO AWESOME!!

I love the way this guy explains things. Very quality video!

A big "thank you" !. Now I love the cross ratio. Next video : how to shoot like Roberto carlos...

I'm so glad you made that 'cross' pun at the end. I was waiting for it the whole time.

Sometimes, surprisingly often, in fact, viewing a numberphile video solves a problem I've had months or years ago in programming and this is one of them. Now I'll go back to years old code and completely change it and make if WAY more optimized, thanks a lot!

@numberphile

6 жыл бұрын

You’re welcome.

Nice one! Could be estimated in 5 seconds by counting the stripes in the gras (they are equal distance :-) ) 6 Stripes to the 16,5m line plus about 6.1-6.2 stripes to the ball :-) Nice Vid. Thanks!

this was amazing and you explained it exceptionally well. thank you. subbed

Fuck me, I love this channel! Everything is so fucking interesting and the answers are so satisfying and I get so happy every time I watch a video from here! Keep it up, Numberphile! Never change! :D

The best one in a while! Cool guy!

So much fun! Thank you for sharing.

Grande Federico! Algo que a priori es un muermo, ha hecho que me mantenga pegado a la pantalla hasta el final

I still come back to this video because the cross ratio seems like such a powerful tool! Fascinating

This is wonderful. I love this.

I was looking through videos from this channel to help out with a programming project of mine involving 2D-to-3D-to-2D projection, and this might be exactly what I need to study.

That was really awesome. Thanks so much.

I tried this by taking a photo of graph paper. I ended up with a 5.3% error. I broke up a line into 3 segments. AB = 3", BC = 3", CD=2.25" (known from the grid on the graph paper). Using Photoshop, I found the pixels between these points to be AB=149px, BC =209px and CD=236px. The cross ratio using the pixels was 1.283 and the cross ratio using the know measurements was 1.273. When I tried to calculate the last distance CD using the cross ratio from the pixels calculations, I got 2.37" instead of the know 2.25".Is this lens distortion or correction in digital camera that are throwing off these calculations?

Federico, buenísmo! Muy bien explicado! Enhorabuena!

That is so accurate! if you look at the footage of the goal one can see that x is about the size of the box just by counting the clear and dark sections on the grass pattern.

Robin Hartshorne was my shakuhachi teacher for a while. Thank you for mentioning him

😮 Wow! Thank you so much! That example beautifully illustrates the motivation for the use of the most peculiar and non-obvious cross-ratio theorem which has confused me for days. Goal!!!

Beautiful! I have to watch again, I missed one bit...but this is fun.

This was interesting. do more!! I'd like to see how this is derived!!

Amazing! I already loved math and geometry, now you made me love soccer (well, football, I'm from the other side of the pond)

This video is like 3 months too late - just took a final year module in Projective Geometry and this stuff confused the heck out of me (and I consequently nearly failed the module)! Makes a whole lot more sense now!

Why don't they show the actual goal, just an animation?

@Sitzriese

6 жыл бұрын

lawsuits...

@Bill_Woo

6 жыл бұрын

What he said. Try v=3ECoR__tJNQ

@naota3k

6 жыл бұрын

They don't want to be le sued by the FEDERACION INTERNATIONALE DE FOOTBALL ASSOCIASION *twirls moustache*

@SlingerDomb

6 жыл бұрын

DEMONITIZING

@NonDelusional74611

6 жыл бұрын

Bill Woo Incroyable!!

That was damn interesting!

What a brilliant video.

Great stuff

This is so satisfying to watch

Fantastic !

If I could choose an original brown paper I would pick this. Beautifully written, beautifully drawn.

GEOMETRY BABYYYY!!!

