And welcome to Lillian's Vegan World, the only show in Honolulu where we talk about vegan lifestyle and plant-based diet. I'm coming to you live from the Think Tech Hawaii Studios in gorgeous downtown Honolulu and the times for my show have now changed from Thursday to Thursday, 1pm every second week. So looking forward to seeing you at this time again. I'm so excited to introduce my guest today from Vegan Hills, General Manager Kelly Camau. Yeah, hello. Welcome Kelly. Thank you. Thank you so much for taking out the time in your busy schedule and busy restaurant to come and talk to us. Thank you. It's exciting to be here. I know, do you know that when I first came to Honolulu for the first time two years ago, my first, the very first restaurant I went to was Vegan Hills. Nice. I was talking about how awesome the food was and stuff. So I do have sort of a little soft spot in my heart for your restaurant. And I did visit Vegan Hills a couple of weeks ago with my husband, who's a non-vegan, who's what I would call the opposite of a vegan. And he loved it. Good. Awesome. He honestly did. He loved it. I thought it was fantastic. So yeah, welcome. Thank you. How about it? How's it going? So you recently started working there as the general manager this year? Yeah, this year. I officially became general manager I think in June, but I was co-running it since the end of October of last year. Okay, great. And I've been a waitress there since the first month it opened. Wow. How awesome. Thanks. And you've worked your way up to now really running the show there, which is fantastic. And you're doing a wonderful job. Thank you. Tell us first of all for people that don't know what Vegan Hills is. What is Vegan Hills? Vegan Hills is an all plant-based vegan restaurant. We strive to showcase how eating healthy doesn't have to be eating boring. It don't have to go without flavor to eat a full balanced diet basically. So yeah, we just show that you can eat really, really good food. If you want to go vegan or if you want to go plant-based, and we've had so many people like your husband who is completely not vegan come in, I'd say like 85% of our return customers are not vegan. And they keep coming back. Really? Yeah, because they love how the food makes them feel. They love how it tastes. They say that they can't believe it's not vegan. And we've had a lot of people be like, man, I could go vegan if I could eat this food every day. That's amazing. Yeah, and except Tuesdays. Except Tuesdays? Yes. We're closed on Tuesdays for now. We're working on being open seven days a week in the future, but we want to really hone in ourselves first before we can expand to all the hours. If it were feasible, I would love to run like a 24-hour, seven-day-a-week vegan place that anyone could go into at any time and get whatever they want. That would be my dream. It's hard work. So for hours at the moment, Tuesday you're closed and you're not open for dinner at the moment. Not at the moment, no. Yeah, we're leaving those open more for special events. We do have a couple of those coming up at the end of the month, but we can talk about those later. Definitely. Yeah, so I'd love to hear about them. Yeah, I personally love eating dinner, of course. So we do have it in the works and we're trying to come up with the right menus for it, putting together the recipes, which is a lot of fun. But still, even if you're not open in the evenings, you still cover all of those food groups. You can have like an awesome breakfast at lunch. You can have dinner at lunch because your dishes are just sort of, you're covering all the gorgeous comfort food that you can think of and you've veganized all these dishes. So my husband in particular was just amazed at the pulled pork that is made from jackfruit that you served. You did some sliders, I think they were. Yeah, the barbecue jackfruit sliders, and we make our own coleslaw for that as well, using an almond milk-based mayonnaise kind of, but it's not like thick like mayonnaise. My boyfriend who hates mayonnaise, he really likes that slaw. And the jackfruit, we just marinate it with some barbecue sauce, a little bit of liquid smoke to give it that flavor, and it is insane how much it mimics that. As an animal lover myself, I used to be completely not vegan. The day before I went vegan, I was eating collo pork quesadillas, and then the next day I was like, nope, no more. So, someone who really misses those flavors and those textures, it's really exciting to be able to find out how to recreate those with plants without having to harm any living animals, and to be able to reproduce it for the public, basically. Exactly. I mean, how far has vegan food come? Even just in the past few years, it's just incredible, and I love that you're serving stuff like that, because going back to what you mentioned earlier on, vegan food is not only for vegans. I drum this through everyone's brains, everyone I meet, I actually don't cook for vegans. I cook for non-vegans, not because I prefer to, just because people are curious. We're at the age where people are curious. They are looking for healthier options. They are wanting to try these new things that are just not going crazy all over the world, or these new meat products and stuff, but feeding back to that jackfruit, those sliders. Definitely I hope that viewers go in and check out vegan hills. Kelly, I just want to step back a bit.  How did you end up? Yeah. It was pretty crazy, because for my entire life, basically, it was always like, I'll never give up. Me, vegans are crazy. How could they? The cheese, the wabachis, and that's so true. Yeah. I was at a point in my life where I was just eating out of control. I was working late nights as a waitress in restaurants in Waikiki, and then I'd get out of work, and then we'd just go eat bar food, and I'd eat 20 chicken wings in one sitting. Typical bar foods. Yeah, typical bar foods. I was just the heaviest I'd ever been, and I was not able to perform my work well. I wasn't feeling great, and I was just like, man, I really got to find something that will help me curb how I'm eating, because this is definitely out of control. I had kind of been thinking about it for about a week of what I could do to change my eating, because I was like, I know myself, I'm all or nothing. If I don't cut something out, or if I don't drastically change something, I'm not going to change. Then I saw this five minute video on YouTube called Dairy is Scary, and it's about the life of a dairy cow and how inhumane it is, and I always called myself an animal lover and environmentalist, and those facts in my face like that, when I was already in a very susceptible mindset, I was like, ooh, this is not something I want to put my money towards anymore. And I was like, wait, I'm looking for a way to eat better. I don't like what dairy is doing to the animals. I wonder if maybe vegan is right for me. And then I was like, I'll try a vegan challenge for seven days. I'll go complete cold turkey, so to speak. Yes. And I'll just try for seven days if at the end of the seven days I can't do it anymore than I tried it. And then at the end of the seven days, I had watched all of the documentaries on Hulu and Netflix and YouTube. I had read so many articles. I looked at the nutrition behind it. I looked into the environmental impact. And by the end of that week, I was so turned off by meat and dairy and eggs that I just never turned back. But it was pretty crazy. But yes, it only took you a week to realize how really you don't need it. That's just the fact is that you can live a very healthy life on a plant based diet, and you become aware of so many things, don't you? These documentaries that you talked about, there are so many now to even mention, but they are very eye-opening. And I do recommend people take a look at them. Why not? You've got nothing to lose. Exactly. You might just change your life for the better. Yeah. Exactly. Change mind for the better. It's definitely one of the best decisions I ever made. I'm so glad to hear that. Yeah, thanks. This was about three and a half years ago that I first went vegan. Yeah, February of, I can't math right now. Whenever three and a half years ago. So you never went back? You never cheated? No, there were a couple of times where I would accidentally be served something with it. And instead of letting myself get upset about it, I took it as a learning moment to observe how my body reacted. Because during that first week when I tried it, what I really noticed was how amazing I felt after just seven days. Like I felt amazing within the first couple of days. Then after seven, when my body had really fully detoxed all of the things I had eaten before, I was like, wow, I feel so good. And so anytime I would accidentally consume it, because with quitting anything, you have those withdrawals. And like, oh, if only. And the biggest thing was going out with my friends. Because they all still wanted to go to those same places after work. And I could no longer have. Most places you go, the only option for vegans is fries or rice, which is. Yes, it amames. As much as I love potatoes, I know only so many french fries you can have late at night. True, yes. That was the hardest part. But I only ever thought about it. But then I would even look at my friend's food and be like, maybe