Math Antics – Working With Parts

Now that we know how to use fractions to represent parts of a whole, there’s a few different things we can do with them. First of all, we can compare fractions. Comparing fractions means checking to see if one fraction is greater than, less than, or equal to another fraction. That’s pretty easy if we think of fractions as parts of objects, and draw pictures to help us see what we have. Here’s an example… Which of these fractions is greater: three-eights or five-eights ? To answer that question, let’s start by drawing two rectangles and divide them into eight equal parts. Since the rectangles are divided into eight parts, each of those parts are called an eighth. Now, let’s shade three parts of the first rectangle… or three-eighths, and five parts of the second rectangle… or five-eighths. Our picture makes comparing these fractions easy. You can see that five-eighths is greater than three-eights, because more of that rectangle is shaded. Okay, now let’s try one that’s a little harder. Which of these fractions is greater: three-fourths or four-fifths ? Well, let’s start again with two rectangles, but this time, we need to divide them up differently because the fractions we are comparing have different bottom numbers. The first rectangle will be divided into four equal parts, and we’ll shade three of them to show three-fourths. The second rectangle will be divided into five equal parts, and we’ll shade four of them to represent four-fifths. Now to compare, all we have to do is see which rectangle is shaded in the most, and that tells us that four-fifths is greater than three-fourths. There… that wasn’t so hard after all. Alright, let’s try one more example. Let’s compare the fractions one-half and two-fourths. Again, we start by drawing rectangle and dividing them up into parts: two on this one, and four on this one. Next, we shade the parts of the rectangle according to our top numbers: one on this one, and two on this one. Now, all we have to do is compare. Well, what do ya know? The same amount of each rectangle is shaded. That means these two fractions are equal! It might seem strange that two fractions can have totally different numbers and still represent the same amount, but they can! Fractions like that are called ‘Equivalent Fractions’. Equivalent fractions have different top and bottom numbers, but are equal in value. We’ll learn more about equivalent fractions later in this video, but for now, let’s find out what else we can do with fractions. Another thing we can do with fractions is add them together. Any two fractions can be added, but for now, we’re only going to add fractions if they have the same bottom numbers, because those fractions are much easier to add. …like these two fractions: one-fourth and two-fourths. Let’s add them together. Again, we can use drawings to help us solve this problem. Looking at the rectangles for these two fractions, we can add them visually just by rearranging the parts. Because all of the parts are ‘fourths’, our answer will also be ‘fourths’. We can just take this one-fourth from over here, and combine it with these two-fourths, and ta-da… three-fourths! So, one-fourth plus two-fourths equals three-fourths. Let’s try another one. Let’s add three-eights to five-eights. We can use any shape we want, so I’m gonna use a circle this time. So we have three out of eight here, and five out of eight here. Just like our last problem, we can add these by combining the parts. So let’s put these three over here with these five. Well what do ya know?… That fills up all eight sections. So three-eights plus five-eights equals eight-eights (or one whole circle). In those examples, you might have noticed a pattern. The bottom number of our answer was always the same as the bottom numbers of the fractions we were adding. And the top number of our answer was just the sum of the top numbers of those fractions. Well, that’s how it works. That’s the procedure (or set of steps) for adding fractions that have the same bottom numbers. That’s important, because if you can remember that procedure, then you won’t need to use drawings to help you add fractions. And that’s a really good thing, because what if you had to add these two fractions together: fifteen-hundredths and ten hundredths. It would be WAY too much work to draw rectangles and divide them up into 100 parts! Fortunately, since we know the procedure for adding fractions, we can do it without the drawings. First, let’s write out the problem. Now… because we’re adding fractions, we know the answer will also be a fraction. The bottom number of our answer will be the same as the bottom number of the fractions we’re adding: 100. And the top number of our answer will just be the sum of our top numbers: 15 plus 10, which equals 25. So as you can see, adding fractions with the same bottom numbers is easy when you know the procedure. All of this brings up a really important point. When you’re first learning about fractions, drawing pictures and imagining that fractions represent parts of cookies and candy bars can be really useful. (And it can taste good too!) Thinking of them that way can help you understand how simple fractions work, and it can even help you solve some basic math problems. But soon, you’ll have to do harder math problems! And to solve those, you’ll need to stop thinking about fractions as just parts of things, and start thinking about them in a different way. And that’s what we are going to be talking about in the next section. Before we move on, let’s review what we’ve covered so far. We can draw pictures to show how fractions represent parts of a whole. Using drawings, we can compare fractions to see which one represents the greatest amount. If we compare two different fractions and find that they represent the same amount, then we call them equivalent fractions. We can also use drawings to help us do simple addition by combining the parts. By doing this, we learned that the procedure for adding fractions that have the same bottom number, is to just add the top numbers and keep the same bottom number in our answer. Now to make sure you understand how to compare and add fractions visually, be sure to do the exercises for this section. learn more at

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Space Station Transiting 2017 ECLIPSE, My Brain Stopped Working – Smarter Every Day 175

Five, four, three, two, one, transit. Hey, it’s me. Destin. Welcome back to Smarter Every Day. August 21st 2017. It’s Matt Whitman, It’s Trevor. We’re in the middle of Wyoming. This is a really special spot. You have to understand we’re on a reservation, and you’re not allowed to come here. – No you’re not allowed to come here. You’re just not allowed to come here, but the Lady that delivered Matt’s son? -Yep. Midwife, owns this land. It’s a big deal What’s a really a big deal Is that Trevor did the math We started talking months ago. The international space station is going to transit in front of the eclipse, right? It’s going to happen. This is the spot. Like this is the exact spot I’ll let you know what the farmer said earlier today when we told him that his land was the spot. This is Mr. Fleming and this is his land. He leases this and he farms it and You just found out that the space station is going to transit here? Just found out. Didn’t know anything about it. We had some extra solar film so we set up Mr. Fleming’s old binoculars so he could watch the eclipse. We then took a few minutes and chatted about why we thought Crow Heart, Wyoming was the best possible place to watch it. You have the best spot in the U.S. -X marks the spot right here Yeah, it does so here’s the deal So in Kentucky, they’re going to get to see the eclipse longer Twenty seconds. Twenty seconds longer But here you get to see it for almost two and a half minutes and you get the transit of the international space station. We don’t have anybody in Crow Heart, Wyoming. Really? No. But this is a lot of activity for us here. Really? Like the most that you’ve seen in your lifetime? Yeah, since I’ve lived here. And how long is that? I’ve been here 30 years. Yeah. Are you ready for us to leave now? As soon as this is over. *laughter* We’ll get out of your hair, man. [Laughs] We got Mr. Fleming set up with his binoculars here. So you’re good to go Is that a decent trade, letting us use your land to get your binoculars set up? Good trade. (Laughter) There’s good people all over America. You know it. Oh, yeah. After 4 hours of setup we were super happy that the clouds cleared out, so we could finally see if the math was right. Hey come double check my focus if you get a second. – yeah, we got a minute. literally Literally a minute now. 59 Seconds. You can see the sun spot sharp. That’s good. 40.
– 40 seconds Look at everything and make sure that everything is rolling. Final touch of my camera. 20 20 seconds. 10 9 8 7 6 5 4 3 2 1 Transit. Oh! It happened! I saw it! I totally saw it! Dude! -Did you see it?
-No, I didn’t see it.
-It happened. I saw in the monitors. -Go back to your recorder. It happened. -You saw it? -I swear. I saw it. -Six, five, four, three two, one Transit. -Oh! It happened! I saw it! I totally saw it! Dude! (Music) four, three, two, one. Transit There it was! That is so awesome! Dude! Yes! (Music) The math worked. This excitement from successfully getting the transit continued up until the point of the total eclipse where I was so blown away I totally forgot everything I had practiced. It was insane. My, my brain was saturated. This is my shadow band…peice, uh… Did you come up with something? -No, no I -Bump it up. -There it is. There they are. Look at that That’s weird. -One of the gopros has died. -That’s cool What? (Camera clicking) (Sounds of excitement) That is crazy. Oh, my word! Take your glasses off! (Kids laughing) That is bizarre! The stars! Oh, wait… It’s overhead sunset. I had made up my mind ahead of time that no matter what happened with the photography at max eclipse I was going to drop it all and go spend time with my family Which is exactly what I did and it was really cool. This is awesome. I did so bad at this I’m just going to give up.I did so bad. –What do you see? Look around That is crazy. That cool look around behind you ain’t that wild Look around guys look around ain’t that cool Look at the sun. I love the fact that this was so amazing to me that it just overrode my brain I just wanted to experience it. I didn’t I didn’t think about making videos or taking good photos anymore I just wanted to look at it 20 seconds, I totally bought stuff all about photography. I don’t really care Yeah, here it comes What? Wow Watch it fade on I totally forgot about the drone. I never took my filter off So yeah, I didn’t get good photos. I didn’t pull the filters off because I was kind of in the moment And I really enjoyed it with my family Trevor however did get excellent photos and you can go check those out on his website I’ll leave links below, but man this was this was quite an experience Do you want a piece of gum? Would love a piece of gum What did you think about the eclipse? -It was awesome Was it awesome? Cool, thanks for coming with me -You’re welcome What you think sir? Well, I want to thank you for these. -Yeah? They made all the difference in the world Did they really? -Yeah That’s awesome, man -Yeah Thanks for letting us hang out here! Pretty weird -Yeah what’d you think? Did you cry? No, I didn’t cry- You didn’t cry? -I was glad to see the old sun come back [laughs] -a hardened man like you? ‘No, I’ve never cried” I kind of I kind of got a little weird I got a little weird Did you get weird? -I got weird. Do you wanna see the space station transit the sun? -Yeah I’d love to *three two one* *Transit- Oh it happened I saw it!* -there it was, size of a football field just flew in front I’ll be damned. huh. That’s great Follow my nose. I can’t believe how fast that was. -yeah *-one, Transit- Oh it happened I saw it!* [Laughter] I know you normally don’t put swears on your videos But you would be an idiot to leave that off I’m a professional minister, and I don’t think that even counts as a swear because he’s a cowboy What did you think? -I cried, tears of joy I couldn’t not cry Really? -Really. It’s beautiful What’d you think [unintelligible]?
-I get goosebumps Okay, that’s it. I am Destin. This is Trevor- Trevor where can they find your stuff?
-patreon, slash your name my name alright I’ll leave a link below and this is Matt, ten minute Bible hour That’s it the stars aligned, or ‘the’ star aligned with our stuff. That was cool. Anyway, I’m Destin You’re getting smarter every day. Have a good one Can’t believe my brain quit working. -I can. -Why? -Weird that you can’t believe it. -Really? It’s weird that you’re surprised about it. Of course your brain quit working That was awesome