This was fantastic

There are some bright and dark green strips on the field. You can clearly see the ball is sitting on the edge of "3rd" Bright strip starting from the D-Box. We can calculate the width of each strips by measuring how many there are in the D box. Once we know the width of a strip, suppose "x" meters, then the ball is sitting at a 6*x meter distance from the outer edge of the box. Its much simpler this way

Your handwriting is amazing! :D

wow.. thats so accurate from that picture only. And you can really visually see that it's correct if you count the light green and dark green grass lanes. Inside the box there are 6 and outside the box until the ball there are also 6 plus a fraction. Then you'll see that he might've been mistaken only by a cm or 2. Amazing.

This is terrific! It looks, at first, to be bonkers, but it's great.

I also didn't know that the arc outside the penalty box makes a circle centered on the penalty spot. Pretty cool!

I took a load of inspiration from these types of Numberphile videos and made a probability-based football video on how likely you are to win the Super 6 jackpot!

This is awesome! I gonna try to applicate it in some photos

Very interesting video, thanks

great explanation!

You were teasing us so many times with that goal xD

0:38 Just look at the stripes in the grass. There is 6 of them in the penalty box, and 6 more(and a bit) from the penalty box to the ball. So you can conclude it's about 2*Penaltybox_Length. You can even draw a line parallel to the sideline and get an exact measurement.

@derflo3181

6 жыл бұрын

Last sentence is wrong

@derflo3181

6 жыл бұрын

But the rest seems smart ;D

@MrCyanGaming

6 жыл бұрын

it's not

@MrCyanGaming

6 жыл бұрын

you already know the length of the pitch, so you can just divide it by the number of stripes. It will let you get an exact measurement

@derflo3181

6 жыл бұрын

The parallel line will be slightly longer than the sideline because it's nearer to the camera

"I must say, frankly, that I cannot visualize a cross ratio geometrically" We need to get 3Blue1Brown on this.

That was beautiful!

I am intrigued. I'd love to see more about how this quantity was discovered and if there is some kind of elegant proof I'd watch that as well.

Awesome!

I didn’t think I’d like this one, because I’m not into soccer. I almost didn’t watch. But I’m glad I did-I was pleasantly surprised! Cross ratios are very interesting; I had never heard of them before

One intuition for the cross-ratio is to consider it as a ratio of ratios. If you modify slightly, this quantity is the ratio between to ratios with geometric interpretation. One is the ratio in which B is send to D as an homotecy of centre A. Whereas the other is the ratio in which B is send to D as an homotecy of centre C. And if the cross ratio is -1, it can be seen as if C is a centre of homotecy, then A is its inverse in the sense it has the same constant of homotecy but different sign.

I thought the video was going to be about crosses into the box, and the actual topic was somehow even cooler

Very interesting video, especially in light of the likely rematch between France and Brazil. Would have liked to have actually seen the video replay of the goal, though.

Beauty and elegant explanation...

Geometric construction of cross-ratio, using compass and straightedge, is straightforward when one sees it is equivalent to the ratio of two rectangles' areas. Rectangles can be constructably squared. Translation and rotation are also c&s construction. Ratio found by placing edge of each constructed square on a ray, with a corner concurrent and each external to the other, say. A ray through similar corners, not on the base ray, is invariant to their ratio. Do the same for any other line divided by a common projection point, and compare.

i found this deeply satisfying

Came here after coming across the term "cross-ratio" in a grad school book and frowning so hard it hurt

At 14:48, you can see that there are five "landscaping" lines in the grass from the goal line to the top of the penalty box, and then another five "landscaping" lines in the grass from the penalty box to the ball. This confirms the calculations.

I learned about this recently and used it to help use a reference image to recreate something in 3D.

Can you talk more about the cross ratio and how it lets you handle geometry of any curvature equally? (The cross ratio can, along with some quartic, actually be used to present any kind of hyperbolic, euclidean, dual-euclidean, or spherical space. Which space it is depends on the chosen quartic.)

Ok, now looking forward to a video on WHY the cross ratio works. And a Sixty Symbols video on HOW Carlos made the shot. 😀

nice handwriting!

What I particularly like about ratio is it is absolute and honest.

