The Neat Alignment of the World's Biggest Antiprism
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- เผยแพร่เมื่อ 27 ก.ย. 2024
- Check out Jane Street paid internships www.janestreet... (or pass on to someone you think would benefit from it).
Huge thanks to Laura Taalman for joining me for a day of walking and math. You can see all things Taalman on her website including excellent 3D print files: mathgrrl.com/
Next "An evening of Unnecessary Detail" show is 20 November 2023 in London. More shows in the future in the UK and USA, ticket links always here: fotsn.com/tickets
Thanks to "biludavis" for the 3D model of the WTC: www.thingivers... All other 3D models and prints were designed by Laura.
Huge thanks to my Patreon supporters. They put the twist in my antiprism. / standupmaths
CORRECTIONS
- None yet, let me know if you spot anything!
Filming by Alex Genn-Bash
Math by Laura Taalman
Produced by Nicole Jacobus
Stills photography by Truman Hanks
Editing by Christopher Brooks
Sound mix by Steve Pretty
Music by Howard Carter
Design by Simon Wright and Adam Robinson
Written and performed by Matt Parker
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinran...
UK book: mathsgear.co.u...
“Proof by not on the internet”
“Proof by thinking about it for a minute”
I’m really liking these in depth proofs that were getting in maths now
In other walks of life there exists the "proof because I want it to be true".
When I was a little kid I employed what I later realized was "proof by poking you until you agree with me"
Petition to turn "proof by thinking about it for a minute" into a real thing
And "proof by not on the internet".
a "parker proof", if you will
A petition to completely dismantle science? That could catch on. Oh wait...
I think these would be more like your everyday kinda proof. Nothing you write down in your science paper but something you say when someone just keeps talking nonsense because they acually havent thought about it for a minute
Nah, the UK government would say it would distract drivers too much.
Architect here. One of the best arguments for making an Anti-Prism shaped building is designing around lateral loads (more specifically wind loads). As you build taller, wind loads becoming a much larger factor in building design. One way to design for the increased wind load without adding more bracing, is to rotate the structure (This is also why a lot of skyscrapers "twist"). With an anti-prism, you break up the large surface of each facade, and achieve a similar effect.
Much nicer than the corkscrew fins found on large chimneys. I hope chimney designers could use these antiprisms.
Architects don’t design big buildings they are unqualified. You need a structural engineer.
@@NeverFinishAnythi Different Architect Here: Architects are very much involved with the design of buildings, both big and small, including skyscrapers. However, except for the smallest of buildings, we never design a building alone. There will always be a team of consultants who work together with the architect to make a building, with each consultant focusing on their area of expertise. One of those consultants is always a structural engineer, who is primarily charged with keeping the building standing. Other consultants include electrical engineers, plumbing engineers, HVAC engineers, civil engineers (for site work), landscape architects (for plants and irrigation). Additional consultants can be brought in if a project needs it. The architect works to organize the consultant team and drive the overall design, while also choosing finishes, dealing with life safety items, dealing with accessibility items and numerous other things. The larger and more complicated a building is, the more the architect needs to rely on their consultants and the less likely an architect can just do whatever they want. So an architect likely came up with the design for the anti-prism shape, then worked with the structural engineer to figure out how best to achieve that look.
And seemingly by using an anti-prism you get almost the same internal volume you'd get from just using a rectangle.
@@NeverFinishAnythigenerally they're designed by architects then the engineers fix the designs.
These units are mind-boggling. Matt converting his feet to inches and then back to another kind of feet.
And I still have no clue what the volume of the building is, 33 million cubic feet? 🤷♂️
And thats quite a feat
@@kj_H65f and thats quite a feet
neatly, this is basically one million cubic meter, which is 0.001 cubic kilometer !
@@DantevanGemert It's funny how it round down to a million cubic meter.
I wonder if the official figure of 200 feet is actually a measure of the internal dimensions of the building, so the extra 5 feet or so that Matt and Laura measure would be the combined thickness of the walls on either end. Or I guess it could just be that 200 feet is a very rounded (or truncated to 1 sig fig) measurement.
Same thought. Their measurements are more trustworthy than a 'front page' published size.