The Future of War, and How It Affects YOU (Multi-Domain Operations) – Smarter Every Day 211

– Hey it me, Destin. This is hard to explain. So lemme just start here. Everyone has a unique world view, and that world view is shaped
by different perspectives. Perspectives are shaped by how you choose to spend your time. For the past 15 years, I
have been a federal employee, specifically something called a Department of Defense civil servant and as part of that, I took an oath to defend the Constitution against all threats foreign and domestic and I’ve been using math
and science to do just that. I’ve been doing that in the role of something called a Missile
Flight Test Engineer for the Army Test Evaluation Command. We, the men and women of the
Army Test Evaluation Command have been tasked with a
job of protecting soldiers by testing their equipment
to make sure that if someone’s gonna use an
offensive system against them, they can defend themselves and in order to do that, you
must first become an expert on the offensive systems. Does that make sense? This is the part that’s
hard to explain but, for the last however long, I’ve been learning how
to use these systems. For example, this is me,
operating a mortar and I have to learn everything
about this thing. This is me operating a mini gun. Now, it looks like
you’re just shooting but that’s not what’s happening here. This is me learning failure modes, how to operate it, the capabilities and
limitations of the system. This is me learning
common jams that happen, how to clear those jams. There’s a lot of stuff going on here. It’s not just shooting
out of a helicopter. (machine gun firing) This is me in a Bradley
infantry Fighting Vehicle, firing the M242 25 millimeter cannon. That’s something I spent
a lot of time doing. The thing that I spent
the most time on was missles and rockets. This is me participating in a test of a javelin missile system. – [Soldier] Windows open. 3-2-1 FIRE! (missile hissing) (missile exploding) (missile exploding) – Footage like this goes on forever and it’s hard to believe that I was blessed enough to do this job. I took it very seriously and my teammates who are still doing it are some of the most amazing
people I’ve ever worked with. The men and women there are brilliant. Now you know me. The world is interesting and amazing and one of my favorite things to do is to be handed a complex question and work with a team of
competent individuals to arrive at a solution to that and then afterwards, I love sharing that path of discovery with others. I was talking about perspective
earlier and this is why. Over 15 years you can kind of understand major trends and shifts
that are happening. For example, in 2003 the main
issues were terrorism, okay? Slowly, that started to shift and I noticed that over time, here in the last couple years especially, the US is mostly concerned
about major state actors. You can imagine how difficult it’s been, going into work everyday, working on huge problems that matter and that have huge implications and then not being able to
talk about it with anyone. Well, this video changes that. I’m going back to school, right? On my way out the door, I get this call from a
four-star general and he said, Destin, would you like to
film your last mission and after your mission come to
my office and interview me, a four-star general, and
we’ll just talk about what you’ve been doing all this time and how your little piece of the puzzle fits into the overall bigger picture? And then he told me that we can make that a publicly releasable video. So yes, I’m excited about this. Let’s go to Hawaii and
let me share with you my last mission. For my last mission, I was told I’d be filming
a mission in the Pacific, off the coast of Kauai
as a part of RIMPAC. RIMPAC is the Rim of the Pacific exercise. It is the largest naval
exercise in the world and it happens every two years in Hawaii and involves countries
from all over the world. Often with RIMPAC, they’ll
do what they call a sink ex. It’s an exercise which involves
actually sinking a ship. In this case, the USS Racine. My job was to fly out in a helicopter, over the Pacific, dozens
of miles away from shore, as they’re hitting this thing
with missles and rockets and bombs and torpedoes
and all that stuff. I’m gonna be in the helicopter up top and my job is to get footage of impact. So, that’s why I had to
go through dunker training with the Marines because that helicopter, if it goes down in the ocean,
there’s nowhere to land, so you have to be able to figure out how to get out and survive. It took us about an hour
to get out to the ship. We flew out over the water,
which felt like forever and we finally got there and we realized that we have some challenges. Number one, we had to
loiter at 10,000 feet, which meant we were like
looking through a soda straw ’cause we had our lenses zoomed all the in and with the rocking of the helicopter, that made it really hard
to focus on the ship. So I was really concerned that my shot would be out of focus. Another issue is we didn’t know exactly when the impact was going to take place. We knew that there was
this window of time, which meant that me, on
the high speed camera, and the Barking Sands
Missile Range guy to my left, who was operating a stabilized camera, we had to just focus on
the ship the entire time, but we couldn’t look at
it with our naked eyes. So we didn’t know when
impact was gonna happen. We had to rely on the pilot to cry out, impact, impact, impact,
whenever he saw smoke. So thankfully, he was very attentive, he did his job well, we did our job well and we were able to capture the images and get the engineers
data that they needed. They wanted to know what
the warhead effects were, they were trying to do a specific thing with this specific
Norwegian strike missile. It’s a Norwegian missile that
was operated by the US Army, which in itself, is a fascinating thing, using targeting data from other systems. So this is really a big
interoperability test. It was fascinating. That was the first thing that happened and then over the course of the day, the Racine just got walloped. It was monitored by a US Army Gray Eagle. The Australians hit it with a Harpoon. The USS Olympia, a submarine,
they hit it with a Harpoon and then the Japanese Defense Forces, they hit it with a
surface to ship missile, so I had another opportunity to film that in slow motion as well and that was hard because again, we had the same problems. We didn’t know exactly
when it was gonna hit because it was so far from shore. I almost missed it. We got it right as it
approached the ship but I was very thankful we got
the data that we needed. I wish I had triggered it just not even a hundred milliseconds earlier, but thankfully, we did get the bare-bones data requirement that we needed. The USS Racine had a really bad day. After we hit it with all
these missiles and stuff, some Apaches came all
the way out from shore and they did some work
on the deck with rockets and 30 mike-mike and then
after that the grand finale, a Mark 48 torpedo fired from a US sub and this is something I will never forget. In a helicopter, over the USS Racine, was can get down closer
because there’s not a huge safety fan issue like there was with missiles flying at you. You can be right there on it. I chose to get the 4k camera
and just get right on that ship and just hold it as best I could and boom. You can see it happen. The cool thing to me is you can see that big air bubble form under the ship and then it just breaks the keel. It’s like breaking the spine of the ship and that thing is done. Later it was finished off
buy some aerial stuff, but the USS Racine is no more. I’m gonna level with you here. I speak rocket, I speak missile, and I’m not intimidated by those things. This was an intimidating
exercise for me personally, because I don’t understand how to coordinate battlefield maneuvers. Now I understand that
people in the military do know these things. I don’t. I asked my friend Captain Mackenzie Harp to help me get smarter on the topic. So she took me to speak with someone from the Asymmetric Warfare Group, which is a part of TRADOC. Alright this is Lieutenant
Colonel Davis with TRADOC, Training and Doctrine Command, right? – That’s correct. – How does the battlefield
work these days? – So right now, when you think
about the modern battlefield, what you’re thinking about is your thinking about chess
pieces moving on a map. Could we create new version of that, that is three dimensional? Some of us grew up on Star Wars, even my kids grew up on Star Wars. There’s a famous scene
that keeps getting repeated where they are playing chess with these little creatures on a map and they’re in three dimensions and they’re playing this
three dimensional chess. Well when I was a kid,
I always wanted to play three dimensional chess. That’s what we’re trying to
do now with this battle space. So there are five
acknowledged domains right now and arguably there’s sixth but the five domains and just go from lowest to highest ’cause it’s typically easiest that way. There’s sea, there’s the land, there’s the air, which are
our traditional domains which is why we have
Navy for the sea domain, the Army for the land domain and then the Air Force for the air domain. And what we’ve realized in
the last couple of years is that there’s also the space domain and then we also have the cyber domain. Because as we all know, the internet has changed the
connectivity of our planet and it’s also changed not only the way we exchange information, but the way that we are going to interact with our environment. – [Destin] So you’re saying
the battlefield is not land, with Abrams tanks or sea with a ship or something like that or an airplane flying overhead. You’re saying it’s more than than now. – Yes. – [Destin] The battlefield is different? – The operational space has changed, just like the world has changed. – Like a futuristic commander, he could be fighting a cyber
war or cyber battle to his left and he can have a naval
battle to his right and he’s also thinking
about the air battle. So he could be fighting multiple battles at once in the same conflict. – Absolutely. – [Destin] And that’s called what? – Multi-domain operations and then you would be doing cross-domain maneuver, within that multi-domain operation. – [Destin] Explain a
cross-domain maneuver. – We in the military
would define a maneuver as the ability to shoot and move, and often we add communicate to that because you can’t do
one without the other. The ability to shoot, move and communicate within multiple domains. Air, land, sea battles,
our traditional way of thinking of warfare, is a cross-domain maneuver when we are shooting from the sea
through the air to the land. That is a cross-domain maneuver. Now we have soldiers
who have the ability to see through the cyber domain, by using the space
domain, i.e. a satellite, in order to conduct a ground movement inside of a city. – You are working with how people interact with each of these domains. I need to find the person who is basically playing the chess game. And who would that be? – So, this is Four Star
General Robert Brown, which is a really big deal, Sir, thank you. – Well I don’t know about that but, yeah. – For even considering talking to me. What I’ve noticed, when
I started 15 years ago, we were fighting insurgencies, – Right. – Terrorism, things like that. – Right. – It’s changing. – Yeah, sure. – [Destin] So from your perspective, what I’ve learned about
talking to all these other folks that work for you is something called multi-domain – Operations. – [Destin] Operations. – Yeah. – [Destin] And can you explain that to me? – So multi-domain operations,
I will tell you it’s a, I am positive, one thing I am positive, it’s the way of the future and in warfare and we’re either going
to do like we are now, innovate and move towards
multi-domain operations, to either win a conflict
in the future or prevent it because nobody’d be stupid
enough to fight against us, or we’re gonna be forced to do it because we lose someday pretty badly and we’re going to have to do it and I hope that’s not the case. – [Destin] You’re trying
to get ahead of the curve. – Get ahead of the curve,
not fight last fight but fight the future fight and in fact, maybe not have to fight because we’re so darn
good nobody can match us. So this is an evolutionary thing. You go back to the Civil
War, the Battle of Vicksburg, the Army had to work with the
Navy in a river in that case, to win at Vicksburg. – [Destin] Grant actually
diverted the river. – There you go. Exactly. – [Destin] Grant’s Canal. – And so you got domains in play there, a little bit of maritime and land. However, things have changed
as you know as well as anybody. The world’s gotten so complex and now there’re these other domain out there. Cyber, space, I would also
count the human domain as well, I think is the most important. – [Destin] What do you mean by that? – What I mean by the human domain is that when you look at conflict, it’s a contest of wills
and people are involved. It’s about people. I mean, there maybe machines
going against each other some people say, oh it’ll
all be autonomous, AI and all machines in the future. I certainly don’t believe that. There’ll be autonomous
vehicles, there’ll be a lot that’s done but people are
always gonna have to be involved, that’s the nature of conflict and so it’s influence over those people. What we’re seeing today is, you can influence someone before you’re even at conflict with them and you can influence them in the cyber domain and you can influence
them in the space domain and you can influence and
there are no boundaries in these domains. – Forgive me Sir. You’re four-star General, – Yeah. – You’re a very important individual (General laughing) but I’m a simple man and so
I need it broken down for me. – Okay. – For example, I’ve been
keeping up with the news and I know that there’s
huge social media campaigns that occur before an actual
