Welcome back to our lecture series math 3130 modern geometries for students at Southern Utah University. As usual I'll be a professor today Dr. Andrew Missildine. In lecture 11 we're going to continue our discussion of the betweenness axioms which we introduced in lecture 10. Now you'll recall that we said that an incidence geometry which satisfies the four axioms of betweenness is what we called an ordered geometry. Why is it called ordered? What does that order have to do? And so in lecture 11 we want to get through the very technical argument that every line in an order geometry can be given an ordering. Now before we do that in this video we're going to prove the idea of linear density. That is if you take any two points on a line you could always find a bunch of other points related to them with well between this relationship. So more specifically if we have a line L and we have two distinct points on that line call them B and D and this of course is an ordered geometry. Then we can always find three other points A, C and E on the line L such that the point A has B between A and D. The point C will be between B and D and the point E will satisfy the relationship that D is between B and E. So let me try to draw an illustration to illustrate what we're thinking of right here. We have our line L, we have some points B and D like so. Here's L, here's B, here's D. So the linear density theorem tells us we can always have a line, excuse me, we always have a point A so that B is between A and D. We always have a point E so that D is between B and E and we always will have a point C such that C is between B and D. Okay? Now the existence of some of these points are quite trivial. Let's start with A for example. By using the extension axiom which was one of the between this axiom we get that there's going to be a point on the line such that A, there's a point A on the line such that in particular D dash B dash A holds. We also have by of course the symmetric axiom that this is the same thing as saying A dash B dash D which is what we're trying to start off with. So the existence of the point A is given it basically immediately by extension but we're also using the symmetric axiom there. The existence of the point E also follows immediately from extension so we have this point B dash D dash E. Alright so A and E are pretty easy to come by. Why do we have the point C? That one's a lot harder to explain so that's going to be the bulk of this argument right here. So to find the point C we are going to use the extension axiom repeatedly but we have to be a little more clever about it. So to start off with we're going to have to find a point F that lives off of the line L. Now the existence of this point is a consequence of incidence axioms. Given any line in the geometry we can always find a point off of that line called that point F. And then consider the line, let me try that again, let's consider the line that goes between F and B like so. For convenience we will call this line M. Okay? Well apply the extension axiom to the line M right here. There has to be a point, let's call it G, that extends the line from B in the direction of F. Again, extension right there. And so then we want to consider the line that connects G to the point D. Kind of missed that one. Let me try that one more time. By line determination and axiom of incidence there is a line that is determined by the points D and G. We're going to call that line N. Again just for convenience here. And then by using extension one more time there is a point, let's call it H, that lives on the line N like so. Okay? And so really what we're done here is of course we've constructed these lines and these points and I want you to focus on this idea right here. We've constructed this triangle, the triangle G D B. That's what we're playing around with right here. And some of my labels a little bit hard to see but remember here's F here is, oh we didn't actually write its name on the screen did we? So this was our point G, not G, H right here on our graph. Okay? And so the main objective here is then to consider the line segment that connects F and H together like so. Consider this line, the line between F and H. In particular it's the segment that we care about but we really we can think of it as a line right? And so when we look at the diagram does appear to be some type of point of intersection, right? But we can't just assume by the diagram because the lines cross they have to have an intersection there. That's what we're trying to prove right here. The whole point of the diagram is just to help us keep track of what all these points and lines mean, what are the between this relations, what are the incidence relations. But in all reality the proof doesn't come from the illustration. The proof here is going to follow from posh's axiom, another axiom of between this. So let's see how that happens here. Okay? I'm going to scooch my picture up just a little bit so we can see more of the proof here. And so scooch it just a tiny bit down here. Alright, so what's the argument here? So we're going to consider this line determined by F and H right here. And so we know that F is the unique point of intersection between the line FH with the line M, right? Because we have this line right here, the line M. We have this line FH, which I haven't really given a name just FH. But the unique point of intersection is going to be the point F. Because we know from incidence