One of the… “fun things” with engineering projects are the as-builts - the drawings that are updated to show what was actually made
@@dobystoneSome say you can hear the architect screaming whenever you view them. 😂
Matt, I rarely comment on TH-cam but I'd like to thank you for the candid and honest way your videos are made. Other math channels always made me feel a bit stupid and wonder how can they be sooo good at math as to never make any mistakes. You're by far the most sincere math youtuber I've ever watched, for you don't try to hide your mistakes, instead, you show us them and it helps a lot in the learning process!
This is crucial. Mistakes and mis-steps are an intrinsic part of the process. You tidy it all up afterwards. Presenting it as a _fait accompli_ does no-one any favours. To take maybe a cliched analogy, it's like presenting a completed jigsaw and pretending you didn't make any guesses at any stage about which piece went where, they all just slotted in first time in the correct place with the correct orientation :-)
That’s the Parker Brand Promise.
I was coming to comment the same thing, but I'll just reply to boost your comment. It is great modeling that even brilliant people make mistakes, and that those mistakes are not the end of the world. You accept the correction, fix your work, and move forward.
You know, I thought "world's biggest antiprism? how do you know?" and then realized it'd be _really_ obvious if there was a bigger one, so fair enough :D
why obvious?
have you kept track of every sky scrapper in the world?
@NoNameAtAll2 the big ones? Yeah, lots of people keep track of that.
@@NoNameAtAll2It really depends on what he means by "biggest"
In this case, I think he means tallest?
And if he means tallest, then you just have to look up a list of buildings that are taller, and verify they are not antiprisms.there probably isn't very many.
@lunasophia9002 it would be on the internet if there was a bigger one. Proof by ‘not on the internet’ ;)
@@NoNameAtAll2 The point was it'd be hard to miss something that big. Also, no, I don't track every sky scraper (or sky scrapper), but the folks on Wikipedia do!
My first thought was that a true antiprism would have the same volume as a prism of the same height and top/bottom areas. But when Matt and Laura both disagreed with me, I was pretty sure I was wrong.
Actually, my very first thought upon first seeing Matt was "I really need to shave my head."
my first thought was - if i twist something, it contracts either in length or diameter. if you keep the length and diameter the same, the volume must increase.
@@supremecommander2398 I had a similar thought - the volume of a wire frame prism shrinks when you twist it since it get shorter, and even stretched its thinner in the middle because the faces aren't planes.
But when you add a diagonal wire to each face then it rotates into an antiprism with thicker cross sections meaning more volume. neat.
@@supremecommander2398 And if you keep twisting it eventually becomes a cylinder of the same length.
Me, during my whole education: "math is something that happen in dark rooms with old (usually beardy) people scribbling on kilometers of blackboards."
The video here: "let's get tons of people from everywhere and make math video in parks, outside buildings, during conventions…"
We should put more fun in math in school.
"proof by thinking about it for a minute"
exactly how I do proofs
This is how I proved One World Trade Center is rather large
My mental approach was to imagine that the number of sides of the prism (N) increased then the counterpoint of the the anti-prism would have 2N sides and thus be a closer approximation of a circle. As a circle has the highest surface area to circumference ratio the mid point of the anti-prism would therefore have a higher surface area to circumference ratio than the floors at either end.
I would also suspect that as N increases for the starting polygons then the boost in volume/area gained for making an anti-prism (over a regular boring prism) decreases.
I had same exact intuition, my guess is that’s what Matt thought too but didn’t know how to say it.
I also suspected this, and the limiting case is obvious since it doesn't matter how you twist a cyclinder so you get 0 volune gained.
I thought of that too, but the perimeter isn't constant, so there's no reason the perimeter-to-area ratio would necessarily be relevant. The central octagon should actually have the smallest perimeter of any cross-section.
@@evanhoffman7995 As you correctly identified, the perimeter of any cross section being constant is a big part in the proof. But that perimeter IS indeed constant:
All the faces that make up the perimeter are triangles. If you go X% up the structure, then the length contributed to the perimeter from the triangles that start at the bottom is X% of the perimeter of the lowest slice. And the perimeter contributed from triangles that start at the top is 1-X% of the perimeter of the highest slice. If you assume that the top and bottom face are identical, then the perimeter stays constant. Otherwise the perimeter varies linearly from bottom to top.
I was waiting for them to check the math by putting the model in water and seeing the displacement.