battle ever occurs now. – Right. – Is that true? – Well they call it the gray zone effect. You look at what some nations are doing. Russia’s pretty darn good at it. China as well where, in what we used to call
Phase Zero, pre-conflict. I don’t like that term at all. There’s always gonna be competition and we’re in a
hyper-competition now, always. You’re exactly right. They will be working
to shape and influence people’s perceptions before
a conflict even occurs. What happens is, it’s a fait accompli, that before we even fire a
bullet they’ve already won. If you look at Russia and Ukraine. Little bit of that. We call them little green men. They took off their patches, claimed they weren’t Russians, just to put just enough
doubt, cyber stuff happening, events happening and it wasn’t quite, just staying right below the
threshold of conflict but yet – [Destin] And then as soon as
it was time to turn conflict, it was already over. – Already over. They already had the territory, they already had what they wanted, now they hold onto it. So this is new. There are other aspects that are new. I would tell you that the the speed of human velocity,
of human interaction, is that a rate never before seen. You can put out a video
and millions look at it. When I was your age, there was no way I could get millions of
people look at anything. I have a private that does something in, I have 106,000 people in
the US Army in the Pacific. I have a private that does
something that’s on CNN tonight, worldwide impact, influence. Good or bad. That’s changed. The speed of perception. I was talking this one time
and someone confronted me and said, well things moved
fast in the Civil War, by telegraph, things moved fast. They might have moved fast, they didn’t diffuse rapidly. Now you can diffuse. Send a tweet and halfway around the world, billions of people are paying
attention to it, right? So this has an impact in conflict, it has an impact in
everything in our society. Business, sports, military conflict, it’s a huge impact. So you’ve got all the domains,
air, land, sea, space, and I think around the all of them is this human domain,
that is the most complex, the most important. Let’s get back to multi-domain operations, we’re going all over the place. I wanna try and simplify
it best I can for you. It’s an evolutionary process. Over time we’ve had
air land, air land sea. We’re evolving but it’s
a revolutionary impact, where you’re maneuvering in all domains to a position of relative advantage. So what we were just talking about was maneuvering in cyber to a position of relative advantage pre-conflict, maneuvering in space to a
position of relative advantage and to take advantage of
multi-domain operations, if you truly work together,
one of our strengths, we’re more joint than
any force in the world, the US Military. We’re still not joint enpugh. We’ve got to work where
it’s not the exception where we’re working really well together, not a joint integration, I’m sorry joint interdependence, but joint integration. – Lemme ask you this. So yesterday I was over the water, watching an engagement
by a foreign nation, engaging a target with information from US satellites provided on a different type of communication network. I just see the engagement. I’m a missile tester. I test the missile but I don’t get to see the chess games. I’m a little too close to the board, I’m the pawn. – So think about in the past if the services are in stove pipes and you can do things in
your service pretty well but if you have a menu of options of all the service capabilities, like you saw yesterday. There was a missile that
launched from the land, can destroy a ship at sea and it was controlled by an Army element, but using Navy, Air Force,
Marine and as you mentioned, some national satellite means things never before
used to pull it together and enable it to talk and work together, that gives you more options
against an adversary. Anybody would want more options. It allows you to present multiple dilemmas to an adversary instead of a linear, here comes the land force, here’s the maritime forces, here’s the air, we’re working together, it’s very linear and predictable, It’s very unpredictable,
multi-domain operations. – [Destin] So that’s the
goal, is to be unpredictable. – Unpredictable, present multiple dilemmas to you adversary that
they have to deal with and then multiple options
to your leadership and that’s what you saw a
little taste of yesterday, now picture if that ship is trying to skirt into
the littorals to avoid our strength at sea that our Navy, the best in the world has, so they’re trying to skirt around it. Well the Army can engage
that ship now and destroy it so now it has to go back out
to sea and be right in our engagement area, where
we’re going to destroy them. So you’re your pairing
up together, all domains, maneuvering to a position
of relative advantage and each domain working together to create those windows of opportunity where you can dominate your adversary. The other good thing is, and the reason I talk
about the human dimension, the human domain in all this, you can’t do this if you
can’t empower people. – [Destin] What do you mean? – If you’ve got someone
that you tell them a task and that’s all they can
do, this would never work. You have to have the ability to empower, what we call in the Army mission command, but the other services
have similar versions, where I can give someone
an outcome I want. They figure out how to get there, they’ll amaze you with what they do. Talented young leaders, our
people, are our advantage. The best non-commissioned
officers in the world. The best leaders that are taught to thrive in ambiguity and chaos. – Multi-domain battle,
we have multiple layers are playing 4-D chess, but you’re saying it’s like the chess pieces are alive themselves. – Yeah. – and they can think on their own. – Good analogy. – [Destin] What move is best. – Yeah it’s not you. There’s no longer the central
figure moving that piece, it moves too rapidly. Those pieces are empowered
and can move themselves but work together and know how to work so they’re integrated because they practice
together all the time, just like we were doing yesterday and just like you saw,
historically for the first time, doing that on a regular basis. You really develop a web of options that you can use to keep an adversary completely off guard and you’re completely unpredictable on that. So you’re using all
domains, all angles you can, to present those multiple dilemmas and that requires incredible
joint integration. That’s not natural ’cause we
all grow up in stove pipes, 37 years I’ve been Army,
trained in Army schools. I did joint time, I was on
the Joint Staff and stuff but I think of an Army solution first. So you gotta get out of those stove pipes. Maybe there’s a better air solution. Maybe a better cyber solution. Maybe a better maritime solution. – I think I see what’s
happening right here Sir. This is US Army Pacific. – Right. – You’re the guy in charge
of the 4D chess board – Yeah. Well, portions of it maybe. – Portions of it. If multi-domain operations,
we’ve got land, sea, space, air, cyber. Is this video a weapon? – Absolutely. – What? So I just made a weapon? – Yeah pretty much. Well, if it can help folks
understand number one, if it can deter those
that would do us harm, it’s a hell of a weapon and if it can help those
who are working this understand it a little better,
support it a little better, then it’s a heck of a weapon. Yeah I guess you could almost say that, that’s why you’re doing this. Pretty clever with these
analogies here, yeah. – I don’t know what to do
with this information now. That’s pretty amazing. So war is changing. – Yeah, there’s no question,
we’re at an inflection point and some of the things,
the reasons we go to war, maybe not changing, but
the character of war, with technology, I mean, significantly changing. – I’ll be honest, when I was growing up, a soldier, the stereotype
of an American soldier, was a tough person with big
muscles that could run far and pick up big things. – Yeah. – Now it’s a hyper-intelligent individual? – I wouldn’t go away, physical prowess is key for a soldier, ’cause it’s still tough work and I would say in all the services, being physically fit is still key. In fact, we just developed a new Army combat physical fitness
test that’s excellent. It help you prepare you for
what you will face in combat. So you still have to be physically fit, you gotta be more of a holistic person, but you got to learn and I wouldn’t say it’s
hyper-intelligence, I’d say it’s the ability
to solve a complex problem in a timely manner with
a creative solution, the ability to thrive
in ambiguity and chaos, that’s not necessarily hyper-intelligence. That’s the ability to
make the right decision, know when you have enough information, know when you don’t
have enough information. It’s common sense, it is intelligence, but it’s training education
that we have to do and we’re doing different
than we did in the past. We might have told somebody in the past, here’s your task, here are the exact steps you need to take to accomplish it. Shouldn’t do that anymore. Here’s the outcome I want, you figure out how to get there. That’s the difference. – War is changing. Thank you very much for your time. – [ General] Absolutely. – I feel like you’re the
game master or something. – I don’t know about that. – There’s a huge, not a
game, war is not a game. – One thing I am positive though, is if we do this right, if we truly come together and
integrate in a joint manner, as the United States along
with our International, our allies are so key
that’s why as you saw, we had a Japanese ground
self-defense force involved and our great allies around
the world are critical. We’re not gonna do anything alone. But if we come together and we truly get multi-domain operations right, it’ll be an absolute key
to peace and stability in the future, ’cause no
one would be foolish enough, and some of these nations
we’ve been talking about, many out there, they don’t have the people, the ability to empower people, the way we do and it’s that American Spirit, that is a huge advantage. It has been for a long time. We can’t handcuff those folks. We gotta empower them and that’s what multi-domain operations is all about and I sure wouldn’t want to go against us when we do it right. Nobody wants to fight. The best thing would be
again, the deterrence where we never have to fight, because nobody’d be foolish
enough to go against us. So that’s the best thing. I thank you for interested in this, flying around in a helicopter, above a missile being fired, several missles being fired. – [Destin] It was awesome. – Pretty gutsy and coming out and seeing
some of these things, just a small taste of it, got a lot to connect, a lot of work to do, pull all the pieces together
but if anybody can do it, the United States military can. I feel very confident and I’m
very proud United States Army leading the way with
multi-domain operations. – The General mentioned that
this video could be a weapon and I hope it is. I hope it’s a countermeasure. The one thing I want you to
take away from this video is that multi-domain operations
are the future of warfare. I find it fascinating that General Brown, a four-star General, a person who wields incredible military might, is concerned about the cyber domain. You heard him say it, right? Several times. The cyber domain and the human element affected by the cyber domain. Let’s think about what’s
different about the cyber domain. If you roll up into a bay in a a ship and you start shooting stuff, people are gonna notice, right? But how do maneuvers take
place in the cyber domain? They’re more subversive than that, right? People don’t wear
uniforms on the internet. Surfninja385 is not declaring his intent to be hostile before he does something. Maneuvers and actions in the cyber domain don’t make loud noises. This type of conflict
is based on deception and the most important deception is to convince you that
you’re not at conflict. Heck, even a couple days ago, I was tweeting something
asking about this stuff and I was accused pretty
quickly of tinfoil hattery. I think the reason that is, is because we like to think
about this place that we gather, the internet place here, as just a fun, consequence-free place where we can like and up vote and comment and subscribe and all that kind of stuff, but come on. We can all feel it, we just
don’t know how to say it. I feel this us versus them thing and I’ve seen really smart
friends fall into it and I’m guilty myself. Something’s happening. I just had a four-star
General tell me that the cyber domain and specifically,
the human elements, us, affected within that cyber domain, is one of the five most
important components to modern warfare. There’s an old saying
from when I was a kid. Knowing is half the battle
and it totally applies here. Think about it. Just knowing that the cyber
domain is a battle space and I’m in it, is huge. So what about the other
half of the battle? Straight-up. I believe the biggest threat right now is division. They’re gonna find the
division within our society and they’re gonna try to amplify it. I would like to submit
for your consideration, a countermeasure. A way to get through
this modern bombardment, in this new battle space that
we haven’t experienced before. If they’re trying to divide us, I think the way to get
around this is proactive, intentional unity. We can all do this. We all need to be more
conscious of what types of content we’re consuming online. What are we liking? What are we sharing? How is that affecting our minds? Is it affecting the way we treat people, both online and offline? If we extend patience and political grace, not just to those people we like, but also to those with whom we disagree, these maneuvers in the cyber
domain meant to divide us, simply will not work. Political grace. Basically the art of disagreeing well. This is the ultimate countermeasure to this kind of attack. Thanks for watching this video.