This is actually used a lot in 3d computer graphics when drawing angled surfaces onto the flat screen. It's part of how to describe mathematically how a flat piece of paper appears as you tilt it.

I immediately started thinking about image rectification and homogeneous coordinate systems, but this was also a cool algebraic hack.

The fact that "crossing" is a soccer action makes this video's title a pretty clever pun.

I really enjoy Brady's productions and I hope he is doing well from them.

Lazy mode: pause the video at 0:32 and look at the striped pattern on the grass. There look to be 6 stripes inside the 18-yard box and slightly more than 6 stripes between the box and the ball. So the ball is a little more than twice the box distance from the goal line. 2*18 is 36 yards so the ball is a bit more than 36 yards (bit more than 32.92 meters) from the goal line.

When I'm watching soccer live, I look at how many lawn stripes make up the penalty area. Here, there's 6, so 3 yards per stripe. I count 6 (and maybe 1/4) stripes from the box to the ball. (6 + 6.25) * 3 = 36.75 yds, or 33.9 meters. For this application, I'll take being less than a meter off.

Seems related to your video on Ptolemy's Theorem, as to where a geometric intuition may come from anyway.

I never thought I'd hear someone referring to the area as "la dieciseis con cincuenta" as my uncle used to say in a numberphile video.

The stadium is actually 105 m, that means he was shooting from 33 meters 7 centimeters ± 10 cm (angle/ ball error), just had to get the image containing both the circle in the middle and the 'D' and it was easy to get an exact measure of the distance the ball was, i just applied the ratio from the image to the DWG drawing of this field; really nice guess in this video, but sooo complicated :)

I'm not sure, but this guy reminds me of my fellow Colombianos. Great presentation, as always, Numberphile!

pile à la mi temps c'est génial !

@xl000

6 жыл бұрын

la mi temps de quoi ?

@al1b219

6 жыл бұрын

Cette vidéo numberphile est sortie pile à la mi temps du match France Belgique il y a une semaine !

Interesting find. I usually estimate this by thinking that the players are always about 9m from the ball to begin with, so it's easy to just add 16.5+9 and make a reasonable guess for the distance between the 'wall' and the 16.5m line. Or, if the shot is reasonably right on, I can do 11+2x9 and go from there. It's just a rough guess, but I always thought this one was from like 32-33m based on that idea.

The lawn stripes in the video look pretty regular. If the person who cut the grass was careful about it, we can use those stripes too. The ball is at the box + 6.2 lawn stripes (about) And the box itself is about 6 lawn stripes. So 12.2 lawn stripes * (1 box/ 6 stripes) * (16.5 m / box) = 33.55m checks out :-)

Now, the real challenge is to measure the arc length of the trajectory of the ball from its kick point to where it landed in the goal box!!! ;^}

Was anyone else surprised that he explained the cross ratio with segments on the (not necessarily parallel) crossing lines, but actually used segments on the rays in his calculation?

Projective geometry is such an interesting topic. Also, this video would have had 100x more views if the words 'football' or 'Roberto Carlos' were used in the title, which would allow more football fans to appreciate the cross ratio!

Pretty cool!

IT'S COMING HOME

The easier way is you can measure the width of the different grass sections in the video or pictures by comparing how many of them fit into the box then calculate how many grass widths away the ball is from the goal line (pro tip - this is how the newsmedia of the time did it)

I have an easier way of calculating that: Just check the light and dark stripes of grass on the field. There are 6 from the goal line to the big box's line, and 6 + a "tiny bit" from the big box's line to the ball. That tiny bit seems to be about 1/5th of the size of the stripe which is 2.75m (16.5/6), so I assume with a margin of error of tens of centimeters that the "tiny bit" measures at about 55 centimeters. So from all that I can calculate that the ball was roughly 33.55m away from the goal. Not as cool as the way shown in the video of course, but it was something I noticed.

A new video in a middle of a game?! Oh, Comeone! It's a hard choice.