I could take a bite. And then the closer I get to it, I'd be like, nah. I'm good. Yeah. As long as you're balancing it out with all the other good stuff, you have to enjoy your food. And that's one of the things I loved about your restaurant. Really, it caters for every palate possible. So definitely do check out Vegan Hills. Kelly, let's have a look at some of the food that you serve there. Let's have a look at the first one. These are photos that I took from the lunch that I enjoyed there a couple of weeks ago. So I did get the loaded fries, the guacamole and chips. And what else, the sliders down to the left, the bottom left, the jackfruit sliders we were talking about earlier. The gorgeous salad that you recommended, the Brussels sprout. It's called the Brussels. Love it. Thank you. That was beautiful. And come back for more Vegan Hills and more vegan fun stuff. Aloha and welcome. My name is Mark Shklav. I am the host of Think Tech Hawaii's Law Across the Sea program. Every other Monday at 1 o'clock, I am here on deck with various guests talking about different topics of the world and the ocean and international law, different areas where we all have seen and wanted travel to and learn about. Please join me for my next Law Across the Sea program. Aloha. Aloha, everyone. I'm Christine Linders. And this is Think Tech Hawaii. My show is Movement Matters. And this is a show brought to you to talk about how to get rid of things like your low back pain, scoliosis, TMJ dysfunction, ankle sprains, pretty much anything that you can do with your body or hurt your body. I am here to bring to you the cutting-edge strategies that you can do right now easily on your own to help get out of pain and get back to doing what you love. Life is better when you listen to your physical therapist. Tune in Tuesdays at 11 a.m. every other week for Movement Matters. Aloha. Welcome back, everyone, to Lillian's Vegan World. I'm here with my awesome guest, Kelly Camau from Vegan Hills, my favorite vegan restaurant in Honolulu. It is delicious. And I did name today's program, Kelly, Vegan Hills, Leading the Way in Plant-Based Cuisine in Honolulu, Turning Comfort Food into Mouth-Watering Vegan Versions for All to Enjoy. Yes, yes, that's definitely a big part of what Vegan Hills is all about, just showcasing how diverse plant-based foods can be and how it can be made delicious for anyone. Exactly, I totally get your place. I think you need to make Vegan Hills in every little area in Honolulu, so I don't have to go far to get my fix. Yeah. Let's take a look, Kelly, at some more of the photos that you've prepared of these gorgeous dishes that you're serving there. Yes, look at that. Awesome, these are two of our cocktails. The one on the left is our morning moon rise with lavender simple syrup vodka, organic vodka too. We have an all organic bar. Right. Pineapple juice, it tastes like a spa in a glass. It's so good. And then the one on the right is our mega-marry. That's awesome. Yeah, it's like a 30-ounce drink. It's got your choice of either vodka or tequila, and then it comes with two potatoes and a breakfast lighter on top. It's been a dream of mine to have something like that. I don't even like Bloody Marys, or I never liked Bloody Marys before I was vegan. And I saw there are places around here that do like the crab claw, and they do like these massive Marys, and then I'm like, all of a sudden I want one. But most Bloody Marys have anchovies in them in the Worcestershire sauce. So it was always my dream to kind of make like a vegan version of that, that vegans can come in and enjoy and also have that epic time that I feel like a lot of it is like I really want to make vegan versions of epic times that I feel like I might miss out on sometimes. Yeah, that is such a great thing to have on your menu. The only reason I didn't order it is because I wanted to try so many dishes, and that just looked like it was going to be really filling and satisfying. But definitely I'll have to try one of those. I do have a mini Mary as well for people who want to eat more than that one thing. Great, I love it. That's very creative. Thank you. I like that. Let's have a look at another one, Kil. Yes, our taco salad. It's made with quinoa, walnut, black bean, air quotes, meat.  I think I'm going to start dribbling, seriously. I think someone's going to nap. That looks so good. No one's going to miss anything eating something like that. You've got the crunch, you've got the sour, the salty, the savoury, the creamy, but there's so much going on in that bowl that it's just like one huge satisfaction punch. Yeah, love it. What did you call that bowl? The taco salad. The taco salad. OK, awesome. And another one, the pancakes. The famous pancakes that I keep telling everyone about. Yes, these are the banana pancakes. They have some bananas cooked inside it, a little bit on top for decoration as well, or for garnish. It comes with our house-made coconut with cream, chocolate sauce, and peanuts, and a side of syrup. Our pancakes are so fluffy. I love them so much. I suddenly feel like my bowl of fruit after the show is just not going to be there. It's not going to get me there. It's not going to cut it. I tried the berry pancakes with that gorgeous, luscious berry sauce and that coconut cream. I couldn't believe that they were vegan. I actually don't eat pancakes much, but the reason we ordered pancakes was that the table next to us ordered them, and my husband and I were going, oh, they look all right. We were so full by that stage, we were like, OK, well, let's just order some and try them. Kelly, they're the most fluffy, moist, oh, God, delicious pancakes I've tried. Oh, I'm getting hungry. In a restaurant. They are amazing. I'll go there just to eat that. Maybe try one of those mega-marys for research purposes. Yeah, exactly. That's all about the research. Let's do another one. What else have we got? Oh, nice. Yeah, on the left here we have our two-knot melt sandwich with potatoes, and then on the right is the breakfast slider, the one that comes on the mega-mary we were talking about earlier. OK. The two-knot melt is one of our more popular sandwiches. It's made also with jackfruit, and we make a dill mayo in-house. And the combination of the two, the way that we prepare the jackfruit, it tastes like tuna wood. Really? Yeah, a lot of people come in and asking if it tastes just like tuna, and then they try it, and they're like, this is so crazy. It tastes like childhood. It's a melt, so we melt our vegan cheese right on there. Wow. We also made pickles in there, and I was a fan of tuna melts when I was a kid, so to, again, have that in this restaurant is so awesome.  I mostly eat at the restaurant. OK. But yeah, I do. I would work there. Yeah. Just so I could work in there, so I could have the food. Yes. Yeah. So sometimes I do try to recreate the recipes at home. The one that I always have trouble making on my own at my house is the cocoa whip. For some reason, I don't know. Maybe the walls of vegan hills are magic. No, I must admit that that's a gorgeous cocoa whip you've got going there. Thank you. Again, almost scary that it's not, it's the real stuff. It tastes very, very like a normal whipped cream, but it's got that nice, light coconut edge to it. So when you're eating it and devouring it, you don't feel guilty at all. That's another good thing about plant-based food.  It's doing good things for you. You haven't harmed anything to get it on your plate. So it's kind of a very nice experience. Thanks. Kelly, I'm going to show you some stuff that I've been making in the past few months, a little glimpse at Lillian's vegan kitchen. First, no, we do have to show this other slide. I made that one. He's been trying to rip off your dish. You sure you don't want to work at vegan meals? Is this a scramble? Oh, no. It's mac and cheese, actually. Oh, OK. It's really good, another one of our really popular dishes. We've just started trying some variations of loading the mac and cheese. So now we have like a chili mac.  