How many ways can you join regular pentagons?
ฝัง
- เผยแพร่เมื่อ 1 มิ.ย. 2024
- Thanks to Jane Street for supporting this video.
www.janestreet.com/join-jane-...
Curved-Crease Sculptures by Erik Demaine and Martin Demaine
erikdemaine.org/curved/
The original paper: "A Complete List of All Convex Polyhedra Made by Gluing Regular Pentagons" arxiv.org/abs/2007.01753
Get your hexagon equivalent here: arxiv.org/abs/2002.02052
Cookie. Clicker. Like that video is going to happen.
arxiv.org/abs/1808.07540
Huge thanks to my Patreon supporters. They keep all my polyhedra convex. / standupmaths
CORRECTIONS
- None yet, let me know if you spot anything!
Filming and editing by Alex Genn-Bash
Written and performed by Matt Parker
Produced by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b... - บันเทิง
I meant what I said: 50k likes and Cookie Clicker video gets made. But I'm pretty sure I'm safe.
Maybe we'll find out how many cookies Jane Street will sponsor... www.janestreet.com/join-jane-street/
1st
BTW from what I can tell only 13 of your videos have 50k likes so you're pretty safe.
Edit: This will probably age pretty poorly
Edit: the video just reached 31,415 views and it has over 6k likes THIS IS NOT LOOKING GOOD
I will interact and push the algorithm so we get the likes and the video.
I, honestly, would like to know how to mathematically optimize Cookie Clicker, despite not playing myself.
Also, there is one game that I would like to know how to optimize farming in, and that is Bloons TD6, which I play a LOT of.
BEZAN LIKO
Don't underestimate Cookie Clicker players. You'll end up making that video for sure! lol
I hope Matt has underestimated how much the community NEEDS a Cookie Clicker video.
I haven't played it in years, I will totally go back to it if he makes a video on it. So maybe I don't want him to make that video 😂
🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪👵👵👵👵👵👵👵👵👵🖱️🖱️🖱️🖱️🖱️🖱️🖱️🖱️🖱️🖱️🖱️🖱️🚀🚀🚀🚀🚀🚀🚀🚀🚀🚀🚀🚀🚀
This. Cookie Clicker x Matt Parker is like a fever dream you'd think would never happen, it would be so awesome!
@@hujackus 🕰️🕰️🕰️🕰️🕰️🕰️👨💻👨💻👨💻👨💻👨💻👨💻👨💻🧠🧠🧠🧠🧠🧠🧠 (ooh spoiler for the new update) 🙋♀️🙋♀️🙋♀️🙋♀️🙋♀️🙋♀️🙋♀️
🍪
This feels like a rare instance where the cardboard objects aren't meant to represent broader mathematical concepts, but rather its literally about what you can do with cardboard pentagons.
There's a definite discontinuity where Matt goes from saying they have to be planar pentagons, to where he makes them very much non-planar. I get why now, but it felt like it was never explained why the rules can be relaxed.
It wasn't that the pentagons themselves have to stay entirely planar, but that each face (after folding) has to stay planar. It wasn't explained super clearly but the video was fascinating.
Fold lines don't count as curvature, so technically the faces aren't curved. The gauss-bonnet theorem (which gives the second equation i.e. total angle deficit = two full turns) still applies. I do still think it's a bit of a cop-out.
Parker planar pentagons
I was confused by this for a while too
Yeah I think the "gluing rules" is what lets you combine multiple together to make a face
I love that it's called a "degenerate" polyhedra, feels like when maths people call a solution "trivial" but it's even more judgemental about it, like you can almost hear the mathematician saying "yeah, i guess, but you *_know_* that's not the answer i was looking for..." 😂😂
I uttered the phrase "no *you're* a degenerate taco" during this video 😆
My favourite bit of judgemental maths jargon is the name for the transition point between a left-handed and a right-handed helix: a perversion.
In the same sense there is the infinite family of polygons (polyhedra, polytopes) whose vertices are all the same point, probably the easiest way of intuiting what degenerate cases mean
@@hughcaldwell1034 good grief 😂 they could have called it ambidextrous… although that does imply both handednesses (is that a word?), which might not be appropriate.
I am not only missing some properties we would like, I also have some undesirable properties as smelling bad. I am still a degenerate human?
I admire Matt's courage in scoring a bunch of papers straight on the table without any protective surface.
I mean easy to do with no risk of damage if you use the right tools, there's no need for a blade to get a clean crease line, just need a reasonably narrow edge...
I suppose the question isn't how many polygons exist that have pentagons as surfances, but how many polygons can we make, of which all surfaces can be constructed out of uninterupted pentagons.