Yeah, if that's a scale model then Archimedes seems like a great way to get/check an answer - although the physical geometry of re-arranging the quarters is a cool realisation too!
Way too physics for this maths class haha
3d prints are mostly hollow, so it would just float and youd have a hard time getting an accurate measurement
@@sachathehuman4234 It would also fill with water so it wouldn't displace all that much...
You could also check the volume of each using the slicer software used to print it. At least PrusaSlicer tells you the volume, and I assume others do too.
I love seeing math nerds work together. It doesn't have to such rigorous tedious work to just figure something out for fun with a friend. Great video.
New York... the city of architexture
Normal people: New York City
Americans: Nooo yourkh siddee"
And at 1:00 he was "joint" by his friend.
That non metric system sounds feudal.
@@mildlydispleased3221L
as compared to (5/6)A*h, I computed that if the building was a frustum with bottom area A and top area A/2, then the volume would be (1/2+sqrt(2)/6)*A*h ~= 0.7357 A*h
88.3% the volume of the anti-frustum
“No views 46 seconds ago”
I feel so cutting-edge.
You could solve this problem by constructing two watertight models, one being a cube and one being the antiprism. Fill them up with water and then measure the volume of the water.
Should've waited till November 9th to post a very British video about the American One World Trade Center 😂😂😂
My instincts were telling me that going from the prism to the antiprism made the shape closer to a cylinder with diameter square root 2, which is larger than the normal rectangular prism, thus my guess was that the antiprism was larger.
A subsection of the Shanghai tower might qualify as anti prism and would certainly be taller than the anti prism subsection of the owtc
So, you have an anti-frustum with a larger square at one end and a smaller square at the other. To find the volume, you take two perfect anti-prisms, one with the larger square at both ends and one with the smaller square at both ends. You find the volume of each and then average together the two volumes.
More collabs with Laura Taalman please.
++
Great video! I love that you showed the process you went through to calculate the volume!
Yey I guessed it right. Because every triangle tilts outwards it must cover more space.
Conductivity enters the chat to speak to the Fresnel reflection and transmission coefficients to ruin the fun.
As a european person, when I hear or read “feet cubed” my mind is only capable of picturing some weird Minecraft-style foot
Instinctively, it makes sense the antiprism would have more volume, because if you break the normal prism into a stack of infinitely thin layers and give that stack a smooth twist so the top is offset by 45 degrees (or the appropriate angle for a different polygon), the vertical sides of the stack become concave. Antiprisms are convex.
The perimeter is independent of the height.
Therefore the octagonal cross section is bigger than the square section, because it has the same perimeter and more sides.
Incidentally, the radius of a sphere with the same volume as the building is only 5% larger than the width of the base of the building.
A way to quickly intuitively grasp why the antiprism is larger in volume than the prism if you have 3d modelling software handy (or if you are very good at picturing solids in space) is to intersect the two - the parts of the antiprism that "stick out" are visually obviously larger than those of the prism...
Have you checked this?
Just curious and without the software.
"My proof of thinking about it for a minute", is that I'd expect that there are also parts of the prism that would stick out too. 😅
It’s pretty easy to figure out the formula for the area of an octagon by dividing it into a square, four rectangles, and four isosceles right triangles. It’s also a nice and fairly easy calculus exercise to compute the area of the anti-prism and anti-frustum.
Or a square _minus_ four isosceles right triangles, if your octagon is based on a known outer dimension, rather than a known side length.
I watched the building go up, and have walked the underground tunnel adjacent to it, and I've sort of built a model using Magnetiles, but I'd never heard what the shape was called, and I did wonder about it.
So thanks, Matt, from a grateful New Yorker, who is now miffed he didn't know you were in town to attend your lecture.
In the tapered case, can you use the intermediate value theorem to show there is always exactly one horizontal slice which is a perfect octagon?
Would think so, each set of four sides of the octagonal intersection changes continuously (and in opposite directions) from zero at one end to the nonzero side length at the other, so somewhere in between there must be an intersection where they're all the same length...
Yes, it smoothly goes from an octagon with diagonal length of 0 to an octagon with orthogonal length 0.