Yeah, so it's a nice twist to the mac and cheese. Yeah, we really encourage our guests to come in and add whatever they want to the mac and cheese. So what we'll suggest a lot is people add like jalapenos or a tempeh bacon, maybe some kale. We have a whole list of all the additions that we have and people can really build their own mac and cheese, just like our grilled cheese sandwich called Cheese Your Own Adventure. That is so I love the play on words. Thank you. So clever. Yeah, we're all about puns. I love it. No, I did notice on your menu that you do have all these additional add-ons and stuff that you can, so you can really walk in there and turn, yeah, like you said, a mac and cheese into your own funky version of it. Yeah, make it your own. That's what we're about. We love it when people express themselves through food because we're expressing ourselves through food. Definitely, what a great idea. I love that concept. If anyone out there hasn't been to Vegan Hills yet, I totally recommend it. It's in Kaimuki, by the way. Do you have parking available there? There is a paid parking lot behind us. We don't validate for it, but it's only like $0.75 an hour. It's definitely on the cheaper end of all the parking around us. And then there's a little side street across the road from us, and there are other metered parkings around. So no problem with the parking. And do you take reservations? Yes. What's the number? Our phone number to make reservations is<PHONE_NUMBER>. If we don't answer, we can always call back or just leave a message saying how many people are coming, your name, your phone number to reach you at, or you can send us an email veganhills.hi at gmail.com. I'm sure that'll come up at some point. Or you can head to our website, and they have a lot of directions there. We highly recommend reservations. They're not required, but we are a very small restaurant. We only have 16 tables. And if we're wanting to feed as many people these delicious vegan food as possible, we try to make it as organized and easy to do as possible. But we do recommend reservations. So you're at least guaranteed to get a spot right when you want to, because it goes from zero to full in like two seconds. I noticed it was packed when I was there. And yeah, definitely do get a reservation in the hours, Kelly, that you're open. We're open every day except Tuesday from 10 AM to 3 PM. OK. It's a nice five-hour window there. Yeah. We do our last seating and last call around 2.45 ish, but we can still do takeout between 2.45 and 3. So if you're like rushing to get there, just call us ahead and place a pickup order. And we can do takeout, yeah. Yes, I noticed that you're doing takeout as well. So that's like get to vegan hills. Get your vegan fix at vegan hills. Let us feed you. You do have one more photo for us, Kelly, that I'd like to show. Nice. Yeah, there's the tuna. Sorry, the tuna, you get the milty gang. It's just so good you have to show it twice. Yes, it is. And then these ones on the right, those are our wine nachos. We have the ohana size and the mini. The mini is just a third the size smaller, but it's more personal sized. Those are the ones that my parents still ask for me to find a way to send to them. Yeah, it's got like two different kinds of cashew creams. There's one cashew cream and then our jalapeno queso, which has that cashew cream as the base. And then we add some vegan cheese, jalapenos, and all sorts of other delicious spices to it to make it taste like queso. And then it has our house made quinoa black bean chili on there. Oh, yummy. Kelly. Sorry, I know I wasn't hungry before either.  I do what we call intermittent fasting.  Oh, yeah. From about between 4 PM to 8 PM. Anyway, I feel like I'm really, really not going to stick to that today. Well, we are open today. I know. Geez. Well, let's have a look at my video, Kelly. So this is what I've been cooking up in my kitchen the past few months. And we can just chat while I show that. Yeah, oh, these are good. So I do have sort of a strong Japanese influence with my cooking having lived there for 30 years. And that's where I worked as a chef and cooking instructor and stuff. So there's all lots of stuff going on here. Oh, those are some fresh tempura. I remember was it last month that you had a very famous chef from Japan come in and do an event at Beacon Hills? Yes.  Shōjin ryori. Yeah, there we go. It's a type of cooking that's very basic and more honed in on the plants and simplistic eating and minimalist eating. And he did a demonstration on a couple dishes. And we provided a sake pairing with it with the dishes that he had. Yeah, right. How did you like the food? It was good. That food originally comes from the monks, what the monks used to eat. So it's very, as you said, very minimalistic, very simple flavors they don't really do much to the food. But it was crazy how even those, they were very simple flavors. We actually took one of the dishes that he made there. It was three or four ingredients that we already had in the restaurant. And so we asked if we could use that recipe for one of our dishes. And it was the carrot sesame oil one with just cumin seed and salt. And it's just those four ingredients. You just massage the carrots a little bit so that it incorporates the flavors. And it is just so powerful in flavor. Yeah, we were like, this is amazing. And it was really fun to watch all that. He did like a stuffed mushroom. What did he stuff the mushroom with? The cap of the mushroom. And then I think a little bit of tempeh. I forget the other thing. I was passing out sake as he was trying to look and talk at the same time and not get sake on the table. It's completely different to what you're doing at Vegan Hills, that type of cooking. So it's nice that you're having these little events that can introduce people to other types of vegan food. That shorting deity in Japan is becoming popular again because finally in Japan, the vegan movement, if you want to call it, that is starting. It's just starting to happen. So it's so exciting. So they're starting to bring back all of this old traditional way of eating, which is nice. So that's what I've been doing. Lots of stuff going on. I do have my events posted for the end of this month and my October events, which you can find on all of my forums under the Lillian Vegan name. So my Facebook page is under Lillian Vegan. I have about 200 recipe videos on YouTube, all of my original vegan recipes. So lots of stuff going on there. And of course, you can watch this awesome show every second of Thursday, 1 PM, live from downtown Honolulu. We're very lucky to be living here and in a place that allows this vegan food and restaurants like this to flourish. I'm so excited for you. Yeah, thank you. So what have you got at Vegan Hills that's new on the menu? Yeah, we are trying to put out weekly specials.  Say that you have to use everything that we have in the pantry, make me something new. I love it. It's done a lot of fun. But this week, we're actually revisiting an old dish that was really popular. It's called de Calbi. We take soy curls, which is very similar to tofu. But in the last stage, instead of putting it in the block, they freeze dry the soy and then curl it up. And then you just rehydrate it and marinate it. And we marinate it in a way that tastes like Calbi beef. And we put it with this spicy red miso paste and put Korean chili threads on top. And it's a very fantastic Korean inspired dish. That's fantastic. You've really covered all of the food groups, so many cuisines going on there. So Kelly, it's been awesome having you on the show. Thank you so much for taking the time out. You'll be seeing a lot of me. A lot of me. And I'm sure a lot of everyone else. So definitely, if you haven't checked out Vegan Hills in Kaimuki, please do so. The food is fantastic. Kelly will take care of you and make sure that you get lots of plant-based yummies in there. So thank you so much, Kelly. Yeah, thanks for having me. And huge mahalo to Think Tech Hawaii. Look forward to seeing you in a couple of weeks. Have an awesome day. And mahalo.


The world is just about to witness a tragic disaster. Millions across the globe have tuned in to watch the first participant of the Teacher in Space programme ascend into space. Excitement is building as the Challenger Space Shuttle is just moments away from launch. The time is 11.38am and it's the 28th of January 1986. The shuttle is just about to lift off the launch pad at Kennedy Space Centre and although initially off to an apparently good start, just 36 seconds later the craft is hit by a strong wind shear. Not long after a small plume of smoke is seen, at 60 seconds post launch Capcom informs Challenger to go up throttle up. Just 5 seconds later the Challenger began disintegrating and quickly after a fireball consumed for shuttle stack. The world had just witnessed the loss of one of NASA's cutting edge in human space travel. Today we're looking at the Challenger Disaster. My name is John and welcome to Plainly Difficult. So this event happened three years before I was born and my first true exposure to it was via one of my favourite films, Koyane Scotsy. Ever since the first time I saw that film back in secondary school in 2001, I'd been fascinated by this disaster. Something as apparently infallible as NASA easily became the victim of its own confidence. Our story today would even lead to a new term, Normalised Deviance, which probably can be applied to a lot of subjects on this YouTube channel. Right, let's get cracking with a little bit of background. NASA during the design phase of the Apollo missions sought out a more cost-effective reusable heavy spacecraft for orbital missions. The concept was decided upon in the early 1970s and this design was set in March 1972. The design had the orbiter which housed a crew and payload with two separate reusable solid booster rockets and an external fuel tank. During the mid-1970s the first orbiter named Enterprise was constructed. She couldn't actually go into space but acted more as a test bed for the systems needed for the project. Eventually, over the coming decades, six orbiters will be built but for today's video we will focus on just one shuttle, Challenger. She was manufactured by Rockwell International, originally being attended as a structural test article to be completed in 1978, after which she was sent to Lockheed for 11 months of vibration tests designed to simulate entire shuttle flights from launch to landing. We'll pause with the Challenger story just briefly to talk about how the orbiter actually gets into space. So to get into orbit, the