The Math Behind Basketball’s Wildest Moves | Rajiv Maheswaran | TED Talks

My colleagues and I are fascinated
by the science of moving dots. So what are these dots? Well, it’s all of us. And we’re moving in our homes,
in our offices, as we shop and travel throughout our cities
and around the world. And wouldn’t it be great
if we could understand all this movement? If we could find patterns and meaning
and insight in it. And luckily for us, we live in a time where we’re incredibly good
at capturing information about ourselves. So whether it’s through
sensors or videos, or apps, we can track our movement
with incredibly fine detail. So it turns out one of the places
where we have the best data about movement is sports. So whether it’s basketball or baseball,
or football or the other football, we’re instrumenting our stadiums
and our players to track their movements every fraction of a second. So what we’re doing
is turning our athletes into — you probably guessed it — moving dots. So we’ve got mountains of moving dots
and like most raw data, it’s hard to deal with
and not that interesting. But there are things that, for example,
basketball coaches want to know. And the problem is they can’t know them
because they’d have to watch every second of every game, remember it and process it. And a person can’t do that, but a machine can. The problem is a machine can’t see
the game with the eye of a coach. At least they couldn’t until now. So what have we taught the machine to see? So, we started simply. We taught it things like passes,
shots and rebounds. Things that most casual fans would know. And then we moved on to things
slightly more complicated. Events like post-ups,
and pick-and-rolls, and isolations. And if you don’t know them, that’s okay.
Most casual players probably do. Now, we’ve gotten to a point where today,
the machine understands complex events like down screens and wide pins. Basically things only professionals know. So we have taught a machine to see
with the eyes of a coach. So how have we been able to do this? If I asked a coach to describe
something like a pick-and-roll, they would give me a description, and if I encoded that as an algorithm,
it would be terrible. The pick-and-roll happens to be this dance
in basketball between four players, two on offense and two on defense. And here’s kind of how it goes. So there’s the guy on offense
without the ball the ball and he goes next to the guy
guarding the guy with the ball, and he kind of stays there and they both move and stuff happens,
and ta-da, it’s a pick-and-roll. (Laughter) So that is also an example
of a terrible algorithm. So, if the player who’s the interferer —
he’s called the screener — goes close by, but he doesn’t stop, it’s probably not a pick-and-roll. Or if he does stop,
but he doesn’t stop close enough, it’s probably not a pick-and-roll. Or, if he does go close by
and he does stop but they do it under the basket,
it’s probably not a pick-and-roll. Or I could be wrong,
they could all be pick-and-rolls. It really depends on the exact timing,
the distances, the locations, and that’s what makes it hard. So, luckily, with machine learning,
we can go beyond our own ability to describe the things we know. So how does this work?
Well, it’s by example. So we go to the machine and say,
“Good morning, machine. Here are some pick-and-rolls,
and here are some things that are not. Please find a way to tell the difference.” And the key to all of this is to find
features that enable it to separate. So if I was going
to teach it the difference between an apple and orange, I might say, “Why don’t you
use color or shape?” And the problem that we’re solving is,
what are those things? What are the key features that let a computer navigate
the world of moving dots? So figuring out all these relationships
with relative and absolute location, distance, timing, velocities — that’s really the key to the science
of moving dots, or as we like to call it, spatiotemporal pattern recognition,
in academic vernacular. Because the first thing is,
you have to make it sound hard — because it is. The key thing is, for NBA coaches,
it’s not that they want to know whether a pick-and-roll happened or not. It’s that they want to know
how it happened. And why is it so important to them?
So here’s a little insight. It turns out in modern basketball, this pick-and-roll is perhaps
the most important play. And knowing how to run it,
and knowing how to defend it, is basically a key to winning
and losing most games. So it turns out that this dance
has a great many variations and identifying the variations
is really the thing that matters, and that’s why we need this
to be really, really good. So, here’s an example. There are two offensive
and two defensive players, getting ready to do
the pick-and-roll dance. So the guy with ball
can either take, or he can reject. His teammate can either roll or pop. The guy guarding the ball
can either go over or under. His teammate can either show
or play up to touch, or play soft and together they can
either switch or blitz and I didn’t know
most of these things when I started and it would be lovely if everybody moved
according to those arrows. It would make our lives a lot easier,
but it turns out movement is very messy. People wiggle a lot and getting
these variations identified with very high accuracy, both in precision and recall, is tough because that’s what it takes to get
a professional coach to believe in you. And despite all the difficulties
with the right spatiotemporal features we have been able to do that. Coaches trust our ability of our machine
to identify these variations. We’re at the point where
almost every single contender for an NBA championship this year is using our software, which is built
on a machine that understands the moving dots of basketball. So not only that, we have given advice
that has changed strategies that have helped teams win
very important games, and it’s very exciting because you have
coaches who’ve been in the league for 30 years that are willing to take
advice from a machine. And it’s very exciting,
it’s much more than the pick-and-roll. Our computer started out
with simple things and learned more and more complex things and now it knows so many things. Frankly, I don’t understand
much of what it does, and while it’s not that special
to be smarter than me, we were wondering,
can a machine know more than a coach? Can it know more than person could know? And it turns out the answer is yes. The coaches want players
to take good shots. So if I’m standing near the basket and there’s nobody near me,
it’s a good shot. If I’m standing far away surrounded
by defenders, that’s generally a bad shot. But we never knew how good “good” was,
or how bad “bad” was quantitatively. Until now. So what we can do, again,
using spatiotemporal features, we looked at every shot. We can see: Where is the shot?
What’s the angle to the basket? Where are the defenders standing?
What are their distances? What are their angles? For multiple defenders, we can look
at how the player’s moving and predict the shot type. We can look at all their velocities
and we can build a model that predicts what is the likelihood that this shot
would go in under these circumstances? So why is this important? We can take something that was shooting, which was one thing before,
and turn it into two things: the quality of the shot
and the quality of the shooter. So here’s a bubble chart,
because what’s TED without a bubble chart? (Laughter) Those are NBA players. The size is the size of the player
and the color is the position. On the x-axis,
we have the shot probability. People on the left take difficult shots, on the right, they take easy shots. On the [y-axis] is their shooting ability. People who are good are at the top,
bad at the bottom. So for example, if there was a player who generally made
47 percent of their shots, that’s all you knew before. But today, I can tell you that player
takes shots that an average NBA player would make 49 percent of the time, and they are two percent worse. And the reason that’s important
is that there are lots of 47s out there. And so it’s really important to know if the 47 that you’re considering
giving 100 million dollars to is a good shooter who takes bad shots or a bad shooter who takes good shots. Machine understanding doesn’t just change
how we look at players, it changes how we look at the game. So there was this very exciting game
a couple of years ago, in the NBA finals. Miami was down by three,
there was 20 seconds left. They were about to lose the championship. A gentleman named LeBron James
came up and he took a three to tie. He missed. His teammate Chris Bosh got a rebound, passed it to another teammate
named Ray Allen. He sank a three. It went into overtime. They won the game.
They won the championship. It was one of the most exciting
games in basketball. And our ability to know
the shot probability for every player at every second, and the likelihood of them getting
a rebound at every second can illuminate this moment in a way
that we never could before. Now unfortunately,
I can’t show you that video. But for you, we recreated that moment at our weekly basketball game
about 3 weeks ago. (Laughter) And we recreated the tracking
that led to the insights. So, here is us.
This is Chinatown in Los Angeles, a park we play at every week, and that’s us recreating
the Ray Allen moment and all the tracking
that’s associated with it. So, here’s the shot. I’m going to show you that moment and all the insights of that moment. The only difference is, instead
of the professional players, it’s us, and instead of a professional
announcer, it’s me. So, bear with me. Miami. Down three. Twenty seconds left. Jeff brings up the ball. Josh catches, puts up a three! [Calculating shot probability] [Shot quality] [Rebound probability] Won’t go! [Rebound probability] Rebound, Noel. Back to Daria. [Shot quality] Her three-pointer — bang! Tie game with five seconds left. The crowd goes wild. (Laughter) That’s roughly how it happened. (Applause) Roughly. (Applause) That moment had about a nine percent
chance of happening in the NBA and we know that
and a great many other things. I’m not going to tell you how many times
it took us to make that happen. (Laughter) Okay, I will! It was four. (Laughter) Way to go, Daria. But the important thing about that video and the insights we have for every second
of every NBA game — it’s not that. It’s the fact you don’t have to be
a professional team to track movement. You do not have to be a professional
player to get insights about movement. In fact, it doesn’t even have to be about
sports because we’re moving everywhere. We’re moving in our homes, in our offices, as we shop and we travel throughout our cities and around our world. What will we know? What will we learn? Perhaps, instead of identifying
pick-and-rolls, a machine can identify
the moment and let me know when my daughter takes her first steps. Which could literally be happening
any second now. Perhaps we can learn to better use
our buildings, better plan our cities. I believe that with the development
of the science of moving dots, we will move better, we will move smarter,
we will move forward. Thank you very much. (Applause)