In fact, since the missing shape is a pyramid, the change is linear (since the side length of a pyramid changes linearly with height), so:
d=1/2w*x
o=w*(1-x)
Where d is the diagonal length, o is the orthogonal length, w is the width of the base and x is the percentage up the tower, set d=o and 1/2x =1-x, or x=2/3, so at 2/3rd up the tower the floor area is a perfect octagon.
Never thought I'd say this, but that is actually a pretty neat shape
The fastest ethos to calculate the volume is the create it in autocad (or solids if you’re into that) and calculate the volume and other useful geometric properties
I've found that in so many things, just breaking down whatever it is into smaller, more manageable chunks (think: simpler shapes here), makes figuring out the "big picture" much easier.
One interesting fact I found: Starting from the base, the first 185 feet (85 m) of One World Trade Center are actually a perfect cuboid. Only after that does the actual anti-prism (and tapering) start
My instinct about the anti-prism vs cuboid was that it was the same size, but I knew I wouldn't trust my mental calculations. So I paused, and opened geogebra, learned to use it, and found the area grew quite a lot in the central octagon.
I initially agreed with Laura's guess and thought it was obvious. When it was shown to be wrong I realised that my mistake was probably that because the top is basically the bottom twisted 45 degrees, my instinct was to think of how if you actually twist the regular prism, the connecting vertical edges will cut into it and reduce the volume. I didn't think about there being two connecting edges for each vertex, not one, and how the second edges would add volume back in.
Love how excited you are! Wishing you luck on your presentations!
You can prove that the antiprism has greater volume without using the formula for the area of an octagon. The similar triangles argument can be used not only to find the side lengths in the middle cross-section, but it further shows that the cross-sectional perimeters are constant! And of course such an octagon will have greater area than a square with the same perimeter.
Excellent observation!
The area function turns out to be _A(z) = -2(√2 - 1)z² + 2(√2 - 1)z + 1_ at height _z = [0, 1]_ So if you graph {x=height from 0 to 1, y=area}, you will see _a parabola_ going through points (0,1), (0.5, 1.207), (1,1).
My first solution was, you already have a 3D print, submerge it and a normal prism with the same sized bases. water displaced will be the volume. But you are Stand-up Maths and I expect you'll create a math proof either way.
My other question is how this effect structural integrity as a building. (With the Frustum version, since a true anti-prism wouldn't be made because of the overhang.)
One in Hong Kong, one in London, one in New York.... *Setting reminder to investigate the basement of Jane Street office buildings in search of any wierd fiery portals, whips, giant creatures or flying red capes.*
Given how architecture and geometry have gone hand in hand for all of human history, and the new methods/materials that we have to build with, the next 100 years or so is going to be such an exciting time for geometry
I was waiting for someone to cover this for a long time, so great video!
We're doing an egg drop competition at work and the maths for it is interesting answering questions like: what will the impact velocity be when dropped from X height? What is a safe velocity for the egg to impact the surface? What is the maximum safe pressure to apply to the egg shell? With Y crumple zone dimensions, what material properties are required to dissipate the energy and ensure a safe egg?
The results of the calculations have steered us towards an unexpected solution. I realise this is like applied maths/physics, but it is 95% maths but knowing what maths to do.
When you twist the prism the sides of the rectangles connecting the n-gons are no longer going straight down and thus longer than in the prism. Not a proof but that's how I intuited the volume increasing.
That's the offices where Sam Bankman-Fried worked at before moving on to FTX! (Jane Street NY)
I had a slightly different intuition for it being bigger with the octagon middle - the closer a thing is to a circle, the smaller it's surface area to volume ratio is, and then the more volume it has relative to surface area (i'm a biologist, so this is like, the one math thing i know). Since antiprisims will always have a more circular middle than top or bottom (since the middle profile is always a 2n polygon from the top and bottom surfaces) it should have a greater volume with roughly the same surface area
When Matt was off by 3 orders of magnitude, all i could think was "not only is it an antiprism, it's also the anti-tardis."
He could have left the wrong answer in and almost nobody would even have been any the wiser, if it was all in American units.
* It is not an square-antiprism frustum.
A frustum of a regular antiprism has trapezoid faces not triangular faces, and the top has eight sides of alternating lengths (or a regular octagon if bisected).
0:40 For an irregular-shape based antiprism, there would be no "perfectly offset" rotation. So, I guess, we get an infinite family.