orbiter's engines are first to fire up and a few seconds later the solid booster rockets also start. This is enough to get the spacecraft off the launch pad. After liftoff, the stack goes inverted. The solid rocket boosters and the shuttle with its external tank separate. This is roughly two minutes after the flight has begun. The solid booster rockets deploy their parachutes, land in the sea and they are recovered, sent back to land and refurbished for reuse later on. So the orbiter and the external tank continue to ascend, with the orbiter's main engine still firing. The external tank is separated from the orbiter. The external tank, after being jettisoned, disintegrates in the Earth's atmosphere. Ok so that was a very basic description of how the orbiter gets to orbit. For more information I'd recommend watching maybe a curious droid video. Apparently people think we're related, we're not. So after its mission the orbiter was designed to re-enter Earth's atmosphere and upon re-entry glide to Earth and land much like a conventional aircraft. Well wasn't that little side tangent exciting? Well let's get back to the history of the Challenger. It was decided that Challenger should have a new lease of life after her experiments, and was to be converted into a full blown orbiter. She had her refit for spaceflight beginning in 1979. Many of her parts were sent back to her manufacturers for reworking. She also even had a new fully functional crew module added. The refit took around three years with Challenger rolling off the production line in July 1982. Her first flight was as STS-6 lifting off on the 4th of April 1983. She was taken to space over her operating life, the first African American, first woman, first Dutch person and be the first night launch and landing. She was used more often than Columbia between 1983 and 1986 making her the go to orbiter amassing 10 flights and over 62 days in orbit. But we'll have to talk about an issue with the shuttle program and it was around those solid booster rockets, the SRB. Those two solid booster rockets were vital to launch as they provided the most amount of thrust at the beginning. They were built by Morton Fire Corps. Each one consisted of multiple sections. Each section was constructed in Utah separately to be put together at the Kennedy Space Center ready for use for blasting an orbiter into the sky. Of course this requires a joint between the sections and this came in the form of a Tang and Cleavish joint. They were free in total for each SRB and they have several vital, especially important for this video components. So the Tang and Cleavish joint works like, hang on with me this is relevant to the disaster, like this. The Tang being the upper segment connecting the two sections by fitting into the lower Cleavish section. There is a retaining pin, clip and band placed to hold the two sections together. The joint is also externally wrapped with fiberglass reinforcement tape. To create the seal between the sections two rubber o-rings seals are employed. They run the entire circumference of the SRB and they are bonded to the case using an adhesive. The seal design was originally considered to be redundant as if the primary first o-ring failed then the secondary one could also be relied upon. As such the seal was rated as critical but a redundant setup. The gaps between the components were then sealed by zinc putty. The o-ring seals are vital in containing the hot gases produced from the SRB's motors during burning. During operation they would be compressed sealing the surface, thus stopping any unwanted flames out at the side of the boosters, which you know could be a little bit catastrophic but the system design was flawed. Issues were discovered in the early 1970s. This was that the o-ring wouldn't always compress and would instead extrude. NASA thought this was acceptable but the issues didn't end there. In 1977 a test discovered that the Tang and Cleavish joint could bend away from each other. This weakened the compression on the o-ring and thus reduced its seal effectiveness. The outer spec situation then allowed combustion gases to erode and damage the seal causing more gases to escape. The issue was further exacerbated when in lower temperatures as like with everything different materials expand and contract at different rates. This greatly concerned NASA engineers and they requested the redesign of the joint to include a shim to close the gap in 1978. One year later and not having received a response to their 1978 memo a second one was sent continuing to object to fire called solid rocket motor joint seal design. A test in 1980 showed yet again in a Tang-Cleavish relative movement was greater than originally predicted. In May 1980 the verification certification committee recommended that NASA conduct full scale tests to verify the field joint integrity including firing motors at a mean bulk propellant temperature range of between 40 and 90 degrees Fahrenheit. NASA replied NASA specialists have reviewed the field joint design updated with larger o-rings and thicker shims and found the safety factors to be adequate for the current design. Reanalysis of the joint with larger o-rings and thicker shims is being accomplished as part of the lightweight case program. The joint has been sufficiently verified with the testing accomplished to date. The first shuttle mission took place on the 12th of April 1981 all went well but just on the second mission STS-2 o-ring erosion was noticed but was found to be an infrequent problem over the coming years. O-ring was found in March and April 1983 and in February and August of 1984, the latter of which had discovered soot had blown past the primary o-ring. Even though after 1982 the o-ring's designation of redundant was removed and with evidence of blow by shuttle launches still continued, NASA would push the design beyond its temperature limits when in January 1985 the launch of STS-51C being the coldest space shuttle launch to date. The air temperature was 62 degrees Fahrenheit or 17 degrees Celsius and it was estimated that the o-ring temperature was 53 degrees Fahrenheit or 12 degrees Celsius. Surprise surprise both SRBs experienced o-ring erosion and blow by. Morton fire coal determined that the low temperature reduced the elasticity of the o-rings which in turn resulted in the gap not being fully filled which would then in turn lead to blow by of soot and gases. Nearly all of the 1985 launches resulted in o-ring deterioration. This prompted Morton fire coal to redesign the case however it was going to take some time and as such the original design was to be used till the replacement was available. This would cover the missions that would eventually run into the start of 1986. STS-51L So I'm not going to go into too much detail about the background of this mission however it was quite important in terms of boosting the public's opinion of the shuttle program. The orbiter Challenger was mated to her solid rocket boosters and external tank at NASA's Kennedy Space Center in Florida on the 17th of December 1985. The mission had been pushed back to January 22nd 1986 after a number of cancellations beginning in July 1985 then due to payload issues in November 1985. As such NASA really wanted to get Challenger into space she was built and ready to go and her crew were trained and eager to get on with it. One of the crew of STS-51B was the first participant of the teacher in space program, Krista McCulfey. Again the mission found itself delayed in January 1986 and was pushed back to the 28th. However there was an issue the air temperature was forecast to drop to 18 degrees Fahrenheit minus 8 degrees Celsius during the night before the launch date. This would mark a significant drop in lowest temperature the shuttle would ever be launched in. As such this actually caused some concern with Morton Firecalls engineering team. Blow by of the vital o-rings had occurred in warmer temperatures after all. A conference call was set up between NASA and Morton Firecalls engineers. In the call Morton Firecalls employees Robert Lund the vice president of engineering and Joe Kilmenster the vice president of the space booster programs advocated against launching until the temperature was above 50 degrees Fahrenheit or 12 degrees Celsius aka the lowest the shuttle had previously been launched in. But something about the mood had changed in communications between the Marshall Space Center and Morton Firecalls. Usually it would be a case that Morton Firecalls would have to prove that the SRBs were safe to fly. Understandable in an environment of if in any doubt don't risk it. However the Marshall Space Center had moved its position of to prove that the shuttle can't fly almost as if any doubt was no longer acceptable. Robert Lund would later say we had to prove to them that we weren't ready. So we got ourselves in the thought process that we were trying to find some way to prove to them it wouldn't work and we were unable to do that. We couldn't prove absolutely that the motor wouldn't work. He would also continue to say later on well but it's the kind of mode we got ourselves into that evening. It seems like we have always been in the mood in the opposite mode. I should have detected that but I did not but the roles had kind of switched. As such after a 30 minute break in the