What’s so special about Euler’s number e? | Essence of calculus, chapter 5

I’ve introduced a few derivative formulas but a really important one that Ieft out was exponentials. So here, I want to talk about the derivatives of functions like Two to the x, seven to the x, and also to show why e to the x is arguably the most important of the exponentials. First of all, to get an intuition, let’s just focus on the function two to the x. And let’s think of that input as a time, “t,” maybe in days, and the output, 2 to the t, as a population size perhaps of a particularly fertile band of pi creatures which doubles every single day. And actually, instead of population size, which grows in discrete little jumps with each new baby pi creature, maybe let’s think of 2 to the t as the total mass of the population. I think that better reflects the continuity of this function, don’t you? So, for example, at time t=0, the total mass is 2 to the 0 equals 1, for the mass of one creature. At t=1 day, the population has grown to 2 to the 1=2 creature masses. At day t=2, it’s t squared, or 4, and in general, it just keeps doubling every day. For the derivative, we want dm/dt, the rate at which this population mass is growing, thought of as a tiny change in the mass divided by a tiny change in time. And let’s start by thinking of the rate of change over a full day, say, between day 3 and day 4. Well, in this case it grows from 8 to 16, so that’s 8 new creature masses added over the course of one day. And notice, that rate of growth equals the population size at the start of the day. Between day 4 and day 5, it grows from 16 to 32. So that’s a rate of 16 new creature masses per day. Which, again, equals the population size at the start of the day. And in general, this rate of growth over a full day equals the population size at the start of that day. So it might be tempting to say that this means the derivative of 2 to the t equals itself. That the rate of change of this function at a given time t, is equal to, well, the value of that function. And this is definitely in the right direction, but it’s not quite correct. What we’re doing here is making comparisons over a full day, considering the difference between 2 to the t plus 1, and 2 to the t. but for the derivative, we need to ask what happens for smaller and smaller changes. What’s the growth over the course of a tenth of a day? A hundredth of a day? One one-billionth of a day? This is why I had us think of the function as representing population mass since it makes sense to ask about a tiny change in mass over a tiny fraction of a day but it doesn’t make as much sense to ask about the tiny change in a discrete population size per second. More abstractly, for a tiny change in time, dt, we want to understand the difference between 2 to the t plus dt and 2 to the t, all divided by dt. A change in the function per unit time, but now we’re looking very narrowly around a given point in time, rather than over the course of a full day. And here’s the thing: I would love if there was some very clear geometric picture that made everything that’s about to follow just pop out, some diagram where you could point to one value, and say, “See! *that* part. That is the derivative of 2 to the t.” And if you know of one, please let me know. And while the goal here as with the rest of the series is to maintain a playful spirit of discover, the type of play that follows will have more to do with finding numerical patterns, rather than visual ones. So start by just taking a very close look at this term 2 to the t, plus dt A core property of exponentials is that you can break this up as 2 to the t times 2 to the dt. That really is the most important property of exponents. If you add two values in that exponent, you can break up the output as a product of some kind. This is what lets you relate additive ideas things like tiny steps in time, to multiplicative ideas, things like rates and ratios. I mean, just look at what happens here. After that move, we can factor out the term 2 to the t. which is now just multiplied by 2 to the dt minus 1, all divided by dt. And remember, the derivative of 2 to the t is whatever this whole expression approaches as dt approaches 0. And at first glance that might seem like an unimportant manipulation, but a tremendously important fact is that this term on the right, where all of the dt stuff lives, is completely separate from the t term itself. It doesn’t depend on the actual time where we started. You can go off to a calculator and plug in very small values for dt here, for example, maybe typing in 2 to the 0.001 minus 1, divided by 0.001 What you’ll find is that for smaller and smaller choices of dt, this value approaches a very specific number, around 0.6931. Don’t worry if that number seems mysterious, The central point is that this is some kind of constant. Unlike derivatives of other functions, all of the stuff that depends on dt is separate from the value of t itself. So the derivative of 2 to the t is just itself, but multiplied by some constant And that should kind of make sense, because earlier, it felt like the derivative for 2 to the t should be itself, at least when we were looking at changes over the course of a full day. And evidently, the rate of change for this function over much smaller time scales is not quite equal to itself, but it’s proportional to itself, with this very peculiar proportionality constant of 0.6931 And there’s not too much special about the number 2 here, if instead we had dealt with the function 3 to the t, the exponential property would also have led us to the conclusion that the derivative of 3 to the t is proportional to itself. But this time it would have had a proportionality constant 1.0986. And for other bases to your exponent you can have fun trying to see what the various proportionality constants are, maying seeing if you can find a pattern in them. For example, if you plug in 8 to the power of a very tiny number minus 1, and divide by that same tiny number, what you’d find is that the relevant proportionality constant is around 2.079, and maybe, just maybe you would notice that this number happens to be exactly three times the constant associated with the base for 2, so these numbers certainly aren’t random, there is some kind of pattern, but what is it? What does 2 have to do with the number 0.6931? And what does 8 have to do with the number 2.079? Well, a second question that is ultimately going to explain these mystery constants is whether there’s some base where that proportionality constant is one (1), where the derivative of “a”to the power t is not just proportional to itself, but actually equal to itself. And there is! It’s the special constant “e,” around 2.71828. In fact, it’s not just that the number e happens to show up here, this is, in a sense, what defines the number e. If you ask, “why does e, of all numbers, have this property?” It’s a little like asking “why does pi, of all numbers happen to be the ratio of the circumference of a circle to its diameter?” This is, at its heart, what defines this value. All exponential functions are proportional to their own derivative, but e along is the special number so that that proportionality constant is one, meaning e to the t actually equals its own derivative. One way to think of that is that if you look at the graph of e to the t, it has the peculiar property that the slope of a tangent line to any point on this graph equals the height of that point above the horizontal axis. The existence of a function like this answers the question of the mystery constants and it’s because it gives a different way to think about functions that are proportional to their own derivative. The key is to use the chain rule. For example, what is the derivative of e to the 3t? Well, you take the derivative of the outermost function, which due to this special nature of e is just itself and then multipliy it by the derivative of that inner function, 3t which is the constant, 3. Or, rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t and how that intermediate change nudges the final value of e to the 3t. Either way, the point is, e to the power of some constant times t is equal to that same constant times itself. And from here, the question of those mystery constants really just comes down to a certain algebraic manipulation. The number 2 can also be written as e to the natural log of 2. There’s nothing fancy here, this is just the definition of the natural log, it asks the question, “e to what equals 2?” So, the function 2 to the t is the same as the function e to the power of the natural log of 2 times t. And from what we just saw, combining the facts that e to the t is its own derivative with the chain rule, the derivative of this function is proportional to itself, with a proportionality constant equal to the natural log of 2. And indeed, if you go plug in the natural log of two to a calculator, you’ll find that it’s 0.6931, the mystery constant that we ran into earlier. And the same goes for all of the other bases. The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base, the answer to the question, “e to the what equals that base?” In fact, throughout applications of calculus, you rarely see exponentials written as some base to a power t, instead you almost always write the exponential as e to the power of some constant times t. It’s all equivalent. I mean any function like 2 to the t or 3 to the t can also be written as e to some constant time t. At the risk of staying over-focused on the symbols here, Ireally want to emphasize that there are many many ways to write down any particular exponential function, and when you see something written as e to some constant time t, that’s a choice that we make to write it that way, and the number e is not fundamental to that function itself. What is special about writing exponentials in terms of e like this, is that it gives that constant in the exponent a nice, readable meaning. Here, let me show you what I mean. All sorts of natural phenomena involve some rate of change that’s proportional to the thing that’s changing. For example, the rate of growth of a population actually does tend to be proportional to the size of the population itself, assuming there isn’t some limited resource slowing things down. And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water. Or, said a little differently the rate at which that difference changes is proportional to itself. If you invest your money, the rate at which it grows is proportional to the amount of money there at any time. In all of these cases, where some variable’s rate of change is proportional to itself the function describing that variable over time is going to look like some kind of exponential. And even though there are lots of ways to write any exponential function, it’s very natural to choose to express these functions as e to the power of some constant times t since that constant carries a very natural meaning. It’s the same as the proportionality constant between the size of the changing variable and the rate of change. And, as always, I want to thank those who have made this series possible.

