The Cross Ratio - Numberphile

[@Petch85]

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

[@99joery]

Null Blank the imperial system is defined using the metric system

[@quintrankid8045]

FFF System.

[@JorgetePanete]

Null Blank It's*

[@mal2ksc]

Should have done the whole thing in smoots.

[@rillloudmother]

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

[@eac-ox2ly]

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

[@AbhijayPaul]

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

[@AshishGupta-ql9lq]

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

[@deniztuncer296]

i had the same thought as well

[@Triantalex]

??

[@nymalous3428]

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]

Thanks. Glad you liked it.

[@Triantalex]

??

[@gerardmarquinarubio9492]

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

[@miro007ist]

correct

[@ZoioGame]

Only americans use It right?

[@kristianwilliams441]

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

[@00BillyTorontoBill]

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

[@blindleader42]

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.

[@alexmcgaw]

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!

[@judo-rob5197]

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

[@evaggeloskotoulas8412]

this video is a great introduction to photogrammetry

[@redguy333]

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

[@PinochleIsALie]

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

[@JorgetePanete]

PinochleIsALie you forgot a cool and refreshing question mark

[@PinochleIsALie]

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

[@GeneralPet]

that's a very egotistical thing to say.

[@ObjectsInMotion]

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

[@JorgetePanete]

Objects in Motion No, it's*

[@Sam_on_YouTube]

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

[@MattMcConaha]

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

[@xooperz]

+Matt McConaha the outcome would hardly change

[@subschallenge-nh4xp]

Sam g

[@subschallenge-nh4xp]

Johnny Lee y

[@letsseepaulallenscard.6604]

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

[@seanehle8323]

"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.

[@unknown360ful]

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

[@venkatakarthikbandaru8326]

Clever way to teach cross ratios.

[@General12th]

Agreed. This approach is brilliant!

[@xintongbian]

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

[@grexursorum6006]

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!

[@lumpi806]

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

[@thecakeredux]

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]

You’re welcome.

[@AtlasTheGarbage]

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.

[@siren5742]

Belgium won because they watched this.

[@MiguelPereira-yw8rg]

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

[@Sitzriese]

lawsuits...

[@Bill_Woo]

What he said. Try v=3ECoR__tJNQ

[@naota3k]

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

[@SlingerDomb]

DEMONITIZING

[@NonDelusional74611]

Bill Woo Incroyable!!

[@benhood7680]

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

[@numberphile]

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

[@QuasiELVIS]

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.

[@manugo4]

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

[@marksimonelli9717]

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?

[@alexeifando747]

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!

[@glum_hippo]

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

[@joseleperez8742]

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.

[@Lioness99a]

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!

[@nicholascase371]

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.

[@andres927909]

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.

[@r0z3d]

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

[@Doktor_Vem]

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

[@corytoews5222]

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.

[@jawadanwar6684]

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

[@BagelBrain]

*BEFORE YOU COMMENT:* 1. They can't show the live shot because of copyright and 2. He should apologize for using the metric system because the playing field is defined in yards, not meters. Stop flooding the comments with these two questions.

[@vidaroni]

The people writing those comments aren't going to read your comment either, unfortunately. :-/

[@jyrikgauldurson8169]

No, he shouldn't apologise for use the metric system, the playing field is also defined in meters and everyone understands them

[@samus88]

"defined in yards", you don't watch soccer now, do you?

[@xl000]

fair use much ?

[@michaelgian2649]

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.

[@robin888official]

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?

[@MrCyanGaming]

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]

Last sentence is wrong

[@derflo3181]

But the rest seems smart ;D

[@MrCyanGaming]

it's not

[@MrCyanGaming]

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]

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

[@avecus]

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

[@timberhoff]

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.

[@lunardancer6047]

An easier way to do it would be just count the number of strips from the lawn mower. Each strip is 3 yards, and there's 12 and a bit strips between the ball and the goal, so it ends up being about 36 and a half yards.

[@ako969]

If you go easy route, just count how many stripes of grass (from the goal to penalty box = 6 stripes ) and 6 more stripes from penalty box to Roberto Carlos. So just double that 16.5m and you get 33m. But of course, you won't be using that cool cross ratio. And we'd have to assume all the grass stripes are the same length and also would have to use that other perspective.

[@rickseiden1]

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.

[@minatalaat7268]

another easy way to calculate it is that the field ground is divided equally with those horizontal rectangles, the penalty box length fits for 6 rectangles (look at 14:48 ) so the rectangle height is 16.5 / 6 = 2.75 m then there is also 6 rectangles from the outside of the box to the ball meaning there is 12 rectangles from the goal line to the ball, the last step is just multiply 12 * 2.75 = 33 m it's not that accurate because the ball isn't precisely on the edge of a rectangle

[@Elitematt74]

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

[@aakksshhaayy]

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)

[@KungFuBlitzKrieg]

That was damn interesting!

[@mjswart73]

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 :-)

[@feliciabarker9210]

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

[@iwersonsch5131]

To the animator: You drew the line from the MIDDLE of the football, however it was actually the BOTTOM that was on a flat plane with the other points used. So drawing the line from the bottom of the ball was necessary here in order to even use those theorems.

[@freotr9955]

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 :)

[@KarlLew]

😮 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!!!

[@SchlafliedSensor]

This is so satisfying to watch

[@Porglit]

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

[@dattanandraykar]

That handwriting is pleasing

[@TheBloodsuger150]

15:20 “In mathematics you don’t get to chose which field you work on” he invented a better quote than he one he read out

[@shruggzdastr8-facedclown]

Something which further complicates finding an accurate measurement for this particular problem has to do with the photograph itself, and that has to do with the lens used in the camera which was used to take the photo. Now, if that particular shot used a lens-body with a normal focal-length, then there's no problem, but if the camera in question used a telephoto lens mechanism, there can be a problem given that telephoto lenses compress distances. This distortion gets more and more pronounced the further the camera is away from the subject which is being photographed. So, the lines that we see in the resulting picture used in this demonstration to make the calculation(s) may not have the correct proportional relationship had a normal focal-length camera been used.