conference call to confer amongst Morton Firecalls staff management came to the conclusion that flying in low temperatures should in theory cause no more issues than when the shuttle flew at a temperature of 53 degrees Fahrenheit. When the conference call resumed Firecall came back saying that there was substantial margin in the event of failure and gave their final approval for flight on the 28th of January. Measurements overnight recorded lows on the left SRB of 25 degrees Fahrenheit or minus 4 degrees Celsius and the right SRB at 8 degrees Fahrenheit or minus 13 degrees Celsius. Ice had formed all over the fixed service structure. This caused concern with the orbiters manufacturer Rockwell International. They were concerned that ice could damage the orbiters thermal installation. However engineers at Kennedy and Johnson Space Centers advised that there was no issues with the ice. However an hour delay was ordered to allow the temperature to warm up a little bit. The launch was set for just after half 11 in the morning of the 28th of January. At 20 minutes before liftoff ice was seen to be melting and clearance for flights was given for 1138 a.m. The air temperature now was around 36 degrees Fahrenheit or 2 degrees Celsius. Time for liftoff. Six seconds before 1138 a.m. Challenges three main engines ignited. At T minus zero seconds the two solid rocket boosters ignited and challenger started to move. Almost immediately a gray puff of smoke was observed around the right booster's joint near the attachment strut. This was hinted at gas blow by due to the improper sealing of the O-rings. Just two seconds later however the smoke stopped. This was due to some of the solid fuel falling within the booster and temporarily sealing the gap. At 36 seconds post liftoff challenger experienced a severe wind shear. To counteract this the shuttle attempted to correct its trajectory. The change in yaw and pitch dislodged the solid fuel that had been acting as a plug and this was at an altitude of 3,000 meters or 9,800 feet. At around T 58 seconds another plume of smoke appeared around the same spot as before on the right SRB strut. Two seconds later the plume begins to touch the external tank. At 64 seconds a hole could be seen being burned into the tank. This caused a leak of liquid hydrogen fuel. Challenger's computer tried to counter the now lost in thrust from the right SRB. Capcom and the crew aboard the orbiter were still unaware of the unfolding disaster and at 68 seconds the instruction of throttle up was given. At around 72 seconds the crew felt a lateral load. Pilot Michael J Smith was recorded as saying uh oh. The pressure was falling in the external tank and thus was affecting the main engine's performance. At 73 seconds the aft dome of the external tank failed and at around the same point the right SRB rotated away from the strut. Moments later the whole vehicle began to break up. The altitude of Challenger was around now at 47,900 feet. The external tank was now fully disintegrating. Challenger was pushed against the airflow and was hit by aerodynamic forces resulting in a load factor of up to 20g. The orbiter separated from the external tank and continued to fly for another 30 or so seconds. The shuttle stack had been completely separated. In sections of the now disintegrated orbiter flying about in different directions hurdling back towards Earth. The crew compartment remained relatively intact. The force at separation was estimated to be between 12 and 20g but quickly dropped down to 4g and then free fall. It could then be seen falling through the sky. The crew cabin fell for approximately two minutes and 45 seconds after its separation eventually slamming into the Atlantic Ocean. At 89 seconds post launch the video Challenger was seen in mission control and now the full weight of the disaster began to sink in. NASA Public Affairs Officer Steve Nesbit would announce the explosion when just a few moments before he had been reading out stats from the flight data. He would say, We have a report of the flight dynamics officer that the vehicle has exploded. The flight director confirms that. We're looking at checking with recovery forces to see what can be done at this point. Recovery operations were dispatched immediately with the ship's MV Freedom Star and MV Liberty Star heading towards the crash site. However, due to falling debris recovery couldn't begin until 1237. Understandably hopes were held for any survivors but with an estimated impact force of the crew compartment with the sea being 200g it was very unlikely. By 7pm some recovered debris from the orbiter and external tank was found. With surface operations concluding on the 7th of February, subsurface operations ending the next day. However submarines and submersibles would scour the seabed around the impact area for SRB and crew compartment debris the months to come. Pieces of Challenger would continue to wash up ashore even as late as 2022 when a pretty large piece of around 20 feet long was discovered off the coast of Florida. The crew compartment would be found on the 7th of March 1986. The remains of the crew were removed on the 4th of April. However, one of the bodies floated away. This was Gregory Jarvis. His remains were recovered just over a week later. But what the public really needed to know now was how could such a tragedy happen to NASA of all organizations? Finding the cause. Now I'll say straight away as this video outline has hinted at that we know what the cause was. It was the poor decision to launch in cold weather but have reduced the effectiveness of the o-ring seal. But in January 1986 that wasn't exactly obvious. After all blow by had occurred several times before and the orbiter had completed many successful missions. The shuttle had over 2.5 million parts and it was the world's most complex flying machine. The investigation would be no small task. The presidential commission on the space shuttle Challenger accident was set up in February 1986 and it was tasked with sifting through thousands of documents photographs and statements. Although the failing o-rings was a big contender they did have to look at every potential avenue for disaster. The board included Chairman William P Rogers Vice-Chairman Neil Armstrong and then David Aikerson Eugene Covert Richard Feynman Robert Holtz Donald Kutcher Sally Ride Robert Rummel Joseph Sutter Arthur Walker Abbott Whelan and Chuck Yeager. The investigation would gain the name the Rogers commission after its chairman. The committee was split into four panels the accident analysis panel development and production panel prelaunch activities panel and the mission planning and operations panel. Each panel will contribute to over 160 total interviews over 35 sessions and would involve over 6000 employees from NASA and its main contractors over a period of four months. They were zero in violent extensive footage of the launch on the cause of the plume of hot gases that appeared on the right SRB next to the strut. The evidence was pretty damning of an instance of blow by. On the 11th of February during a televised hearing board member Richard Feynman produced a sample of SRB O-ring and placed it in a cold glass of water. Oh I took the stuff that I got out of your seal and I put it in ice water and I discovered that when you put some pressure on it for a while and then undo it it maintains it doesn't stretch back it stays the same dimension in other words for a few seconds at least and more seconds than that there's no resilience in this particular material when it's at a temperature of 32 degrees. This memorable moment helps solidify in the public's mind of the poor oversight that NASA had demonstrated. Feynman had been at odds with other members of the commission including his chair Rogers due to his harsher approach to investigating the evidence. He was much more critical of NASA management and their apparent lack of understanding of the risks that they had taken with not only the O-rings but other parts of the shuttle program. NASA's standards had seemingly dropped in order to keep up the busy schedule of orbit emissions which fostered a management culture of quantity over safety where blatant red flags were ignored. We can see this in Robert Lund's statements after the disaster. I mentioned earlier that he felt the culture had changed from prove safe to prove it won't be safe. Feynman's criticisms and investigation would appear in the report's appendix F. His conclusions were pretty harsh. Feynman would say if a reasonable launch schedule is to be maintained engineering often can't be done fast enough to keep up with the expectations of originally conservative certification criteria designed to guarantee a very safe vehicle. In these situations in these situations subtly and often with apparently logical arguments the criteria are altered so that flights may still be certified in time. They therefore fly in a relatively unsafe condition with a chance of failure of the order of a percent. It was a clear case as of what sociologist Diane Vaughan would coin of normalised deviance which is the normalisation of clearly unsafe practices. Exactly what happened with the first case of O-rings blow by. Just because it didn't cause a disaster doesn't mean it was safe but NASA a molten fire call carried on regardless. The disaster would result in