I. M. Pei at MIT – Tech Day 1994

speakers will, in fact, talk about a specific
relationship between mene et manus, between
art and engineering, in the world of architecture. We are very honored
today to have with us architect IM
Pei, class of 1940, formerly with the firm of
Pei Cobb Freed & Partners. Of course, Mr. Pei
is very well-known for many, many buildings. And I will not list them all. Some of them are particularly
important for the arts, including, of course,
the Louvre, the East Wing of the National
Gallery at Washington DC, and many MIT alums are familiar
particularly with the Meyerson Hall for the Dallas
Symphony Orchestra, a remarkable building. At MIT, Mr. Pei has not only
designed the Wiesner Building on Ames Street but the arch
to that building, which is a gateway into East Campus
as you cross Ames Street, also– not, I think coincidentally– when you are walking
toward the campus, frame his three buildings
in the main quadrangle with, in perspective,
the MIT dome overall, the three buildings on
the main campus with their very stark geometry of verticals
and horizontals, triangles and rectangles. Most recently– or not most
recently, but among Mr. Pei’s most recent works– are the Bank of
China in Hong Kong and the Four Seasons
Hotel in New York City. And I think we will be hearing
about some of these works today. Mr. Pei has also been
affiliated with MIT since 1972 as a member of the Council
for the Arts at MIT. And we appreciate his years
of membership on that group. Talking to Mr. Pei today will
be Dean William Mitchell. Dean Mitchell has been
at MIT since 1992, and he is particularly
well-known for his book, published by The MIT Press,
The Reconfigured Eye, which deals with digital
photography and, in fact, visual images that do not
represent the real world. [LAUGHING] Dean Mitchell is
working very hard in the School of Architecture to
lead the School of Architecture into the 21st century
with his Virtual Design Studios and The Design
Studios of the Future using computer technology
to create a seamless process from the initial designs
through to the communications with clients,
engineers, and others. Please help me to welcome IM
Pei and Dean William Mitchell. [APPLAUSE] MITCHELL: Redoing the Louvre– that sounds much better. Redoing the Louvre
is an extraordinarily large and complex project
involving technical problems, organizational problems,
political problems, certainly, problems of very complex
cultural symbolism. So what we thought
we’d do today is to ask Mr. Pei to describe
a little bit of his thought processes, what the problem
was about, the solution that he finally derived. And then, we’ll take some
time to discuss this truly magnificent project. So Mr. Pei. PEI: Professor Morrison talked
about hundreds of centuries. I’m talking today on the
subject of the Louvre about eight centuries,
the history of 800 years. But these are very important 800
years for the French, at least. [LAUGHTER] It’s the building of the period
during which French nation was born. It’s a period that,
I would say, that’s roughly parallel the
recent French history. In fact, it’s also a wonderful
symbol for them, at least, of French civilization. So therefore, the problem of
the Louvre for the architect is not just a technical
problem, architectural problem. But it’s a problem that
has many, many challenges. I would like to start
off by telling you something about the
history of the Louvre. Now, I don’t have too
much time, but I’ll try to make it very brief
and, therefore, incomplete. It started in 1202 by, I think
that you can call him a French King, Philippe Auguste,
who built this fortress on the right side of the sand
to protect the Ile-de-France, which is now the Il de la
Cité, which is the place where the heart of French life at
that time in Paris took place. It was built as a fortress,
le donjon, as they call it. And it’s important
because I wish I’d taken some slides to show
you what it is like today. But anyway, it was
a wonderful building where they put prisoners,
where they put ammunition. It’s a very important
place to protect, let’s say, something
that they considered to be very important. It lasted as a donjon,
or as a fortress, for roughly, I would
say, maybe 200 years. It did not become really
a place for the Kings to live in until about
the end of 14th century. And I think Charles V was the
king that one should remember. Because that’s the
beginning of Louvre, not only as a place for the
king to live in but also the beginning of French art. The first library
of France was there, and the collection of many,
many objects of Charles V was displayed there. Now, for how many
years since then– 1400 to today? I would say 500
years, 600 years. Nearly all the French kings that
you know of of any importance have either lived there,
died there, married there, and born there. So therefore, it’s
an important place. So therefore, this is
not really surprising that when someone like myself– I guess I can consider myself
an American of Chinese descent– has to tamper with a
very important part of French history. So I’m going to go
through this in two phases to show you what I went through. Phase one lasted five
years, from 1983– six years– to 1989. And during those
six years, the plan was laid, how to deal
with the problem. It’s a very complex problem. And the pyramid was built. But before the pyramid was
built, we had a lot of trouble. We wasted two years. It’s a media that I had
to deal with at that time. And I was totally unprepared. And my French language is just
not adequate for that purpose. But I had to deal
with the media to try to convince the French that
this was the right thing to do. So therefore, even though
it took six years, but only four of those six
years were devoted to architecture and the
building of the phase one. Now, phase two is
less spectacular in the sense of
public information but is perhaps the more
important of the two. Because it completed a
wing called Richelieu. And without that wing, there
would be no Louvre today. And we couldn’t get
that wing until 1985. The wing– I tell
you what the wing is. The wing was occupied by
the Ministry of Finance ever since, oh, I would
say Napoleon III. [LAUGHTER] Only 200 years– 150 years. No, less– 1856– 150 years, yes. And they refused to move. [LAUGHTER] And at that time, the
pyramid is already discussed. And people say, OK,
if you must have it. Yes, Mitterrand must want it. OK. But I said, look
here, there’s no point to build the pyramid if we don’t
have the Ministry of Finance. So that gave the president
another headache. [LAUGHTER] But long story short, we
moved them out in 1989. So the second phase is the
building of the Richelieu wing. It’s architecturally not,
of course, spectacular but very important. Because why? Because I have to
keep the facade. The only thing I
could do is inside. And we demolish
everything inside except a big suite of
Second Empire rooms– very beautiful rooms,
you must go and see it– and two or three staircases. And the rest of Richelieu
was completely gutted. So it was a major
engineering project but not very spectacular, except there
are some interesting things inside, which you will
see when you come. So because of time, I’m going
to start with phase one quickly. MITCHELL: You have
your slide changer. PEI: Oh, yes. I’m going to have to
learn how to use this. MITCHELL: Press the top there. PEI: Nope. Oh, yeah. MITCHELL: Yes. And this one for the other side. This one here. PEI: Excuse me. MITCHELL: We have to sort
out the technology here. PEI: Oh. Oh, I see. MITCHELL: There we go. PEI: Oh, I see. Now I understand. OK. Now Louvre, as you
know it, perhaps without too much debate among
architects and planners, is perhaps the most important
urban composition in the world today. There’s no other that
can compare with it. It’s history– 800
years in the building. It started as a
fortress and then was added on by kings after kings. But the important
kings to remember that built the
Louvre are probably, I would say, Charles V,
Francis I, Henry II, Henry IV, Louis XIII, Louis
XIV, and eventually, Napoleon I and Napoleon III. And there are many other kings
in between that added something to it. But those were the
important kings. And it can also be
said that, perhaps, every important French
architect– nearly every, from Lescot to Mansart to
Le Vau under Louis XIV– many, many others– and then
eventually Lefuel, Visconti– they all participated
in the building of it. So therefore, it’s
not surprising that if you come into this
wonderful complex that’s already formed and to try
to do something with it. Now, the reasons why something
had to be done to the Louvre is for the reason that you
know, that it was built first as a fortress, and
it was added on and added on to try to make
it more comfortable for king after king. And to turn it into
a museum in 1793, the convention is a
move that was correct– interesting– because
it became Louvre. It became a public museum for
the first time 200 years ago. But it was not at all
suitable for a museum because it was meant for
life, for kings to live in. So the Louvre has never
really worked as a museum. I was there for the first
time on a fellowship– not MIT but Harvard fellowship. [LAUGHTER] I did receive MIT fellowship,
but it was during the war. I couldn’t do anything with it. I live across the
street from the Louvre in a very tiny, little hotel. And I went over there
every day to look at it. And I’ll tell you,
in those years, you really have to have time
to see the Louvre because you get lost in it and you don’t
know where anything was. And there are no
toilets, no restaurants– nothing of that sort. But it had a
wonderful collection, and you have to go back time and
time again to find surprises. And that was the way Louvre was
to all of you, to many others, until something
happened in 1989. So therefore, Louvre is a
wonderful complex of buildings. But Louvre Museum was not. Louvre Museum happened to
be a tenant in the Louvre. That’s all. It was occupied by the
Ministry of Culture running the Museum of France. It was occupied by the
Ministry of Finance, occupied by many, many others. Louvre Museum occupy a long,
long wing along the Seine. And it’s about, I would
say, 800 meters long. And that wing was
almost impossible to go from one end to the
other without going up and down and there. So therefore, most people
who went to the Louvre, as I did, probably only
saw, maybe, 25% of it. And the rest, you just miss. You have to have a guide. And I didn’t have a guide. So therefore, I had to
go back time and time again to find new
things, new surprises. So Louvre did not
work as a museum. And the French knew it. But they wanted, finally,
under Mitterrand, to do something about it. And the one move that was
perhaps the most important move of all I mentioned earlier. It was the recapturing
of the Richelieu wing from the Ministry of Finance. [LAUGHTER] Now, this is perhaps
not a very good slide. I apologize. But you can see the
importance of Louvre in the heart of Paris. Because it’s situated perhaps
in the most important– really in center– heart of Paris. And yet, it separated the
left bank from the right bank. There’s no way to
penetrate except by car from left bank, which is
below to the right bank. So it became,
actually, a barrier. And that became an urban design
problem of first magnitude. Because we have to
open up the Louvre. To open up the Louvre, then you
rejoin the two parts of Paris, not by car– you can always drive
around– but on foot. And this is one of
the challenges that is least talked about but
has, in my opinion, the most important. The access of the Louvre leads
all the way to Saint-Germain. And it all, probably,
is, again, when I say the most important
urban composition, I really should say that
it’s the most important axis of the world. There’s no axis
like that that leads all the way from the
Louvre to Saint-Germain. It went through the Garden of
the Louvre, which you all know, Place de La Concorde, up
Champs-Élysées to the Arc de Triomphe and then on to La
Défense and then La Défense [? beyond. ?] Now, this is a
diagram of the Louvre to show you the gray area on top
was occupied by the Ministry of Finance. [LAUGHTER] The ochre, the yellowish
color wing, 800 meters long– half a mile long– occupied not just by the
museum, by the museum and by the museum
administration of France. And French bureaucracy, I tell
you, takes up a lot of space. [LAUGHTER] So it’s not all museum. You can see that people come
by one side of the courtyard and enter there
while you’re waiting in line if you’re lucky– a good day. You don’t mind,
but then you’ll be pestered by people trying to
sell you all kinds of things that you don’t want. But nonetheless, the people– three million people–
continue to come. Why? For this collection. But you really have
to go through, really, a very uncomfortable
experience in the process. Now, what I saw
at that time when I was asked by the
French government to say, can you do something
about the museum? I said, I don’t know. But anyway, I’ll try. And the one thing that
became apparent to me, when I look at this problem,
I said the Ministry of Finance has to go. [LAUGHTER] And I said, the
reason is simple. I said, when you want to
build a modern museum, Louvre, why it is not a modern museum– for simple reason. And that is in modern museum– which I knew well then because
I finished the National Gallery already– about 50% of exhibit
space has to be matched by 50% of supporting spaces– reserve, conservation
laboratories, restaurants, auditorium, lecture halls,
public reception spaces, toilets, things like
that– which it did have– tiny little toilets. In fact, I remember very
well that when I went there, I frequently had to leave
the Louvre because I have to find a place to go. And then, when you
leave the Louvre, some people don’t
come back again. [LAUGHTER] So they lost a lot of people. And it’s not really surprising
that the average stay of visitors to the Louvre
is only one hour and a half, whereas the National Gallery
is three and a half hours. Metropolitan Museum–
about the same. So I said that the
key to making Louvre function as a museum is to
change the Louvre from 800 meter long– up and down, up and down– to something very compact. But you can give up this
wing, the floor where the Spanish paintings were– or still are– and
give it to other uses. But you must move the
Louvre to that part. And that way, you
accomplish two things, which is essential to making
Louvre Museum function. Number one, you can
excavate that court– which is in gray area– the Napoleon court. We can go down two levels. It goes to the
level of the Seine, which is about 10 meters down. You can go down two levels. And you can recapture half a
million square feet of space just by excavating that. And why not? You can use that
space for reserve. You can use that space
for all the infrastructure support that a
modern museum needs. Because under the old Louvre,
there’s no foundation. There’s no space. They have some pipes
and that’s about all. But nothing there– there’s
some sewers that go through. But that is the key
to making the Louvre into the modern museum. Second reason is that if you
put the center of the Louvre not on one side, down
below, but really in the middle of that
gray area, you’ve got the center of gravity
of the Louvre right there. And from that point, you
can go to the three wings. Louvre had three wings. Do you know? Venus de Milo is there,
and Mona Lisa is there. And to Richelieu and to
Sully, very short distance– not 800 meters anymore. It’s going to be only
about something that’s 50 meters, the difference so
that you can divide Louvre into, really, three parts, even
though they’re interconnected upstairs– three
parts, and each part is at least several days’ visit. So a big museum, you
may wonder whether you need a museum this size. But be that as it may,
it’s a big museum, and that’s the only way
to solve the problem. So therefore, I
told the president, I said, you’ve got
to do two things. You’ve got to let us
dig under the court. Because there may
be relics there. The project could have
been stopped if they found something important there. And then, you also
have to find a way to move the Ministry of Finance. [LAUGHTER] At this point, it’s not
so funny at that time. [LAUGHING] But this tells you something
about the man who is still the president of France. I don’t know of
any heads of state in the world that has the kind
of breadth of understanding of the history and
culture not only of France but of the world. He’s really a remarkable man. He may not be the
greatest politician. I don’t know. Time will tell. But he was and he is a really
exceptional leader when it comes to art and culture. And I could not have done
anything there without him. That’s obvious. The court, before we
did anything with it, was a parking lot. It’s a parking lot for the
bureaucrats that occupied the Ministry of Finance. [LAUGHTER] The court, after excavation– one year, archaeologists were
digging, really, literally, with brushes. And they found
things, all right, but nothing terribly important. Because this court,
unlike the court which I wish I had slides to
show you with the marvelous donjon foundation,
is now on display, which, when you
go there, you see. This court was inhabited by
people who served the kings. They’re bakers. They’re cobblers. There’s a little church. And that’s about all. And therefore, there’s
nothing that is so important that they say stop. We were very fortunate. From ’84 to ’85,
that year, we were waiting to see what happened. We wish they discover nothing. And the archaeologists,
of course, wish they had found something. But anyway, they found
nothing of importance, and we proceeded to
get the green light. Now, here I want to show you
the organization of the Louvre. This is the court. The rest of the
Louvre’s not shown here. It’s only the Napoleon Court. By putting an object– something– in the
center of this court– let’s say Napoleon Court– you can connect
the three pavilions in a very short and direct
