How many ways can you join regular pentagons? 

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 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 😂

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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) 🙋‍♀️🙋‍♀️🙋‍♀️🙋‍♀️🙋‍♀️🙋‍♀️🙋‍♀️

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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 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 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...

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)

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.

Maybe all you need is colored cardboard pieces.

"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.

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

Matt & crafts 😂

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.

Optimization problems are the most satisfying math problems. Nothing is more satisfying in math than finding the optimal solution to something.

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

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.

Русский замечен

Ah true, that is an elegant use of the Euler formula for polyhedrons

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?

No, it's too mundane a failure.

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.

Parker Pentagon

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!

Are you referring to the dedekind construction of the real numbers, or to something else?

If you’re using phrases like “the spirit of the shape” then maybe mathematics is not for you