the redesign of the SRB where the joint's tang would have a reduced risk of rotation. NASA also reevaluated other parts of the shuttle project which wouldn't have saved the Challenger's crew but would overall improve safety of future crews in a multitude of different events. And now for the slightly grim part of when did the crew perish. Although the exact moment isn't known it's likely some survived post separation as some switches have been operated that couldn't have been done so by the impact. The crew were failed that day by a culture of safety ignorance. Of course space travel can't be 100% safe but Challenger really shouldn't have flown that day. Feynman in his conclusion to appendix F would rather eloquently say I think it best describes the situation. For a successful technology reality must take precedence over public relations for nature cannot be fooled. Pretty poetic if you ask me I definitely recommend reading the full Feynman appendix F section as he was obviously a genius. So that's it my video on the Challenger disaster. Although it is a little bit longer than my usual videos I still feel as if I've barely even scratched the surface. I highly recommend reading the Challenger launch decision book by Dianne Vaughn. I'll put an Amazon affiliate link in the description if you decide to buy it it helps out the channel with a little kickback. So if you want me to cover the Columbia disaster let me know in the comments but I do also fear that it will be quite a larger video. Please feel free to comment like and subscribe hit the bell and all that jazz I'm really trying to push for a million subscribers because why the hell not. And all I have to say is thank you for watching and Mr Music place out please.


Hi, I'm Zor. Welcome to a new Zor education. Let's talk a little bit more about graphs. We used to draw graphs of function of one variable which looks like this. On the coordinate plane where this is x-axis, this is y-axis, and for every x from the domain where this function is defined, we have the point which has coordinates x and y equals to f of x. Now, with every x on this particular line, well, whatever belongs to the domain of the function, we can draw this particular point. And then all these points we can basically connect together, and that's what basically forms the graph. All right. Well, this is not the only form which is used when certain graphs are presented. I would like to have a little bit more general case where this is actually a particular case of that more general case. And here is the more general case. What if instead of expressing our function as some kind of algebraic expression with one variable x, and the result of the algebraic expression is y, which is coordinate on the y-axis, what if I will have an equation like this, where f is the function of two variables, whatever the algebraic expression is. Well, let's just remember the definition of the graph. The graph is, in this particular case, it's a set of all points with coordinates x, y where if you substitute the x into this formula, you will get the coordinate y. Well, for this particular equation, the definition is exactly the same. So a set of all points on the coordinate plane with coordinates x, y, if you substitute x and y into this formula, you will get 0. This is basically the same thing, and any function expressed in this particular way can be expressed as this, where this is a function of two variables, and this is a particular case of this more general case. So that's all there is. It's just a small generality which I would like to introduce into this picture. Now, if you can resolve this particular equation for, let's say, y, and get y as an expression with one variable x, then you will basically reduce this case into this case. Like, for instance, if I say x plus y equals to 0, this is my function of two arguments. Yes, you can always resolve it using this, and this is a familiar function which we dealt with before. But sometimes it's not possible, and you would still like to present the dependency between x and y expressed as this particular equation in the graphical form. And that's what I will be talking about today. Okay, so we understand what this particular kind of graphs are all about. These graphs represent a dependency between coordinates expressed as this particular equation, where f is a function of two arguments. We are talking about real values, obviously, x and y are real names. So let's just have a couple of examples. For instance, the first one is this. Well, this is a typical example of an equation of two variables, where it's really very easy to resolve it for one of them. So we will probably resolve it for y is equal to minus 2 third x minus 4 third, right? And the graph of this function is very easy to draw. We know that y equals to minus 2 third x is a straight line, which goes through the beginning of the coordinates. And it goes from the second to the fourth quadrant with certain coefficient. This minus means it's this direction rather than this direction. And minus 4 third, which means we go down by 4 third. So altogether, it will look like this. This is y equals to minus 2 third x minus 4 third. So this is easy. And it's absolutely straightforward. There are no underwater currents or any kind of peculiarities of this. This is simple. Can we do it differently, this particular graph? Well, we probably can. Here is one of the ways you can just consider drawing this particular graph. Well, first of all, you understand that this graph is supposed to be a straight line. Just because this is x to the first degree, this is y to the first degree. So everything is just linear. Linear dependency between x and y is expressed as a straight line. So for the straight line, all we need is just two points. And then we will be able to find out exactly how the line goes. Well, let's do it this way. Coordinance x, y, 0. Let's find where is the intersection of our line, which we don't know yet, with, let's say, an x coordinate. Well, if it's intersection with x coordinate, that means y should be equal to 0. Now, if y is equal to 0, then x must be 2x plus 4 equals to 0. So x is equal to minus 4. That might minus 2, sorry. So this is minus 2. So our straight line always goes through the point where x is equal to minus 2 and y is equal to 0. Okay, now let's find intersection of this line with the y axis. Now, in this particular case, if our point is on the y axis, it means x is equal to 0. So if x is equal to 0, what follows is 3y plus 4 is equal to 0. So y is equal to minus 4 third, which is here. So x is equal to 0, let me just put it differently. And y is equal to minus 4 third. So this is another point. And obviously, our line goes through these two points. So no matter how we draw this using this technique or that technique or anything else, we will come up with something like this. So this is an easy case. Now, let's consider a little bit less trivial, because this is trivial because it's resolvable for y. So it's not really very interesting to represent it in this particular form, because it can be easily transformed into another more familiar to us. How about this? Well, this is kind of strange. What is the graph of this particular equation? I'm not saying function, because you will see that this is not really a graph of any function in a classical meaning of that word. But what is the graphical representation of this dependency between x and y? Well, I'm not going to do something very naive. Let's divide both parts by x. And I will get y equals to 0 divided by x is 0. So I will have this. And what is y is equal to 0 on the coordinate plane? It's basically the line which coincides with our x-axis, because only all the points in this x-axis have the y-coordinate equal to 0. Well, that's obviously not exactly the right way of doing this. Because you can not just divide by x and not considering what if x is equal to 0. So what is that if x is equal to 0? Well, if x is equal to 0, y can be basically anything, which means any point where x is equal to 0 and y is anything, which means any point on this particular y-axis also belongs to our graph. So our graph is the set of all points which lie on both coordinate axes. Well, obviously, this is not a graph of any function, because our function in our classical definition function is some kind of correspondence between a single value of one variable, which we call the argument. And the result will be one concrete value of the function. Now, in this case, if argument is somewhere here, then yes, the function is always equal to 0. But if the argument is here, there is no concrete value of the function. Anything can be a value of the function. So this is not a function, so to speak. This equation does not represent a function. It represents the pendency between x and y, which is graphically represented as the set of all points on both axes. But it's not a true function in the classical sense. All right, but that's fine. So equation is not necessarily a function. And in this particular equation, that does not necessarily define a function. But it's nevertheless a certain dependency which you can find from some practical experiments or anything like that. Some equation between x and y can be derived in whatever way you want. And then it can be expressed graphically again in some way, which does