way and very understandable. You can see it. When you are in the court
or in the hall below, you can look out, and you
can see Denon, Richelieu, and Sully. And that clarity of orientation
is key to a big museum. This is an urban study. So to try to ventilate
Paris, you must first ventilate the Louvre. To ventilate the Louvre, you
have to do several things. You can come over from the left
bank over Pont des Arts, which is a pedestrian bridge– a wonderful bridge,
architecturally. From engineering point of
view, it’s a wonderful bridge. And then you enter into Cour
Carrée and then turn left. But you have to do two things
in order to make it truly open. One is that you have to open
that passage under Richelieu, called Passage Richelieu,
to lead you to the Place du Palais-Royale, which is right– and then, from then
on, to Palais-Royale, and then to Pompidou
Center, which you know. The second thing
that you must also do is to open up the garden. A new bridge is
being built. I think it’s about to start construction
here so that you can walk from Orsay, which is where
the wonderful Impressionist collection is today, from Orsay
all the way to the Louvre. And that will open
up the Louvre. And once that Louvre is opened
up, it no longer is a barrier. It becomes a connector. This was a drawing done
by Steve, who’s here, Steve [? Oves. ?] I show this
drawing to the president. And he was not shocked. Perhaps he was the only
man that was not shocked. But when I show it to the
press, they were shocked. And I tell you, from
that point on, as I say, 18 months of just nothing
but harassment by the media. But still, we got
the green light. And the president, as
well as his official, who was in charge of the
Louvre– an important man, Biasini– said, let’s dig. Because the archaeologists
has already said it’s OK. Let’s dig. And this was what you see. And that was the beginning
of another furor and people demonstrating and everything
except they didn’t do anything really serious
like committing suicide and like that, but very close– very close. There were a few. [LAUGHTER] Now I’m going to go
quickly because of time. Quickly, to show you how
the building process did. And the building of this
palace was remarkably fast. The French contractors
are incredibly good. They’re very, very good. These are slides that
shows the construction. This is Napoleon
Court, by the way. This is not the rest. It’s only one part– the important part. This is the part where
the pyramid must emerge. And the square grid that you
see, the structural steel member there, they’re
all on bearings. It can move. [CHUCKLING] It can move. And they’re supported on
four columns, and that’s all. We had about 60 or some slides
because from one vantage point, they took pictures
every so many months. But I’m only showing you a few. So that’s very quick. But it took– [LAUGHTER] [APPLAUSE] Thank you. It took 18 months
to reach this stage. And at this point, I would
say, it’s the end of phase one. It was open in spring 1989. And Mitterrand was
there to cut the ribbon. And it was rather
well-received by then. But still I would say
there were, maybe, 30% or 40% people still
not very happy with it and for any number of reasons. But nevertheless, it was built. And since this is MIT, I thought
I have to bring this slide. If I show it to
another audience, they would not be
interested in it. And I’m going to show
you some of the slides of the construction. But one thing that
should also interest you that the French government
make no secret of it, they want everything
to be made in France. And the only exception,
which I asked the president to allow us, is to have all
the tension elements made in Massachusetts by a small
rigging company that did the America’s Cup, the yachts. I don’t know whether
Bill Coke is here or not but he would know. [APPLAUSE] And we are the best. And I showed him some
samples of the rigging, how the 10 buckles fit into
the cable and all that. And he was fascinated with it. And because he liked it so
much that he didn’t even bother to say, is there a
French equivalent to it? We just say, go ahead. So we did it. And on the other hand,
the glass problem. You see, glass has to be
white, has to be clear. If the glass is not clear– this is laminated, you
know, it’s double layer. It’s very thick. It’s about almost
3/4 of an inch thick. If it’s not clear,
it will be green. And you will see it through
the corner of this glass. It will be very dark
green, bottle green. So therefore, that is not
acceptable because the Louvre, you see, the composition
of the Louvre must be seen. And the ochre color stone
shouldn’t look green. So therefore, I requested
the French manufacturer, Saint-Gobain, to
make this glass. They say no. We no longer make them. And then, finally,
they say, well, if you built 1000 pyramids,
we’ll make them for you. So I didn’t report
this to the president. I went to a German
firm, Schott Glass. I say, can you make it? Yes, we can. Saint-Gobain said,
we’ll make it. [LAUGHTER] I’m going to go very quickly– the making of the pyramid. The pyramid has one virtue. It’s a very stable form. You know that. And consequently, it requires
the least amount of steel in order to support it. And consequently, if you
use the finest technology available to you,
technologically, to build it, it will be the most
transparent form. And it must be the
most transparent form because you want to
see that composition. That composition is so
important to the world, not just to France, So you may ask, do you need
something that projects? Many architects in
France, as well as abroad, suggested, why not just have
a glass sheet on the ground, and you bring light
in just as well? But I said, no. I said, you have to have space. You go into the lower level 10
meters down, nine meters down, and you don’t want to
have a glass ceiling. That word may mean
something to some. [LAUGHING] You don’t want a glass ceiling. You want space. And so something has to project. And I defended the pyramid. And the pyramid, for those
of you who know France, is a very important
symbol to French. I think the French are probably
more conscious of pyramid as a form than any other people
because of Napoleon, I guess. They really went to Egypt. They took a lot of
things away from Egypt. So the pyramid form is
necessary to give you space. So give you light, give you
space, but at the same time, transparent. You can see through it. And also, you need a symbol. Because if this is going
to be the main entrance to the Louvre, it cannot
be just a subway entrance, so simply cannot be. Glass was put in this manner. And for a long time,
there was a debate. Can you clean the glass? And we tried different ways. They hired some Indians
from Canada to clean it. They even tried robot. Eventually, we clean by robot. But finally, we
got some alpinist to get up there, hang the rope
from the tip of the pyramid, and wash it. And it only took
two days to wash it. But now, it took less. Now, one day is all we need. So the washing problem
no longer a problem. But for a while, that
became also a problem. Because the French want to find
any reason to object to it. And cleaning the pyramid
also kept us quite involved. Now, you’re inside the space. And you can see the Louvre. You can see the
Louvre through it. Ah. That was D-day. [LAUGHING] But no booing– no booing. Lots of applause, no booing. Ah. I like that photograph. You don’t see anything. There’s a story
about this statue. During Louis XIV’s time– I’m short of time. I would try to go quickly. Louis XIV’s time, they invited– and this shows that in those
years, the kings of France are already very
cultured people– he invited Bernini
from Rome to come. Bernini had just finished the
St. Peter’s arcade, that wing that enclosed that space. Perhaps [INAUDIBLE],,
he and Borromini were the two most important
architects of that time. And the French king
wanted the best, so he invited Bernini
to come to do something in the back of the Louvre–
not the this part– in the back of the Louvre. And he was there six months. And he did not survive
the French architects. And the only thing he left
behind– there a few things he left behind,
not of importance. But the only big thing
that he left behind is a statue of Louis XIV. But that was made in Rome,
and it was shipped to France afterwards. But Louis XIV never liked it
because his image at that time was 25 years ago. He was a young man then. He’s no longer young. So he banish it to Versailles,
and it remained there. And because it was not known,
it was never vandalized and never broke– I mean it was never
destroyed during the war because they just left there. But it was marble. It was vandalized by
a man from Brittany– anarchist, I guess– and
no longer salvageable. But I persuaded the
conservator of Versailles to let us make a cast of it– the only way. And they did. And it’s made in lead. And I thought, we should
put it there to remember him as someone who tried. [LAUGHTER] But that very important,
that location. I needed something there
to terminate the access of Chance-Élysées because
that axis of Le Notre– the Le Notre axis– was not terminated at Louvre. Because the Louvre is a
little bit like this– like this. So it has to be terminated–
not by the pyramid, by something else, and
something that’s strong. And I was fearful of
commissioning a French sculptor at that time. You don’t know what’s
going to happen. I’m perfectly frank about it. You just don’t know what– so that was my escape
from responsibility. Now, the second
phase of the Louvre– not spectacular, not
at all polemical, but extremely important. Because now, it opened
only a few months ago. If you go there
now, you understand why the pyramid was put
there in the first place. And I think the vindication
of the whole plan is now made possible through
the completion of this wing. And for that reason, even
though architecturally, it’s not spectacular because
the facade had to be kept and everything has
to be internalized, done inside– number
two I had to work with two French architects,
which is reasonable. Because after all,
I can’t hog it all. Now, you see under
the Napoleon Court, and this is what it looks like. This is 10 meters below ground. You go up three sets of
escalators to a intermediate level– still below ground– and you can enter into the
three wings of the Louvre. And the fourth wing,
going this way, goes to shops,
parking, bus terminal. Eventually, all the buses
that you see on Rue de Rivoli, as well as on the
[? Quai, ?] will disappear, as they have now. They’re all underground now. So urbanistically, that’s
another very important contribution. And below, at this lower
level, we have auditorium. We have restaurants. We have a reception area
for the young people. And we have a bookstore–
enormous bookstore– and the shops and meeting
rooms and conference center. Everything is there. But below this level,
it’s all circulation. There is at truckway that
connect all the departments underground. And there’s a large reserve
so that all the collection– nearly all the collection of the
Louvre– now comes back home. Now, we have to talk briefly
about Richelieu wing, even though,
architecturally, as I say, it’s not very spectacular. The two courts were proposed
way back to be covered. They were parking and trucking
for the Ministry of Finance before, so not used. By covering it, we can
turn it into an exhibition space for French sculpture. And further, we also proposed
to dig down below the Richelieu wing, as you see, so
that the two courts are connected at the lower level. And this was a
trucking area, which now become exhibition
area for French sculpture. This wing house
four departments– the sculpture department,
the oriental antiquities department, the objet
d’art, which is probably the most important objet
d’art collection in the world, and paintings. I put this in mostly,
again, because this is MIT. This is a pyramid, about
50 feet square, inverted. Why inverted? Because its positioned
at the place where the circular rotary is. And you don’t want to see any
projection there coming up. Because one pyramid
is quite enough. But to use the same theme– and we wanted to
bring light in– this is the intersection. If you go to the bus
terminal, to the parking, and to the shops, you have
to go through this point. So therefore, it’s
nice to have something to make people feel they’re
still in the Louvre. So now, this suspended pyramid
is a major engineering project. It was designed by
a man, Peter Rice of Arup Associates in London. And too bad I don’t
have the drawings. It’s actually a very
brilliant design. The whole thing, there are only
four rods in the center, all suspended. And the rest are all cables. It’s cables and four
rods, and that’s all. And it has one other
very exciting byproduct. [LAUGHTER] Clearly, I was very proud of it. Another very important
byproduct of this is the prismatic
effect of the glass. You see the glass,
because we don’t have to keep water
or rain out of this. Because inside, we can
polish the edges of the glass and bevel it. By beveling it, the spectrum
of the colors came out. So at times you see,
a sunny day like this, it’s just a rainbow inside. It’s really quite spectacular. If you go there, make sure you
go there on a good, sunny day. Now, the Richelieu
wing, there was one thing has to be done
to the Richelieu wing. Because it’s about the
painting collection is perhaps the most important
French painting collection in the world– not
perhaps, definitely. And yet, people don’t go there. Because the French
conservators want it upstairs because of daylight. They are very, very
insistent on using daylight– our conservators are less so– very insistent. And for that reason, they
take the attic space. They have a ceiling
not very high. But to get up there,
you have to walk up 75 feet, vertical space. And most people
don’t walk up there. So they miss a lot of visitors. I was told only about, maybe at
the most, 10% or 15% of people go up to the top floor. And that’s a great pity. So I proposed to
put in escalators. I hate to do that. It’s a 19th century
building, and you don’t install
something like that unless you have a good reason. That was a big battle. But it was won. And today, nobody disagree
that it’s absolutely needed. Because otherwise,
the interconnection up and vertically, it’s
very, very difficult. This is objet d’art. I show you the
sculpture actually. I show you the object d’art. These are the Maximilian
tapestries, never shown before– no place to show it. Now, they have a place. And it’s a must. The Maximilian tapestries
are a must to see. They are very dimly lit because
of the color of the threads. Now, before I go
into this, this work here was done by architect
Willmotte of France– of Paris, France. And my role in the
Sculpture Garden and this is what they call,
really, a [INAUDIBLE].. It’s sort of like a coordinator. I participated in all
the decision-making. I chose them as my architects. I had that responsibility. But they should get
the credit for it. Now, this was old Louvre. The lighting of paintings is a
really, a very special science. It really is. And it’s extremely important. It’s never studied enough. It just hasn’t been. We don’t have good daylight
galleries in America. We don’t have it in
the National Gallery. We don’t have it in the
Museum of Fine Arts in Boston. They’re all like this. They are laylights
with skylight on top, even though it could have
been artificially lit, and you wouldn’t
know the difference. And another disadvantage is that
the skylight is very bright. And therefore, the
brightest is the ceiling. The second brightest
is the floor. The walls, where you want
the light to be, looks dark. And consequently,
we decided this is something we want to
do a piece of research on. And I think one of the
major breakthrough, I consider, in this
wing is in the lighting. The lighting solution is this. We make the ceiling
into three layers. The first layer’s glass,
skylight, with a UV filter, of course. And then, below the
glass is an egg crate. And egg crate is
carefully calculated so that the orientation
is such so no direct sun rays will come in. It would have been better
if they were movable. Because then we get good light
all year round, all seasons. Unfortunately, the
French have experience with maintenance crew. They say it won’t work here. [LAUGHTER] And they are right. They’re right. So we use fixed louvers that
don’t have to be touched. We cut off a lot of
light– a lot more light than we wanted to. But it does remove the
headaches of the [INAUDIBLE] that was doing that job. And the reason of
the big cross is that people have to walk there. So it has to be wide
enough for people to walk, to clean, to relamp,
and that sort of thing. But you can see the sky if
you are walking on one side. But the light, as you see,
now is deflected to the walls. It’s no longer coming
down to the floor. So the walls now are bright. And they get light. And that is turned
out to be something that– the French conservatives
are very conservative, and they accepted this. And they now claim this
is the best in the world. They have to– something
has to be, always. And this, they like. And this is a very
important suite of paintings by Rubens
celebrating Marie de’ Medici’s journey and eventually
to apotheosis. All the way, and
you can see again, the light is no longer
bright on the ceiling. It’s directed to the walls. And this is another
version of it. This is octagonal room. That’s a long room. And the previous one, I
think, is a square room. And this is the way
the light looks– no reflection, no reflection. I guarantee you that– no reflection. All right. Richelieu wing was
finished in November, open in November 1993, exactly
200 years after the founding of the Louvre. And the Richelieu wing, together
with the Napoleon Court, is now complete. And therefore, Louvre
finally functions as the way we had planned to. And some of these
slides are mostly to show you what it looks
like when it’s all finished. And young lady is
celebrating the event. So there you are. I think that’s– oh. I’ve used up too much time. Well, sorry. [APPLAUSE] I’m sorry.