geometry because of line determination that when two lines intersect, they intersect a unique location. Okay? And so the intersection between these two red lines has to be F. In particular, the line FH doesn't contain G nor does it contain B. Because if it did, there would be nonunique intersections. So B and G are not on the line FH. Okay? We can also say a similar thing about the point D right here. Because after all, we know that the line FH is going to intersect the line N at a unique location. That unique location is going to be H right here. And so again, by line determination, lines intersect a unique point. Therefore, the line FH does not intersect the point D. Why are we bothering talking about this? But it's because of Poch's axiom here. By Poch's axiom, we have that we have to consider a triangle. We have to consider a line that doesn't intersect any of the vertices of the triangle. Okay? That's the big assumption of Poch's axiom. Then we also know that this line FH intersects one side of the triangle. I'm going to zoom out a little bit so we can still see the picture but we can also read the proof. Okay? So we know that the line FH intersects one of the sides of the triangle. It intersects the side BG in particular. So by Poch's axiom, since the line doesn't go through any of the vertices of the triangle and since it intersects one side of the triangle, it has to be true that FH intersects the segment BD or it intersects the segment GD. Now, if it intersects the segment BD, which is actually what we want, that's because the intersection, since it's not a B, it's not a D, it would have to be at some point between BD. That's the point C that we're looking for. Okay? But Poch's axiom does have two possibilities. So why can't it be GD? Why can't it intersect something over here? But the reason for that is the line FH we already know intersects in and it happens at H. Lines only intersect at two different lines only intersect at one point. So if there was a point of intersection between D and G, that would be a second point of intersection between the lines in with FH. Well, unless, of course, this point is H itself, but by construction H is not between G and D. H is actually what we do have is that D is between H and G. And by trichotomy, we can't have multiple between this relationship. Notice how all of the axioms between this are coming out. We've used extension a lot. We just talked about trichotomy. We used the co-linearity and symmetric axiom and then Poch's axiom is going to finish the day for us right here, right? And so I want you to be aware that this original construction, these extensions were used exactly so that we had control over the between these relationships. We started off with a point F that was off the line, then we extended it, then we connected the dots, then we extended it again. And so again, this was so that we can construct a triangle that forces the intersection to be not here but over here. And so that then concludes the theorem here. By Poch's axiom, since the intersection of F H on N can't lie between G D, this would mean that the line F H must intersect the segment B D and it happens at a point between B and D and that then gives us the point C. An important consequence, of course, of the density theorem is that in order geometry all lines have infinitely many points. We already knew that by extension because we can continue to extend, extend, extend in both direction. But the reason it's called dense is that it's not that we just make the longer the lines longer and longer longer is that between any two points we can find another point and between them we can find another point and between them we can find another point and we can find another point. So between any two points on the line, we have infinitely many points and that's what density is measuring right here.


I'm Corbin. I'm Rick and you can follow us on Instagram whoa and Twitter I'm all cheers That was weird. We both like subliminally knew the other one was gonna stop and finished it. Oh You love me. Yeah We like finishing each other A dance is it a classical dance? Yes, it's a good and it's a different one that we haven't seen before It's not a dissy and it's not the other one. I can't pronounce. I don't believe so It's person. I think explained it to me. Hold on. He says it's called Kuchipudi a classical dance from Andhra Pradesh Do we know that name under Pradesh? But I don't know Kuchipudi if I'm pronouncing it correctly The dance is called Kuchipudi which of the Classical dance forms. I'm wondering they I would love to see something about the evolution of classical dance in India in which forms came First and informed other dance forms and just didn't inform each other at all Cool. Yeah, so if there's a video like that, please send it my way Yeah if there is a video that gives you the evolution of like or the history of Indian classical dance and the different forms and which one came first and informed the other because I know Differentiations between the two and I'm learning more and more about classical dance And this one I is new to us. So I'm excited and I want to read the title for me Krishna Shabdham Kuchipudi by Sanjya Raju And so that's what that is. I don't really know what those words mean Well, my I don't know which ones the dancer and I don't know which ones the the choreographer Yeah, well the Kuchipudi is the style of dance apparently right and I guess it's Sanjya Raju would be the