Then maybe polyhedra whose planar nets can be constructed from regular pentagons?
@@rosiefay7283 The question remains. Does it also fold in the fourth dimension? Or is the folding of a pentagon just a shadow of a regular pentagon crossing into the fourth dimension which makes it looks like it's folded? 🧐
Nah, it's folded alright. :P
@Dithernoise if we visualize the surface of the final polyhedron as a continuous space, where from the perspective of a 2d entity they can't directly perceive the fold, the pentagons would seem continuous.
This framing of the question makes me feel a lot less deceived! 😂
I genuinely got so upset at the third one because he didn't end up with a shape with pentagonal faces, which seems like cheating (or at least rules lawyering)
For the past two years, I've taken to wrapping my Christmas gifts in custom boxes of various complex shapes made up of various polygons. The box essentially becomes part of the gift, which makes it fun, especially if the gift itself is otherwise boring or expected. This video has given me some ideas for new gift boxes. Figuring out how to wrap them in paper will be especially interesting, though.
The subtle difference between a convex polyhedron made by sticking regular pentagons together, and a convex polyhedron with regular pentagonal faces.
I appreciate how Matt highlights these mathematical papers that we would never see otherwise, describes them in an easy to understand way, and then actually builds the shapes. I doubt with those papers whether any physical copies were made. Bravo Matt for taking something from abstract maths and making it concrete an tangible for all of us.
P.S. I feel like the four pentagon ones are a very elegant and simple example of the same net folding into three different shapes. Definitely simpler than any of the constructions in the video about those. They are also all easily seen to be distinct.
I strive to get as much joy in my life as Matt when he sees colored cardboard pieces
Maybe all you need is colored cardboard pieces.
It's a regular pentagon where regular pentagon is defined as the pentagon that Matt just drew.
Parker Pentagon
Parkergon
@@dleonidae No, he's still here :P
@@OverkillSD budum tss. Now get out.
I need that on a t-shirt now. The Parker Pentagon. Pretty sure one of the angles is divisible by π.
Matt, as someone who's clinically conditioned to click cookies continuously, you don't know how much I need a cookie clicker video.
(I tried to keep the alliteration going, but I couldn't quite conjure continuing 'c' words.)
"you can't comprehend my compulsion for cookie clicker videos" maybe?
@@VaguelyCanadian hmmmm
"As someone who's clinically conditioned to click cookies continuously, your cavalier comprehension of these cocoa-containing crystals conjures commiseration for your conceitedness."?
....continuously, critically consider calming my craving and create cookie clicker content to complement your channel!
@@icedo1013 Comrads! Cease creating crazy comments!
Clearly, commenters covet cookie cutter commitment.
Of course we want a video on the maths of cookie clicker...
For the trio of names I propose: Hamburger, Hotdog, and Pasty. All ways of holding your meal! If you absolutely need to make them alliterative, may I reluctantly offer “Handwich”.
Also I’d love a video on Cookie Clicker!
I think my favorite part of this is that all of the constructions require by definition that the folds join vertices, meaning if you start from a set of regular shapes as you did then all of the folds are simply "fold along the line made by these two vertices". This means that this could become an exercise in classrooms without a lot of hassle, and that's just awesome. I'd have thoroughly enjoyed doing something like this in school.
Finally, the long awaited sequel to "Every Strictly-Convex Deltahedron"
It's nice to see Matt's inner 5 year old come out with making colorful construction paper objects. I liked the video. Looking forward to the Cookie Clicker video!
Oh man, I was playing Cookie Clicker (thanks to the people on the One True Thread of the xkcd forums) when I was in the middle of moving to Austria, and I’d just been thinking it’s been almost 10 years since then… I hope we get that video.
I always love a Maths & Crafts video from Mr. Parker.
Matt & crafts 😂
I mean... I always like Matt's videos, so it's a no brainer that I'd want to see a Cookie Clicker video, even though until now I'd never heard of such a game.
The 5 polyhedra between the "simple" cases look like they could be really cool jewel shapes. The N=6 also kind of looks like a beautiful heart shaped jewel (if you leave all the faces flat).
Plato would probably die instantly if he saw those volumes.
I ABSOLUTELY want a Cookie Clicker video! I usually don't like like-baits like that but that one is just Sooooo worth it!