Constructing the Circumcenter

one of the four main types of points of concurrency that we find in triangles is the circumcenter point of concurrency means you have at least three lines intersecting in one spot so the circumcenter is where the three perpendicular bisectors of each side intersect which also makes it the center of a circle that circumscribed about the triangle so if we take a look at a sketch of what a circumcenter might look like we notice that it could be a really big circle if you have an obtuse triangle and that it passes through all three vertices which means that the center is equidistant from the three vertices but when we ever apply this in real life well a common problem that you might see on a test is if they give you three different points and they say where's the treasure the treasure is located at an equal distance from three random points maybe a tree stump a gravestone and the beach so what you would do to find the treasure is you would have to find the circumcenter of the triangle by drawing those lines so what you would do if we erased this treasure is you would draw in your three sides of your triangle and then using your compass you would construct the three perpendicular bisectors of each side so there would be one perpendicular bisector and then here would be another perpendicular bisector and again this is just an estimate to show you how you would solve this problem and then here you'd find your perpendicular bisector and ideally by definition these points are concurrent as you can see I'm a little bit off but that's just because I was sketching so you would say that the treasure would have to be right here which is the center of a circle that passes through the three vertices

Construct Congruent Angle 2.m4v

let's construct a congruent angle I have angle XYZ it's a fairly large angle the first thing I need to do is give myself a nice straight rate to work from so I'm going to use my straightedge I'm just gonna get myself a nice ray okay so we made ourselves a nice rate of work from this could also be like our endpoint Y here I'm going to take my compass and I need to start from Y open the compass a comfortable amount and draw a nice arc through both rays now I'm going to keep the same compass setting it's really important that I not change that I'm going to come down here put my compass point on my new Y and I need to make a mark that's visibly larger than the other one see how much bigger that is now what I'm going to do is use my compass as a measuring tool and I'm going to measure this arc inside the two rays so I need to really be careful about adjusting my compass and see how I make that nice Construction mark there well I'm going to come down from this point of intersection I'm going to make a similar construction work and now this piece of art and this piece of arc should be the same length what that means is when I use my straightedge and my writing utensil going through this point through the point of intersection I have now created an angle here who is congruent to this angle here and since this is angle XYZ I could also call this angle XYZ because they are the same size they are congruent