choreographer and dancer possibly But what's the name Krishna Shabdham? I guess that's the dance and it's gonna be Krishna centric Gotcha. Okay, just like we've seen the other dances that were Shiva yeah, exactly for the gods. I think this one is focused on Krishna. Here we go My goodness her hair is that her hair? Yeah Yeah Down in front This is definitely the most fluid and quick one we've seen She's using her eyes and there's clearly a story going on. That's some beautiful and Sanjya Raju presents her guru dr of uh, uh, Vinpati China Satyam's choreography of Krishna Shabdham presented live at, uh, Rajapalliam Tamil Nadu on 13th april 2014 Krishna Shabdham is a very typical and popular piece in the Kuchipudi repertoire The young heroine is trying to beckon her unattainable hero. That's what I get sense. Yeah come I want to cook for you come to me.  I saw that she put one of the flowers in her hair Yeah, so she did a fantastic job of just because I got all that. Yeah from a hundred percent from doing it This had uh, obviously some of the differences particularly with the classical dance form Most familiar to me that we have seen and I've paid attention to is odisi and the Footwork on this first of all is obviously so much softer than very odisi footwork as well as even the physicality This is more way, um, not that odisi doesn't have fluidity. It has lots of fluid Yeah, but this is a primarily fluid. Yeah one and it's also deeply of all the classical dance forms We've seen this is the one that has the most mime. Yeah 100 it's like dependent on them the line and the performance Yeah acting behind it. Yeah where in odisi for example, you tell the story through the the you actually like if there's a Rather than being actually taking the flowers off of the the tree. There'd be a hand Movement that communicates the flower and every single spatial movement of the hands and the fingers is depicting the flower Yeah, and so this is more Storytelling and using and props that aren't there. Yeah versus the dance movements informing Yeah, it's really interesting. I do love seeing all the different types though. The difference of dancing I love the difference equally as beautiful as anything we've seen Uh, and I was he was captivating because I could just I could watch her just the entire Yeah, doing I just wish it ended at the right time Yeah, really and I'm sure the same thing I saw her doing some things particularly with I'm sure this too like odisi hand Gestures will convey meaning as well It just seemed that this dance formed owes a lot more of its storytelling aspects to this mind Yeah, this performance this performance of it. Yeah, we don't know And we also wouldn't know the discipline well enough at all to say how good she is compared to other dancers of this this dance form but It's I've never seen this dance form the the dance form I've seen the most even before this channel Was the one I I knew was more symbolic and more odissy odissy style where the hand gestures and postures are Conveying the story more than you would this is the most I've ever seen mine Indian dance. Yeah 100% Uh, and I loved the the costume was beautiful. And of course the two times she did little Yeah, I don't even know how she did that. It was beautiful. So fast.  Yeah Like there's a lot but she had a lot of her movements were Yeah, time to that and even that little nuance where she was doing the step at a certain pace and then the the rhythm It sounded like a tablet going dig dig dig And she was walking walking and went dig a dig an instrument With her feet But the softness of the feet was really fun to watch compared to odissy Which is so strong and planted and pounding onto the ground. It's just beautiful I love every single one of these dance forms we've seen Yeah, me too. So so pretty. If there's others that we need to to to know And bring them all on these send them because we love these and hope you love it. Yeah, I hope you love it I just from from the feedback. I've gotten people people enjoy it. Yeah, I'm glad because we enjoy it We would have done it regardless of if you enjoy it. Yeah So I'm glad that you also enjoy it. I mean, it's not a new experience for you It's a new experience for us It is a new experience for us and I think also we've noticed comments from from stupid babies talking about how they like us And what differentiates us among other things with reaction channels Is that we don't limit ourselves to just like say movie trailers and music videos We really want to know as much as we possibly can about the totality of creative artistic culture in every part of india, which You'll take the rest of your life to even begin if you we could do a whole We could do a channel just on classical dance and classical singing Oh, 100 and spend all of your time on that 100 so it's just beautiful to have such a wealth of rich history of creative expression for more millennia than Any other country that's ever existed really except for you know, people Compete china with india, but I think the diversity of the culture of india There's there's no country that has a longer history of diverse artistic expression in india The most artistically diverse country in the history of the world. Here's your question Are there men that do is there a classical man? Good question. We've only seen the female. Well, yeah As far as the solo in terms of classic Yes, I know they dance. I know 100 100 that there are male odyssey dancers Yeah 100 and that odyssey isn't just on a solo performance odyssey has done As groups or couples or so. Yeah, but as far as like soloing we've only seen females do the solo dancing I would love to see if some of the men are doing some solo dancing as well. Yeah, and uh, All different types.  Yeah, it's his name kale Last name is kale. Yeah So that's please let us know more because we love these types of things Please let us know and then I'll vote the ones that you think are good because that's how I know What is good and so I can do what you guys want us to do. Yes. So thank you