For some reason videos where Matt folds polygons together to make polyhedra are my favourite 😅. I guess it's just fun seeing them being made. Matt, have you ever thought of making a video on Archimedean and Catalan Solids? The Platonics are everywhere but there aren't really any good videos showcasing these other two groups. I'd be very interested in seeing you construct them and perhaps giving us some fun facts about them. And as a bonus, you get to talk about your favourite dodecahedron!
also the kepler-poinsot polyhedra
it's also very fun and satisfying to make them yourself. When I was in high school geometry I had a project that involved making a polyhedron out of card paper and I chose a cube glued to one of the square faces of an anticube/uniform square antiprism, and I really liked it and kept it for multiple years. I think the only reason I don't still have it is it got destroyed when I moved at some point.
I feel like this video was the first time I actually grokked external angles. Somehow the definition got stuck in my head without ever actually filling out as a concept. Ah the random holes in our educational journeys, thanks for patching this one!
I'm not a really a fan of Cookie Clicker personally, though I do enjoy idle games of other varieties.
Regardless, I would find a video into the math behind the optimisation problem of such games to be extremely interesting, so it has my vote. Can't get enough maths!
Optimization problems are the most satisfying math problems. Nothing is more satisfying in math than finding the optimal solution to something.
As someone with a cookie clicker save file so old that it doesn't even include a "date started" value, yes make that cookie clicker video!
Wouldn’t bending the pentagon make it multiple other shapes?
He states the condition 2D pentagons In the first minute. But please investigate relaxing this condition as that is what maths is about.
Parker Pentagon
It's all triangles when you get down to it
@@griffingeodetriangles with a 2D Pentagon constraint
Yes. Triangles are pentagons now.. deal with it.
Called they/them pentagons
I'd be interested to see a video about self-intersecting polyhedra!
I assume you've heard of the video by Jan Misali about the 48 regular polyhedra? I'd be interested to learn more about that topic!
„These are 2. But I promised 8. Which means there are 6 more.“
That’s exactly the hard, cold maths I‘m watching these videos for.
It's the Parker-Pentagonal-Polyhedron! Much love Matt! Keep it up! I love that you encourage us to give it a go!
I was expecting this to be similar to Vsauce's video on the 8 convex deltahedra, where he used expansion, snubification, and another little things to generate them, but this was still a pleasant suprise, new ways to turn shapes into other shapes!
As somebody who makes spreadsheets about games, I'm 100% in for a video about the math for a game.
Of all the videos you've ever made, this one took me the longest to get through. I got REALLY stuck on that first new solid with the two pentagons, stopping and rewinding, advancing frame by frame, trying to figure out how you'd done the folds. I couldn't tell which edges were originally pentagon edges and which were folds... it might have made it easier to see if the pentagons had begun with black marker pen around their edges or something, so that this was more obvious.
8:41 What a nice Parker regular pentagon! 🤭
I commented before Matt Parker saw the typo in the title
I guess you could call it
a Parker title
Fixed now! I appreciate all the ways your comment helped.
@@standupmaths You're welcome
@@standupmaths
how many way do you appreciate it though? ;)
@@standupmaths It's also interesting how the unique numbers of pentagons in the final polyhedra is just twice the factors of 6
2,4,6,8,12
1,2,3,4,6
Back in the 80s, there were paper kits called Fuse Blocks that folded up into icosahedra sans the faces around one vertex; there were also separate "caps" and "seed blocks" to fill in the gaps. They could make all sorts of fun shapes glued together. I still have an unused pack of them. Good luck trying to find info on them online anywhere...
Jan Misali made a similar video to this called “There are 48 regular polyhedra”. He used different definitions hence the different results but it’s still very interesting
I need a cookie clicker optimum strategie guide. Also, I am still in need of an Oregon Trail guide, as well.
Matt wants people to stop prefixing foolish things with Parker, but then he goes on ahead to create a Parker Pentagon at the start of the video 😂
I would definitely watch a video on the mathematics of optimising cookie clicker haha
15:47 “The beautiful square gem” (as @DukeBG calls it) is made of parts I recognize!
@standupmaths, it is possible (as you surely know) to embed a cube inside a regular dodecahedron. Each pentagon contributes two nonadjacent corners and the connecting diagonal to the cube. You can follow four of these connecting diagonals across four pentagons to identify one of the square faces of the cube. If you slice the dodecahedron along the plane of that square, the smaller piece that comes off is (what a friend of mine called) a little “roof“ that’s made of two obtuse triangles and two trapezoids - plus a square base.
So now I see that it appears if you take two of those “roof“ shapes and attach them to each other on their square faces, with an angle of 90° between the top of one roof, and the top of the other roof, this looks to me to be the shape you pasted together whose beauty caught your eye!
😸
I have never heard of Cookie Clicker, but now I want to see the video on it!!
As someone who has been playing cookie clicker on and of since 3255 days ago(8 August 2014, apparently), and is very close to getting all upgrades and achievements(depending on whether or not i ever get a juicy queenbeet), I...