Great video as always, thanks to all involved! I'm not particularly maths inclined, but I enjoy learning a little, even if much of what I retain is just "wow, how cool is maths?!" :)
My very non-mathematical intuition was that if you rotate the top square by 45 degrees, then its corners jut out beyond the area of the bottom square. That extra space (4 triangles' worth) should be included in the volume somehow, so the volume should be larger than a regular prism! It's nice to be right about something mathematical for once. Now it makes sense why the building tapers at the top, too.
TH-cam picked an appropriately prismatic advert to show during this, featuring the prismic truffle variant of a certain triangular Swiss chocolate bar that comes in tapering rectangular prism packaging. This is probably coincidence though and not because their algorithms are matching the maths.
If it's more volume (more floor space) and obviously looks cooler, then why aren't more buildings anti-prisms?
Al-Qaeda just really hated rectangular prisms and they wanted a rectangular anti prism
As a fun aside, a prism and anitprism do have the same surface area
To see this, first notice that the end faces have the same area and perimeter. Let’s say the side length is S and the height is H
Using similar triangles like Matt did, you can see that for any cross section at a height h, there are n sides of length S*h/H, and n of length S*(H-h)/H. Adding these up to get the perimeter we get
n*S*h/H+n*S*(H-h)/H = nS
The same as in the prism.
And because the ends have the same area, and every cross section has the same perimeter, we can show that the entire antiprism has the same surface area as it’s corresponding prism
That would be true if the sides were vertical. They aren't, though. So the _h_ in the _½bh_ of each triangle is a little larger than the _h_ of the prism or antiprism.
Can you imagine working in a simple cuboid?
That must be so depressing...
😂
The Twin Towers (the original WTC buildings) were simple cuboids. Square footprints (now big memorial fountain pools), and rising ~1360 feet above the sidewalk. I imagine it was mostly a pretty interesting place to work, except for a couple of days here and there. And that last one, of course.
I recall a vague theorem named after a old greek mathematician which states that if two shapes have the same cross section for all cross sections in some line, then the shapes have the same volume. Clearly a prisms cross section is the base polygon for all cross sections orthogonal to the hight of the prism. The antiprism also has this same polygon as a cross section, but now it rotates as a function of the hight lf the cross section. We observe that the two shapes are the same for all the cross section, hence the prism and the antiprism have the same volume. :)
Actually no, the shape that I just discibed isn't an antiprism, so this argument doesn't hold
So to definitely answer the question raised at 3:00, one needs to find the area of the pyramidal square frustum with height H, base side length L and top side length L/sqrt(2), which is H/3* (L^2+L^2/sqrt(2)+L^2/2) or about 0.7357 times the base area times the height, i.e. less than the volume of the frustum antiprism found to be 5/6 times the base area times the height. Almost 12% less in fact.
I was guessing more, based on this:
The shortest distance between two points is a straight line. In a cuboid building, the corners are connected with straight lines. In order to connect it as an antiprism, the lines have to be angled away from vertical, which makes them longer. That made me think it's probably larger. But I don't know if that description is true or coincidental
I loved how you searched for a solution, I loved how you dealed with your small mistakes (keeping them in the final cut). We have to improve those 2 points here in France.
But in France, you also have _le mètre_ to make the calculation so much easier .....
But it’s an irregular antiprism (top and bottom are different sizes). Is not a frustum at all. The faces are complete isosceles triangles not trapezoids.
I'm gonna use "Proof by thinking about it for a minute" on an exam
Before I see the answer:
My theory is anti-prism has more volume because it has more faces, making it more “rounded”.
seven minutes in and he's digitally removing the top half of a world trade center building
I am surprised that there wasn't a comparison between the anti-frustram and a truncated rectangular pyramid.
The anti-frustram is always larger as the top floor shrinks until both shaped devolve to a rectangular pyramid.
I didn't even know buildings had feet, let alone 39 million of them. I thought they stayed still most of the time
My first thought was just making a hollow prism and antiprism and filling them with water to find the volume. Which I understand isn't the point but it'd be a cool water bottle.
How I got it:
Folding the alternating triangles of an antiprism to a flat plane results in a parallelogram. So the circumference is the same at any given hight.