not have to be a graph of a function, so to speak. OK, next example. x squared plus y squared equals r squared. It probably can be attempted to resolve it for y. Well, yeah, you will get something like y squared equals to r squared minus x squared, right? But then when you think about the next step, you can just have plain this. This is not right because this is arithmetic square root, which means it's always positive. But y can be negative. I mean, if y is equal to, let's say, if point with coordinates x and y belongs to this graph, then the point with coordinates x and minus y also belongs to the same graph. So it's basically like minus y also would be equals to this. So it's not, again, exactly a function now. Since we have more than one value of the function, which can be the value, it's not really a classical function. So let's forget about reducing this particular dependency to a concept of a function. Let's just try to deal with this as it is, as it is given. And here is what's given. Let's consider this particular point with coordinates x, y. And let's think about what is x squared plus y squared. Well, if you project it down, you will have this as x, this as y. And if you connect it to the beginning of the coordinates, you can consider this right triangle. And this would be the square of the hypotenuse, right? So no matter where x and y are, maybe it's here. It's exactly the same thing. Now in this case, x is negative, y is positive. But we're dealing with x squared and y squared. So it's always the length squared, length of this segment squared and length of this segment squared. That's why we always have the length of this segment squared. So where is this graph? I mean, how are all these points, which belong to the graph of this equation, positioned where are they? Well, obviously they are on the same distance from the center. So any point on the coordinate plane, which is distance, which is distant from the beginning of the coordinate by the length r, belongs to our graph. So it's a circle. That's what it is. So the graph of this particular equation, which represents certain dependency on x and y, is a set of points on the coordinate plane, which are on the distance r from the beginning, which is a definition of a circle, basically, right? So what is a circle with a point o as a center? All the points which are on this particular distance from the center. So this is an equation of a circle. Is it a function? No, it's not a function. Again, same thing. Take any value of the argument x, and you will have two different values of y. If you remember, y is equal to square root of r squared minus x squared and minus, same thing. So two results mean that this is another function. However, it's a legitimate graph, which represents this particular dependence. Now let's do a little bit more technical stuff. The more technical stuff is this. By the way, I hope you are not really discouraged by the fact that I put x squared plus y squared equals r squared. It's not exactly this form, but if you really insist, I can always do this. r is a constant. r squared is a constant, so it's the same thing. So no matter how I write it in this way, or equal r squared, it's exactly the same thing. Same thing stands with this one. I can put it minus r squared is equal to 0, or equal to r squared, same thing. But now, question is, now, what's the graph of this particular function? Well, here, let's just think about. If you remember, when you were dealing with functions, we were talking about that if you have a function like this and a graph of this function, then if you would like to have a graph of this function, you have to shift this one to the right by a. Well, if a is positive, it means to the right. If a is negative, it means to the left. But anyway, it's to the right by a. So in which case, the whole graph would look like this. It comes up like this. Well, it's exactly the same story with this particular case when you have an equation where x and y are mixed together. If you add something to x coordinate, it means the original graph of this function, let's forget this. I put equals to r squared. It's better. So the original graph, and compare it with this one, let's say this is y squared, not y minus v squared, it will be a shift to the right by a. In case a is positive, if a is negative, it would be to the left. But anyway, it's always to the right by a. Why? Well, for obvious reason. Because if the point, let's say, p and q belongs to this particular graph, then point p plus a q obviously belongs to the graph where this is y squared. Why? Because first we subtract a. So p plus a becomes p.  And we know it's already r squared. So if point p q belongs to this graph, then point p plus a q belongs to this graph with this y squared. Now, if this is y minus v squared, I will just make it this. So again, if p q belongs to this, then p plus a q plus b belongs to the graph of this dependence, which means that the whole circle, which is all these points p and q, which satisfy this particular equation, is shifted by a to the right and b to the left. So the center basically will be shifted, but the radius will remain the same. So this is a new center, and the radius will be the same. So from this, we just shifted the whole picture here. So this is also a circle. And this circle represents all points on the distance r from a center at a d. So now we know how to draw a circle algebraically on the coordinate plane. We could not do it, by the way, using function representation because this is not a function. Circle is not a graph of any function. Circle is a graph of a dependency between x and y, which is expressed by this particular equation. OK, what's next? Next is this. What is this? Again, let's go back to the graphs of the functions. And if you remember, if you have a graph of function y is equal to f of x, and then I was asking, so what would be the graph of this function? Let's say p. And we were talking about the whole graph actually being scaled by p along the x-axis. So if I had something like, let's say, this graph of the original function, and let's say p is equal to 2, then it would be this. It's squeezed. y, again, to the same reason. If point p, q belongs to this graph, let's not use the letter p. Let's use the letter k, let's say. Then point p, k, q belongs to this graph. And now I see I made a slight mistake. Because if k is integer, it's not squeezing. It's actually stretching. I should have drawn, instead of this, I should have drawn stretching by k, if k is equal to 2. So when x is divided by k, you're stretching by k times. But obviously, if k is a fraction, let's say 1 half, it would be squeezing. So it all depends on what the value of k is. But for the k equals to 2, it's stretching. For k is equal to 1 half, it would be squeezing. So anyway, obviously, this is a proof of this particular quality of the graphs, f at x and f at x over k. Absolutely the same situation would be here. So what I'm saying is that if point belongs to this particular graph, then point p, x, q, y. Sorry, it's lowercase. p, k, x, q. This point belongs to the graph x square plus y square is equal to 1. So if you have this particular graph, and this is a unit circle, a circle with a radius 1, then this should be actually stretching by p along the x axis. So if this is my unit circle, and let's say p is equal to 2, then x over p square plus y square equals to 1. This would be a stretching by p. Let's say p is equal to 2, so it would be like this. Not very beautiful, but you understand the point. So we are stretching. Now, if this is y divided by q, it will also be a stretching by q upwards. Let's say q is equal to 1.5. So the points will be here. So the result will be here. Sorry, we don't stretch x any longer. So I'm stretching by p horizontally, and I'm stretching by q vertically. And what's the result? Well, it's called ellipse, basically, or ellipse sometimes. OK, so this is an equation of ellipse or ellipse with these factors being the factors where stretching the unit circle horizontally or correspondingly vertically. That's it. So you know how the ellipse can be expressed algebraically on the coordinate plane. What else do we have here? OK, the next one is parabola. OK, well, we know that this is the function graph of which is the regular parabola, like this. Now, this can be expressed in this language, right? So basically, it corresponds to the function of two arguments. And this is the dependency between these two arguments. And the dependency is expressed as a parabola. Now, what's interesting is that let me just consider a similar function. But instead of y, I will use x. And instead of x, I will use y. What's the graph of this? Now, before answering this question, I would like to address the following point. If you have a graph of the function this, and now I have the same function, but I will exchange the arguments. How these two graphs would correspond to each other? Well, let's assume first that this is the equation which can be resolved for y. So it would be y equals to some function of x. Now, that means that this can be resolved for x, right? So it would be exactly the same function, f. But now x is expressed as a function of y. Now, so let's reduce our problem of correspondence between the graphs to these cases. So what if you have an equation which looks like this, and you would exchange x and y places? What actually that would be? Just as an example, y is equal to x square. And this x is equal to y square. This is exactly the case which we need, right? Now, this might not be the same. Now, this is the function. This might not actually be the function, because the function is the y as a formula of x. But that's not always possible, right? But anyway, what kind of a graph we will have if we will try to draw this one? Well, if point x, y belongs to this graph, then point y, x belongs to this graph, right? It's the same thing, formula is the same. But how are these two dots, one represented by coordinates x, y, and another represented by coordinates y, x? How do they position relative to each other? Well, let me just give you an example. If you have point 1, 2, this is 1, 2. Where is 2, 1? 