NEED
that video.
I'd love a video on Descartes's theorem (i.e. the 'missing' angles in a polyhedron adding up to 720°) and its generalization, the Gauss-Bonnet theorem!
Thanks Matt, helps a lot! ..also, looking forward to the cookie cutter video - how exciting
The N = 6 polyhedron is legitimately beautiful
To answer a different but related problem:
If you have a *cubic map* (a map where *every* vertex is shared by exactly 3 faces, so nothing like the four-corners in the U.S.), then you must have that 4C_2 + 3C_3 + 2C_4 + C_5 - C_7 - 2C_8 - … = 12, where C_k indicates the number of faces enclosed by k edges, including the “outer” face on paper (which of course is just any other face when putting regions on a globe). Note that the coefficient of C_6 is 0, so it doesn’t show up. This demonstrates why, for instance, a soccer ball with only pentagons and hexagons has exactly 12 pentagons.
Ah true, that is an elegant use of the Euler formula for polyhedrons
At first, folding looked like cheating, but then it actually turned out quite fun and interesting) thank you
Русский замечен
I have never, ever heard of Cookie Clicker until now - and I've been online since 1995. I liked this video anyway so that I can find out more :)
Love your vidoeos Matt. Making this comment because TH-cam has stopped recommending me your videos, so I'm reminding it how much i like your content. Thumbed, subbed, commented!
I need the cookie clicker video so badly!!!!
I always love it when the rhombic dodecahedron makes an appearance as it’s been one of my 3 favorite Polyhedra for many years. The other two being the standard tetrahedron and the stellated icosahedron
I'm a fan of Escher's solid. It's pretty just aesthetically, but it's also got wacky properties. You can get it not only by stellating the rhombic dodecahedron, but also by augmenting it at a height equal to the distance from the midpoint of each face to the center, just like the rhombic dodecahedron itself can be derived by augmenting a cube in the same way. It can also be derived as a compound of three non-regular octahedra. And it does the last thing you'd expect from such a crazy, spiky shape; it keeps its base shape's property of TESSELLATING space.
Also in the right orientation, each of its normals lie exactly halfway between two cardinal axes, making it probably the coolest shape you can easily build in Minecraft.
I've got a paper stellated icosahedron in my room that I made in my high school geometry class :)
@@dominicpancella3012 I make those on occasion for fun. I also sometimes make the much larger like 900 piece ball with the same pieces (model it after a soccer ball with hexagons surrounded by pentagons)
If we had our collection of regular dodecahedra and joined them face-to-face along a circular path, could we then have a torus made entirely of planner pentagons joined along their edges?
You can also make a transformative one out of deceptagons.
Magnificent video. Thank you :)
The parallelepiped ("hot dog") tiles 3-space, right? Any chance you might make an "infinity lamp" of this polyhedron like you and Adam Savage did with the rhombic dodecahedron?
The hot dog is just a skewed cube.
@@citybadger I'm unaware of a difference between a "skewed cube" and a parallelepiped. It also happens to be the way the solid is described in the paper.
If you allow concave polyhedra then you can trivially make infinitely many chains of platonic dodecahedra.
10th anniversary of cookies clicker!? Heck yes, I want a video on it!
This is so cool, thanks for the video
Currently at 100 quadrillion cookies per second. I love me some cookie clicker 🍪
Soooo, given it is missing some qualities we would ideally want, those two back to back is a parker polyhedron?
No, it's too mundane a failure.
Parker pentagons is really one the incredible videos have watched today....waiting for the cookie cliker video to drop soon 😊
When I played the Cookie Clicker, I always wondered what would be the optimal strategy to get the most cookies in a given amount of time. We definitely need the Cookie Clicker video.
Poor degenerate Polyhedron, he definitely is my favourite
8:42 Parker Pentagon
Parker Pentagon
Good on you Matt, love your videos mate
We proudly present to you: the Parker pentagon. ❤
At 19:18 you say that for polygons with odd vertices you can make degenerate tacos, when you should have said polygons with even vertices
So what's the difference between scoring then folding a polygon and cutting it apart into several polygons and gluing them together to make a polyhedron? At least physically, it seems not like pentagons joined together, but other polygons that together could be assembled to pentagons in the Euclidean plane.
The fact they fold means that there is still a restriction, by deconstructing the pentagon you are instead just constructing with triangles, unrestricted. It would expand what is possible by a lot, to things that would not be possible without cutting
I mean that's the design restraint which makes the problem interesting. Of course you could break all of it down to each face and just cut up Pentagons up to make them. But having the rules of each face having to be a part of an original regular pentagon and then glued to another side of equal length that is also a part of a pentagon sets strict rules for what the final shapes can actually be.