As an octagon is more "circular" than a square and a circle has maximum area packt in, the center cross section and therefor the total volume has to be bigger.
Greetings from germany :-)
The anti-prism is not a prism that has been twisted. It took me a while to realize that. If you take a somewhat flexible and compressible prism and twist it, the volume will decrease. The limit is a very narrow waist. The anti-prism has the new triangular faces, so it can be larger than the original prism.
I hve my extension maths final exam today, I’ve spent 13 years at school building up to this moment but I’m watching fun maths videos instead of studying
Once more a mathematician tries every possible geometric method in their desperation to avoid doing calculus.
I don't want to cry, BUUUT where are our metric units?
How big does the top square need to be, relative to the bottom one, so that the antiprism has the same volume as the prism?
antiprism always bigger no matter how big and how many size they both have. This can be proved easily by calculus.
I mean the shape that is like an antiprism, but where one of the bases is smaller than the other, like one world trade.
@@VanjaPejovic *tl;dr:* Ratio of top square to bottom square dimensions: 87.4%. Ratio of areas is 87.4%² ≅ 76.4%. Let's work through it:
First, make the maths easy: set the base square to 1×1 and the height to 1.
Volume is the integral of cross-sectional area over the range of height. Given an area function A(h): _V = ∫A(z) dz_ integrated from _z=0 to h_
Define _x_ as the top square side length. We're looking for _x_ such that _V = 1._
(Start with a sanity check from what we know already: if _x = 1,_ this is an actual antiprism, with _V > 1._ If _x = √2/2 ≅ 0.707,_ that's the One World Trade Center with _V < 1._ So we know _0.707 < x < 1_ and we'll confirm this at the end.)
So the plan is:
1. Find a function for cross-sectional area: the area cut by a horizontal plane at height _z._
2. To get volume, integrate that with respect to _z_ from height 0 to 1.
3. Set the volume equal to 1, and solve for _x._
*1.* The cross sections are octagons with 45° angles, with alternating side lengths. As _z_ goes from bottom to top, the 4 sides parallel to the _bottom_ square shrink from 1 to 0 with the formula _1 - z,_ while the 4 sides parallel to the _top_ square grow from 0 to _x_ with the formula _xz._
If you draw an octagon with 45° angles and alternating side lengths _a_ and _b,_ you can derive its area _A_ = _a² + b² + (2√2)ab,_ either by adding up rectangles and triangles, or by subtracting the diagonal corner triangles from an enclosing square. Now substitute _a = xz_ and _b = 1 - z,_ then simplify the algebra and so on:
_A(z)_ = _x²z² + (1 - z)² + (2√2)(xz)(1 - z)_
_A(z)_ = _x²z² + 1 - 2z + z² + (2√2)xz - (2√2)xz²_
_A(z)_ = _(x² - (2√2)x + 1)z² + 2((√2)x - 1)z + 1_
*2.* Integrate with respect to _z._
_V_ = _∫A(z) dz_ = _(⅓z³)(x² - (2√2)x + 1) + (½z²)2((√2)x - 1) + z_ with _z_ from 0 to 1
Now for the easy bit: take that with _z=1_ minus that with _z=0._
_V_ = _⅓(x² - (2√2)x + 1) + ((√2)x - 1) + 1_ minus … zero. The _z=0_ half is just zero.
_V_ = _⅓(x² + (3√2 - 2√2)x + 1 - 3 + 3)_
_V_ = _⅓(x² + (√2)x + 1)_
*3.* Final step: set the volume to 1, simplify the algebra, and solve for _x._
_V_ = _⅓(x² + (√2)x + 1)_ = _1_
_x² + (√2)x + 1_ = _3_
_x² + (√2)x - 2_ = _0_
Quadratic formula time!
_x_ = _(-√2 ± √(2 + 8)) / 2_
_x_ = _(-√2 ± √2√5)/2_
_x_ = _(√2/2)(-1 ± √5)_
So there are 2 roots for _x,_ (√2/2)(√5 - 1) ≅ 0.874 and (√2/2)(-√5 - 1) ≅ -2.288.
But side length _x < 0_ does not make sense with our model, so we choose the first solution. *And, sanity check: 0.874 is between 0.707 and 1.*
*Exercise for the reader:* Figure out what the geometry of a negative side length _x ≅ -2.288_ would actually look like. Apparently it has a volume of 1, but what is it? Perhaps it is completely nonsensical and does not have those octagonal cross-sections at all.