2, 1 is here. So obviously, these two points are symmetrical relatively to bisector of the main coordinate angle. Now, I hope it's obvious because we're changing whatever was x here becomes y. Whatever was y becomes x. So we can very easily, purely geometrically prove, look at this. Now, this is a distance, which is a square of this distance is sum of two squares, x-coordinate and y-coordinate. Now, this is y-coordinate and x-coordinate, but this is exactly the same thing. So these two are exactly the same. And so are angles. And so are these. So the congruence of these two triangles is obvious. And from the congruence of these two triangles, follows the congruence of these triangles. So that's why these two points are symmetrical relative to the angle bisector, which means that graphs of this and this are supposed to be exactly symmetrical, as well as graphs of these two, for the same reason. Because if point x, y belongs to this, then the point y-ax belongs to this. So these two graphs are symmetrical, which means these two graphs in our particular example are supposed to be symmetrical. Now, this is a parabola. Now, what's the symmetry of the parabola? What if we will just reflect it relative to this particular angle bisector? Well, it would be this, right? These parabola and this parabola, they are symmetrical to each other. Now, granted, this does not represent a function, because again, for a single value of x, we can have two different values of y. But that's not what we care about. We care about equation, not about the function. So equation, this one, would be satisfied for all points which are flying on this parabola. And again, let's just remember the main principle. If you consider these two equations with the same function f, just exchanging the arguments, then the graph of this and graph of that would be symmetrical relatively to the angle bisector, all right? Now, let me just exemplify it with some other example, which probably would be a little bit easier to understand, because it is about function. So if you have a function this, or y is equal to x cubed. Now, we all know this is a curve which looks like this, right? This is minus 1, 1. This is plus 1, 1. Now, if I will change places, x is equal to x minus y cubed equals to 0, or y is equal to, or x to the power of 1 third. Now, the graph would be, let me get another color, would be this. And as you see, the black and red are symmetrical relatively to this. Now, you remember that the power function, x to the power of something, power of a, it depends on the a. If a is greater than 1, it goes this way. If it's less than 1, like in this particular case, it goes this way. But it's always around the point 0, 0. All these graphs, the greater the a, the steeper goes this one, the smaller the a, the positive obviously, but it goes closer to 0. The closer this particular branch of this curve will be towards x-axis. But in any case, these two graphs, this one and this one, they are symmetrical relatively to the sector, to angle by sector. OK. And finally, just for fun, mass sometimes can be fun, but you have to really struggle to get to this fun. Here is something which I have found on the internet as an interesting curve. x to the force minus x square plus y square equals to 0. So let's try to draw the graph of this particular equation, this particular dependency between x and y. All right. It's not easy. I mean, it's not obvious what the graph looks like. So let's just think about this step by step. Now, the first form, we can represent it as y square equals to minus x to the force plus x square. That's the same thing. Now, this is a positive value, because it's y square. Square is positive. Well, actually, non-negative. Let's put it this way. Non-negative. So this is supposed to be also non-negative, which means what? Which means that x square should be greater or equal to the x to the force, right? For this to be positive or 0, we should have this greater than this. That's obvious, right? Or if you wish, you can always put it minus x to the force plus x square greater or equal to 0. Now, we add x4 to both sides, and we get this. All right, fine. Now, how to resolve this particular inequality? Well, there will be a separate chapter in this course about inequalities. But it's kind of obvious that if one number is greater than another, then we can divide or multiply it by some positive number, and the relationship would actually remain. You can multiply by 3 would be 6 greater or equal to 3. Or you can divide by 3 would be 2 thirds greater or equal to 1 third. So manipulation of division and multiplication by a positive number doesn't really change the inequality. So what do we have from here? Well, obviously, it's either x is equal to 0. Or if x is not equal to 0, then let's divide both parts by x square, which is positive now, right? If it's not equal to 0, it's positive. So I will have 1 greater or equal than x square. Or if you wish, x square less than or equal to 1. Now, what does it mean? Well, actually, this way, it belongs to this, right? x square less than 1, it means absolute value of x greater than or equal to 1. It means that the whole domain for this particular function of two variables should be from minus 1 to plus 1 for x value. That's done. So the main for the x is this particular segment, from minus 1 to 1. x cannot be greater than 1 or less than minus 1. Because then it will force, as you see, y square to be negative, which is impossible. OK, fine. So we have determined the boundaries for our x. Now let's go to the boundaries for the y. Well, we will use the same thing. Now, what's the maximum value for y square? Well, that's what's the maximum value for this particular expression minus x4 plus x to the second. Well, let's do it this way. Let's substitute u for x square. I will have minus u square plus u on the right, correct? Now, what are the values, the maximum value for this thing? Now, this is a parabola, right? You remember that the graph of this function, it's a quadratic function. So it's a parabola with horns downwards because this is minus. And it has two roots, 0 and 1. So it goes like this, with a maximum at 1 half. And at 1 half, it's minus 1 quarter plus 1 half is equal to 1 quarter. So the maximum value for y square is 1 quarter. So the maximum value for y is 1 half, which means these are the boundaries for y. So the whole graph should be within this particular rectangle. Great, we have determined that. Also, let's just have a couple of very obvious straight points. If x is equal to 1 or minus 1 and y is equal to 0, then we have this equation satisfied, right? 1 minus 1 plus 0 is 0, right? So this point, which is 1, 0, and this point, which is minus 1, 0, satisfy the equation, as well as 0, 0, 0. So our graph is supposed to go through all these three points. Also notice that our graph is symmetrical relatively to both horizontal x-axis, because if you will substitute from y minus y, you will get exactly the same. And it's symmetrical relative to the vertical y-axis, because, again, if you change x to minus x, it will also be exactly the same, because these are all squares and fourth degrees of the power. So the graph is symmetrical this way and this way, which means let's just draw this particular graph in one particular quadrant, the first one, where both x and y are positive. And then we will just symmetrically reflect it. Now, how can I draw the graph in this particular area, where x and y are positive? And I know that graph should not really go outside of this rectangle. And I know that it starts here and ends here. So it's probably somewhere here, something like this. That's what I assume. I can probably apply a little bit more logic, but in any case, we have to smoothly connect these two points while being in this particular area. So that's the most obvious kind of a thing. And now we can reflect it down and to the left, and we get almost an infinity sign. So this graph is called infinity curve, just for fun. This is a graphical representation of mathematical symbol of infinity using this type of a formula. All right, anyway, that's it for today. Please go to unizor.com. There are notes for this particular lecture. And I do encourage you to take a look at them. And it would be great if your parents are registered as supervisors. And then you can take exams and basically take the whole course as a course with exams and get some grades, which would probably satisfy the way how you study. That's it. Thanks very much for today.