Mae'r testau yn ymdangosol, ond y byddwch yn ymdangosol y pandemig. Ond mae'n meddwl i'r ffordd, a mae'n meddwl arall oherwydd eu hunain european. Mae'r regerau hefyd yn ymgyrchol yng Nghymru, i 100,000 testau y ddweud ar y ddweud ar y dyfodol. Ond ymddangosol, y ffigur ar y dyfodol ymgyrch yn 18,000 y ddweud, ac mae'n ddweud ar y ddweud, yn 19,000 testau y ddweud. Ymwan yw'r regerau hefyd yn ymgyrchol ydw i'w hyfforddiadol ar y ddweud ar y dyfodol 8 diwrnod yma i'r 180,000 testau y ddweud, i 100,000 testau y ddweud. Ond mae'n meddwl i'n meddwl i ddim yn ddweud, ac mae'r regerau hefyd yn 40,000 testau y ddweud. A'r regerau hefyd yn ymgyrch yn dweud. Mae'r regerau hefyd yn ei ddweud, a'r regerau hefyd y ffigur ar y dyfodol y ddweud? Y regerau hefyd yn ei ddweud, There's capacity for 40,000 tests today and I think it's really important that we fully understand what the first Secretary just said. Because that means that the day before yesterday 40,000 tests could have been carried out, but only 18,000 tests were actually carried out. Now all week, I've heard from the front line, from care workers, those who are frankly desperate for tests for their residents and for themselves desperately... I thought that was pretty good, right?  I was a top QC, I'm the kind of guy who can, you know, ask questions which put people in a difficult situation and I thought he kind of managed it there, so, you know, it made Dominic Ra... He's sort of real Dominic Rabin to saying that, you know, ystyod wrth ei ddweud hyn yn cyfeirio ar holl gwybod. Mae'n cyfeirio ar holl gwybod, ac ar holl gwybod. Mae'r sefyddo chi wedi am fynd eich roi mwy familyllau i wych yn cyfeirio hyn. Felly mae'n cyfeirio ond mae'r cyfeirio ond mae'r cyfeirio yn cyd-i gwaith, yn cy fe fyddwch i'r cyfeirio i hyn, a'r cyfeirio i gael. Ac rydyn ni'n gwybod lle rwy'n ei ein bod yn gallu, oes y byddwch chi'n ddiwethaf, dwi'n golygu proses o'r gaelgol, ac mae wedi gweithio ni'n gwybod. Rwy'n gweithio i chi'n gweithio. Felly, rydyn ni'n aelod? Rydyn ni'n gweithio i chi, mae'n ddiwr. Mae'n gweithio i chi'n gweithio. Mae'n ddweud i chi'n gynghwm i'r Gwyrdyn i chi! Mae'n gweithio i chi'n gweld yn gweithio i'r gwyllgor. Mae'n hon i chi'n gwneud yn ysbryd o'r gweithio ar gyfer ysgrifennu, mae'n gwneud i'r gweithio ar gyfer ysgrifennu. I'm not going to dain to offer an answer now. I think one thing with the reaction you sort of had a lot of sort of blutic centrists or anti socialists. Actually, that's the better word for them.  We deserve what they need to understand is that parliamentary questions. PMQs don't really matter to most people. You know, in my lifetime, there's probably been two or three moments in parliament PMQs where the public really sees what's going on. There's a cut through maybe once there was one interaction between John Major and Tony Blair, there was one interaction between David Cameron and Gordon Brown, you know, there's maybe one interaction between Cameron and Corbyn, where Cameron monstered Corbyn. But Jen and even there, but even then that's like 40 percent of the population kind of vaguely remembers seeing something to do with it. They're not pinned to Twitter or to Facebook or to BBC Parliament watching the stuff. So it's great. He's good at that. It's great. He's authoritative. But that in isolation isn't going to win you an election. It's important that most people don't think that, you know, it is just a select group of several dozen sort of times columnists and evening standard political diarists. I'm going to go to I'm going to go to Lara in a moment. But first of all, I want to go through some of the tweets from commentators and MPs from yesterday because they I mean, I don't know if it was if it was that they all organically came to the same description for Keir Starmer's performance, but their their reviews definitely all looked quite samey. Julia Hartley Brew, a very strong start from Keir Starmer, his first PMQs. All fair and reasonable, put with forensic accuracy. George Eaton, friend of the show. That was pretty forensic by Starmer, Christian Rednedge. Already so forensic on detail. Liam Ford, he's from the Liverpool Echo. Starmer straight into it. Already looking pretty forensic, Ian Martin. Good forensic first questions from Starmer. He's from reaction. The right wing side with this sort of odd name. Lewis Goodall. Again, I like Lewis Goodall. He's also gone for forensic poor wars gone for forensic. And now I think the next one we can look for is what all the MPs have said, which is also very similar. So now you can see all the different labour MPs who've given exactly the same description of all the pundits in there as well. Do you think are you in an EC WhatsApp group? Did anything go around to say, can you please tweet that? Keir Starmer was very forensic at the dispatch box? Or do you think it was a coincidence? Oh, I don't know the. I guess it's good that