[@Pablo.Rodriguez]

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.

[@wisnuops]

Cross ratio in geometry is beautiful. But using it in this context is like using an atomic bomb for killing an ant. The more efficient way is by counting the grass-line. The number of the grass-line between the ball and penalty box is the same with the number of the grass-line inside the penalty box. Since the penalty box is 16,5m (18 yards), therefore the ball is about 33m (36 yards) from the goal. The articles you show in the beginning of thr videos is actually quite accurate. 30m is 33m rounded down. 40 yards is 36 yards rounded up.

[@itaialter]

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

[@Coltranized]

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

[@МихайлоСєльський]

Shortcut: at 14:38-14:48 it's obvious the ball is almost exactly 6 grass stripes away from the box and the box itself is too 6 grass stripes. Done)

[@KanishkaWijesekara]

There's a story of how one guy determined the size of an atomic bomb just by studying two pictures of the blast radius taken at different times that got published in a newspaper. The newspaper later retracted the story.

[@amirseifert7470]

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

[@Mik1604]

If you count the grass “lanes”, you see the penalty box is 6 deep and the ball is 6 and a little bit further. So the answer is correct.

[@jflopezfernandez]

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

[@NotHPotter]

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

[@Koisheep]

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

[@lowisao]

Look up Fifa’s official dimentions. It was a game in preparation for the 98 world cup in France. Therefore, this stadium was in compliance. Fifa determines the EXACT measurements.

[@Grozdor]

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

[@Kram1032]

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.)

[@courtney-ray]

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

[@das250250]

If you understand anything about mathematics you cannot help but love it ..

[@elprincipitoserio7287]

The best kind of matts leave the details to the devil.

[@claasbehrens2554]

i found this deeply satisfying

[@joninurmela1291]

I thought this had something to do with crossing the ball into the box but ok. Still a great video

[@rebilacx]

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

[@koenth2359]

Great! I hate to spoil the party, but because I'd like to see how any measuring errors translate into an error estimate of the distance I did some calculations. Assuming that the field lines are accurate enough not to contribute to the error, and assuming 1mm measuring error in each of the measurements, this leads to a cross ratio of k=1.251 +/- 5.0% Furthermore differentiation gave me dx/dk = 999/(2.7^2) so dx= 133 dk (Using the calculated values) Final result: x=33.1 +/- 6.7 m. Hmmm..

[@EverythingEcon]

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!

[@purple-sky-ro]

If you don't have the 4th point, can't you just choose in the middle of the a distance you already know? So for the 16.5 just add another point in the center and you are left with two segments of 8.25 each

[@berni1602]

:o I usually love your videos, but although I'm not a soccer fan, my mind was blown with this video!

[@brianpso]

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.

[@mow184]

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

[@timermens350]

Both the penalty area as the distance from the penalty area to the ball are 6 lawn mower strokes wide... So seems pretty accurate.

[@Spongman]

wait. the red ratios are only the same as the green ratios if there's no spherical distortion in the camera lens. for a fish-eye lens (the extreme case) those lines are no longer parallel - they meet off at infinity in both directions.

[@jeanbonnefoy1377]

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

[@dimitristryfonos9680]

Well you know the lenght of the whole field and you can measure the nuber of green stripes (darker and lighter ones) which are pretty much consistent in size and calculate the distance in terms of stipes. And you asume he was perpendicular to the goal line so assuming the stripes are perpendicular too you wouldn't deviate to much. This wouldn't as acurate but it would be accurate enough (approximately to +- 1.8-2 meters)

[@No0dz]

Engineers are aghast that you are using a flimsy plastic ruler instead of a proper scale ruler for taking precise measurements on paper. OTOH videos like this make me want to dive further and further into math :D

[@numberphile]

Great to hear.

[@shortcutDJ]

Good luck Brasil, May the best team win. a Belgian

[@williamdelatorre9991]

I like the usaquen buseta sign , I used to live in Bogota in the seventies

[@alberthall6375]

Great application of math to real life!

[@raketnight]

As someone whos from the physics scene, i'd just count the number of rectangle that goes across the pitch. So 12 with 6 for the d box and another 6 from the d box to the ball and so just use the squigily equals then write 33 and be happy with it. But im sure the maths crowd will experience sweaty armpits after reading this.

[@markfischer5044]

Look at the mowing lines in the video at the end...6 bands from the goal line to the front of the box...then six more from the box to the ball. This roughly confirms the calculation.

[@alazrabed]

And now to compute the distance traveled by the ball. You'll express your answer with not least than five significative figures.

[@alephnull4044]

What a brilliant video.

[@sair4269]

VIVA COLOMBIA!! Sí alguien no lo sabe, el que está hablando en el video es un matemático Colombiano

[@turtlellamacow]

I'm missing some logical step here because I don't understand how the cross ratio theorem applies to this situation. He asserts that when four rays emanating from a point cut two other lines in a plane, the lengths of the segments into which the lines are cut have a fixed cross ratio, cool. How does one go from that to claiming that a line in real 3D space and its projection in a 2D photo obey the cross ratio rule (and the four cutting lines are parallel in real life, so they don't even emanate from a single point)?

[@OriginalPiMan]

So reports that said 35 yards were the most accurate, followed by reports that said 35 metres.

[@maiku20]

109 people are geometers at heart who will hate the cross ratio their whole life.