Did you know that a Prism and an Antiprism have the same volume?
No, I didn't
That's crazy, I would have thought one of them would have larger volume...
How did you get to that conclusion?
@jansalomon5749 It's on the internet, it must be true. (Disclaimer: this can indeed be true, but only if the prism and antiprism have bases of differing sizes)
@@Maelwys Well, all Matt said was: if it's not on the internet, then it's not true. It does not follow that the inverse is true.
IIRC, the volume of all shapes with parallel top and bottom (connected by straight lines) is (A(top) + 4A(middle) + A(bottom))*h/6
Today I learned:
For the past 20 years I've been saying the word "incorrectly," adding an apparently-nonexistent "r," thinking they were "frustRa" or that it was a "frustRum"
After about the 10th time of y'all saying "frustum" (no "r") I looked it up, and dang, there's no R!!
It makes a lot of sense when you think about the limits. As you start going up, you barely take anything out of the corners, but you're adding a whole lot of side. Therefore thee area of a slice must be bigger.
Wikipedia says it cost $3.9 billion to build... $100 per cubic foot.
Geoff Marshall was just in NYC as well.
You could also argue by preservation of perimeter. Two of the triangles in the antiprism = 1 rectangle in the prism so the perimeter of the cross-section must remain the same. Given fixed perimeter, the max area is a circle, and an octagon is much more circular than a square.
I knew that the surface area was affected by twisting, from noticing that when you rotate a bread loaf bag, the bag's height reduces. So for the top to stay stationary, it would need more bag height and surface area to have the same untwisted height. So it also extends to the volume of the shape up to a point (like Laura said) where the straight edges formed still "aim" outward from base shape.
If you just twist the prisms top instead of changing the sides (like you do when creating the antprism), I think Laura is correct that the volume would decrease.But that's a different situation ofc. In the limit i think you would get two pyramids stack on top of each other, one upside down. Like an hourglass like shape
Thanks Matt, thanks Laura, helps a lot!
This might just be me being a nerd, but I thought it was pretty obvious you simply subtract a square pyramid from the whole prism once Matt stated that four of the triangles were perfectly vertical
i’m sorry, Matt, but the quote "it's smaller than it looks!" made this whole video. Good maths, very entertaining, but that quote was so unexpected... P.S. please come to St. Louis, Missouri, USA!
I think it’s easy to prove if you imagine the volume of the same building with parallelogram sides. Which would be equivalent to the cuboid volume. So by cutting the the parallelograms across the diagonal, you create a bulge whose edge bulges out more than the planar surface of the parallelogram.
Super clever idea by Laura to split the prism into four quadrants like that(!)
It is rather obvious that the volume of a digon antiprism is larger than the volume of the corresponding digon prism.
I would argue that for each triangle the base would be in the plain of the square of the prism building, but the point would be at the same height but not in the same plain. So two times the triangle surface is larger than the square surface. The surface of the antiprism obviously does not cave in, so the volume has to increase
that was a great show!
matt. we need the follow up video for "how thick is a three sided coin". ive been waiting.
Honestly, all these mathematicians doing things the hard way!
All you need to do is seal up the entrances, fill up a container of water with a known volume. Then you just drop in your object, make sure it sinks completely, and now you can just use a tape measure and measure the displacement of the water!
Someone in NY get on that to double check their calculations, okay?
I can't be the only one disappointed to hear you measure a building in feet?
well technically he did measure it in feet
If the building was designed in feet it doesn't make sense to work in metric if the numbers become messier. No decimals no rounding error.
"proof by thinking about it for a minute" is probably the strongest, most logical proof I can think of
The people in the construction crew that built it must be really angry at its architect, David Childs. I think he made their lives miserable.
Do bad we cannot name future insights after Parker, because famously his name is already given to the most famous Parker Square
Stopping at 4:24 to register my guess. Based on the shape, it seems like antiprisms would pack less efficiently, thus there would be lower volume.
My gut feeling was that it was twice the triangles and thus we're more approximating a circle and a circle has the most volume for a given radius... So it should have a larger volume. Not sure that makes sense but I suppose I got the right answer