this kind of thing is going around like I saw people going, why is everyone complaining because people were saying things about Corbyn that were bad, like good things are going around now. Just leave it and I get it. But I mean, I wasn't as impressed as as as others of on his PMQs thing, because I think Rab kind of fucked up in a bit. Oh, can I swear on this? Yeah. OK, he kind of messed up because he got into that trap, you know? So I think it was like, OK, exchange, but I don't know. I agree with Aaron that not many people watch it. So we've got to be aware of that and how it how it fits into a wider kind of political scene that we're creating. And I think my highlight of PMQs under Jeremy Corbyn was him reading out name things in terms of people's stories and how politics and policies were affecting them and how he got angry about situations. And I really want to see that passion and anger from Keir about injustice and what's going on. So, yeah, I think it was OK, but I don't think it was amazing and forensic. And I think there's this kind of image that people have set for Keir as like this amazing lawyer and, you know, hopefully he plays up to it. But I'd like to see some of that rather than just sort of the lawyer side, you know, the passion about that he says, you know, about doing the right thing, about justice, the reason why he's explained that he went into law and stuff. So hopefully we'll see that. But yeah, I don't think people watch PMQs and I don't think they will.  So, yeah, I didn't think it was that amazing. Was it too constructive for you? That's sort of like some of the conversation answers that this is Keir showing off what constructive opposition can look like. I mean, I was kind of annoyed because I feel like the the left critiques of Keir Starmer were portrayed as we want him to be more angry. Whereas my problem with Keir Starmer was I wanted him to ask genuinely difficult questions, which I what the reason why I thought his PMQs there were quite good was because the questions he asked were hard and they were about the things that people care about. So, yeah, there wasn't there wasn't that image of him standing up and saying, you know, like I'm the tribune of the people and I can speak in a sort of passionate, inspiring way. But he did get Dominic Raab to answer in an unconvincing way about testing. He got Dominic Raab to answer in an unconvincing way about the depths of care workers and he got Dominic Raab to answer in a very unconvincing way about personal protective equipment, which was a massive improvement on the week earlier where he was saying, can you publish your exit strategy and the Tories were incredibly comfortable because they were saying, no, we're not going to publish it because we're in the middle of a lockdown and most people listening to that were like, that sounds reasonable. It sounds like the Tories want to protect the NHS and the Labour Party are being a bit difficult, whereas I feel like you couldn't portray him as that yesterday, so I thought that was a real. Yeah, we've definitely improved from last week, but I think it's pretty low bar like I think the Tories are doing disasterously in this situation and if we're able to call them out on their failures like they have been actually lying at some stages and I know you can't say that in the House of Parliament, but yeah, so I think this is this is the bar where we shouldn't be at like here making talking about the mistakes that they're making, so it shouldn't really be that hard to make them fall into traps because they're making the traps themselves in literally lying in their cotton. So I don't think that's a particularly hard thing to do. Yeah, it was bad last last week or whatever, and we've moved forward from there from talking about the exit strategy, which I just don't understand. I literally have not heard an NHS staff member talk about that. They've all been talking about testing in PPE. So I'm glad that he's shifted his focus, but yeah, the bar, we're now doing well instead of bad, but not amazing. Can I say also a lot of the people that really adore Starmer are obviously kind of like big Blair fans and Blair excelled at PMQs when he was passionate. You know, Blair wasn't this forensic. Blair was like a bit of a populist. And this one was weak, weak. Yeah, weak. Yeah, it was like he just went all, he just like absolutely went all in on John Major as a human being. It wasn't like on there and on page 52, he didn't do that. That's not what he was good at. That's what Gordon Brown was good at. And there's like a lot and he's just been expunged. This kind of this idea of Blair, the populist has been expunged from the kind of centrist psyche. I'm not that's not maybe Starmer cat. I think personally, Starmer looking the way he does being very polite. I think you said it before, Michael, I agree with it. It buys you permission actually to be more radical than you otherwise can be as a party potentially. I buy the argument. So the social movement behind Keir Starmer, the political party behind Keir Starmer, the trade unions behind Keir Starmer have permission to be that bit more radical if they if they play it cleverly with that more intelligently, if they have that guy at the front. I buy the argument and say it's going to work, but it's a logical, consistent argument. But the idea that this is a continuation of sort of the successful sort of, you know, Blair, Blair kind of era of PMQs is just not accurate.