But I think the real beauty is in the uniqueness he revealed at the end; that the pentagon is the only regular polygon that has a finite solution set which isn't either completely degenerate (heptagons or bigger), or infinite solutions (hexagons, squares and triangles i.e. shapes that tile the plane).
And I personally really like when a problem has an unexpected result that mathematically shows that something is unique. In this case that a pentagon is the *only* regular polygon that can fit three around a vertix while not tiling the plain.
The net is constructible from pentagons is the difference
@@noone-ld7pt Thanks! I wasn't saying that the problem was without merit at all, there was just something about folding pentagons and still claiming the resulting polyhedra to be made out of pentagons that rubbed me the wrong way. Your comment made the interesting part click better.
@@Starwort That's a great, succinct way of putting it. Thanks!
YES PLEASE GIVE US THE COOKIE CLICKER VIDEO
Fold a square in half and you and up with what is essentially two rectangles stuck together. Regardless of vertices, it is no longer a square in the spirit of the shape, and so by the same analogy, those folded pentagons are also just a bunch of triangles making a different 3D shape.
You can achieve and make up anything when you make up the rules to suit.
If you’re using phrases like “the spirit of the shape” then maybe mathematics is not for you
I don't get why the degenerative taco (folding a pentagon across a symmerty line and glue it together) doesn't count.
For a pentagon, the taco has to fold across the middle of an edge. The rules the paper's authors used only allows folds from corner to corner. You can make degenerative tacos with shapes with an even number of sides because you can draw a line of symmetry from one corner to another.
Me understanding half of what he says but still listening because it makes me feel smarter
Crafts with Matt. I need more of this.
So we've got the cracker, the last extra-crispy fry in the box, the cheeto, the hamburger, the hotdog, the samosa, the cheese puff and the meatball, and Matt's favourite is the samosa. Neat.
C'mon guys lets put our cookie clicker skills to use, click that like button.
Heya, i was thinking of this channel yesterday when i heard of a new kind of number "dedekind" numbers.
there was some new discovery or one and i literally cant get my head around them and thought
"i hope Stand-Up Maths sees this news and does a piece on them" cause you one of the only channels able to explain complex number stuff in a way my thick head understands lol
Are you referring to the dedekind construction of the real numbers, or to something else?
10000000000% YES I want a Cookie Clicker video!
I love how infectious his excitement for maths is! Been hooked for years
It's not really a pentagon any more if you fold it is it? I mean it just becomes a bunch of triangles. Totally cheating.
Relaxing the convex requirement gives you some lovely ones like the great dodecahedron
I NEED THAT COOKIE CLICKER VIDEO
I love the pattern he has on the 5x5 cibe on top shelf. I developed that independently after learning how to solve cubes while away from the internet, love seeing other people give that pattern some representation.
Some really interesting shapes there!
8:40 Behold! The Parker Pentagon!
i think those would make really cool 3d-models for gems or crystals in video games
I'm glad you talked about hexagons. They're the bestagons.
Please do a Cookie Clicker video!!! I've been playing for about a year now and I'd be curious what the optimal strats are.
I love your videos, Matt! Your passion for math is fantastic and infectious and I wish I'd had more math teachers like you in school.
One potential correction: At 8:00, you mention that "...and x is always 3" when you meant to say "... and zed is always 3". The text on the board is correct, it was just a simple slip of the tongue.
And I'm doing my part for the Cookie Clicker video!
I understood the paper and tape part. But you lost me on the whiteboard 😂
i've never played cookie clicker but i'm big into games and id be so hyped to see a cookie clicker video! would be legendary
I am so excited to learn about Cookies!
This right here is why I watch Matt Parker.
i've yet to watch this past the first few seconds, but i know this will be right up my alley.
In terms of Cookie clicker: sure, why not. I think that could be a very interesting video.
The Demaine's and origmai math in general is an amazing subject.
I first got interested in folding polyhedra from John Motroll's books; single, square sheets of uncut, unglued paper to make a bewildering number of all types of polyhedra.
Omg yes, a Cookie Clicker video would be amazing!
1. The Pancake
2. The Potato Wedge
3. The Whatever-that-is
4. The Hamburger
5. The Hot Dog
6. The Onigiri
7. The Crystal (not safe for consumption)
8. The Monster Meatball
"We know they exist, but there are four undiscovered gluings for sticking infinite families of hexagons together into convex polyhedrons" ...
You do realize there aren't that many people who can say THAT, and make it sound interesting as you do, right?
This is certainly one of my favorite quotes from this channel, thank you again so much for sharing your passion! 😁