Tilburg University's motto is understanding society and that refers to the legacy of our founding father, Martinus Kobbehaar. The university always had, from its beginning, societal orientation. We educate our students to be able to find the right balance between economic values and societal importance, so whether you study law, management, psychology or economics, we will always consider your discipline as part of a bigger societal picture. For several years now, the university has been voted to be the best specialist university in the country and we combine this with the top international reputation both in research and education. The Tilburg educational approach stresses two elements. On the one hand, academic attitude and on the other hand, the small scale. So let me start explaining the academic attitudes. So when you come to Tilburg, you explore topics with an academic attitude. In other words, you obtain an inquisitive mindset. So how do we do this? Well, we incorporate the latest research insights into our classroom and we combine it with societal discussions. We also invite, typically, guest speakers from practice to share their experiences with our students and last but not least, our students are very active in organizing public debates with well-known political speakers or business people. Secondly, the small scale. You are not a number when you come to Tilburg University. It's a relatively small university with 12,000 students and that allows our faculty and staff to actually know each student. We apply also a mix of plenary sessions in the class with small intensive practice workshops. We have, for those students who need it, an active mentor and coaching program and for those students who want to have more challenges, we offer honors courses and last but not least, we assist students in finding, of course, their dream job. Tilburg University stresses international aspects in its entire curriculum. First, in all our bachelor programs, we build in an international mobility window that enables our students to visit one of our exchange partners, one of our 200 exchange partners in the world and, of course, students from these exchange partners visit the university here. Secondly, we have many international faculty members who share their international experiences and education with our students and all the courses that we offer in Tilburg are also offered in English. Last but not least, we have an important tie with international renowned companies since we are located in the brainport region of the country and companies such as Philips, ASML, Fuji and many new ventures offer internships and also job opportunities for our students. I started working at Tilburg University about nine years ago and the first thing that amazed me was the beauty of the green campus and the amazing sports facilities. It provides the perfect setting for great academic experience. The lively city of Tilburg is within biking distance from the campus and reaching cities such as Amsterdam, Rotterdam or Brussels is really easy because of the great train connections and nice highways. It's not surprising that Tilburg is voted as the top logistic center in the country. The other element that strikes me is the international character of the university so when you walk around the campus you immediately hear different languages you are confronted with many different cultures and all the services provided are bilingual. International students get special welcome when they arrive. Our campus facilities are simply amazing. You have different cuisines and different locations so whether you are vegetarian or prefer Asian food you have plenty of choice on our campus and don't forget that the city of Tilburg is really close so if you want to have the real city experience then there is definitely not a problem there. Our newly remodeled library is open every day from eight in the morning till midnight. Coming to Tilburg University to follow a summer school is a great international experience. Every year we have over 300 students with 50 nationalities over the whole world coming to Tilburg University to follow a summer school. It is a great way to experience the culture and academic climate at our university. We combine a great international study experience with also a fun social program. It is also a way to learn more about the Netherlands as a multicultural society in the heart of Europe. Our summer schools are linked to our core competence at the university. The latest research findings are combined with practical experiences so often the format is that you have a morning lecture on a topic that is followed by an afternoon practical case workout. The format varies between one and four weeks and you obtain educational credits for this.


The importance of a family is really going to have a some of the generational family. I do have it. Sometimes I hear of it. I haven't been in Europe yet, so... Yeah, I'd say maybe something similar. But I appreciate it. I appreciate people a lot later. I do. I make this stuff. I think it's really important to reflect on the decision to accelerate the ones who are already here. I'm going to do that. It's been almost 10 years. It's been during that time. I lost my aim. So I was close with the... There are a lot of people who are doing it. There are a lot of people who are doing it. But I don't think that's the way it is. I don't think that's the way it is. I don't think that's the way it is. I think it's the way it is.