/sci/ - Science & Math

In unusually good form for a popsci article, the link was good enough to show an illustration of the other known pentagons, which I reproduce here out of some personal excitement...

moreover, this delightful nigger even has /elementary degree/ angles, and /elementary/ side lengths. I was expecting something onerous at first! Just beautiful.

It makes me want to buy a box of these fuckers and eat 'em up.

>>7466890
vc

what the fuck is even the point of this ?

>>7466924
>this guy

>>7466890
>>7466924

discovering a new motherfucking shape is very, very special and the man who did it deserves a million dollars as a reward.

>>7466941
>>7466938
again...what the fuck..is the point of this shit ?

>>7466905
yes, surprising that it comes out to integer angles, and more surprising all are divisible by 5

Neat!

I would like a larger picture of the tiling... I can't really get a feeling for it, and it looks like the red stripes are not parallel... probably an illusion

>>7466924

For a pure math autist like myself, a real and acceptable answer is: for the fun of it, the poetry, the pleasure of discovery. The first part of this answer tends somehow not to sit well with the general public, unfortunately, and even with other /sci/entists who demand some sort of an application. But it shouldn't take such a genius to imagine that in some space-filling application (chemistry? filling a UPS truck?), it might be nice down the road to know a few extra convex plane tilings than not.

More generally, the problem of tiling the plane with convex figures turns out to be tantalizingly close-to, yet-so-far-away-from completion, Euclid (or someone thinking similar thoughts) having taken care of the 3,4 and 6 cases, and eliminated the higher cases, leaving JUST the pentagon as the incomplete object of study. When something takes your interest, you tend to want to finish it, whether or not you are able.

This whole business of the low-degree cases being long since solved, and certain infinitely-high degree cases dismissed, sounds STRIKINGLY FAMILIAR to what I remember of Perelman's Poincare conjecture, and the various cases of FLT variously treated of by both Fermat and Wiles.

>>7466942
TILING YOU STUPID MOTHERFUCKING PIECE OF SHIT

NEW WAYS TO TILE

ITS FUCKING INCREDIBLE

>>7466954
great. go tell your bathroom vendor the good news.

too bad it doesn't do any good for the rest of the fucking world.

This proves again that studying math is useless

>>7466957

>>7466948

This is easy to do! Also, look up the notion of a "sliding cell", which was described by M.C. Escher and others, to describe a subset of a tiling which does the same job, WITHOUT any rotation in the plane/stretching/reflection (notice that the primitive pentagon and its mirror-image are used in the tiling).
JUST translation.

In this re-work, I have isolated a SLIDING CEL (having 12 primitive pentagons, so my earlier conjecture was correct). Notice how this sliding cell gets the job done by shifting up-and-left, down-and-right, etc.

In fact, now that we have a primitive sliding cell...

>>7466962
3rd post and you still can't give me one good use for it. you should stop posting before you embarrass yourself further

>>7466978

Some quick and dirty (and imperfect, the source image is a lossy jpg but that's a separate issue) scaling in MSPaint becomes feasible!

>>7467003
bruh anyonee can do that

>>7466983
you didn't even think of packing crates

>>7467012
> every pack loses surface area due to retarded design
> edges of the packs will leave uneven empty space which could have been avoided with simple rectangular designs
> you need dozens of units to form them into a compact pack, where rectangular designs only need 4

I hope you were trolling. I would rather prefer to be trolled than encountering to such an imbecile.

>>7466960

Rest of the Sciences,

You're welcome for over half of everything ever.

When you need something new, and are ready to swallow your pride about trivial "applications", and admit that my pentagons are pretty, and do not begrudge me about them (not to mention those parts of Me which you elect to apply to your own ends)...

When you're ready for a new discovery, I'll be waiting. For I am never Invented, but rather, that which was there all along (Me) is simply Discovered. My pretty pentagons, which were of course There All Along though you knew Not, suffice as proof of this principle.

-Regards, Math

>>7466950
Cool story, bro. I never heard about that before. No wonder you guys want to figure out all the pentagon tilings. Got anymore info or links; what does FLT stand for? Finite Lattice Tesselation?

>>7466978
Not sure if I know what a sliding cell is, but I noticed this doesn't translate well on the right and left. Must be some reason why you blanked out those particular shapes, but I don't see it.

>>7467113
FLT = Fermat's Last Theorem

The only natural number n for which there exist natural x, y, and z such that
<span class="math">
x^n + y^n = z^2
[/spoiler]
is n = 2.

>>7467119
that should be <span class="math">z^n[/spoiler]

>>7467119
* non-zero natural

>>7466942
Lattices are how atoms and molecules usuallly arrange themselves in crystals.

The shape of the lattice and what symmetry it has has all sorts of quantum effects which affect the properties of the material

Tiles are a form of lattice, sometimes.

However this particular kind of tiling will certainly never occur in nature so it has literally no point.

It's basically as pointless as a geometries discovering that it is possible to construct a regular shape with 177 edges using just a compass and a straight edge.
It won't lead to real world problem solving (e.g. Physics), and it won't lead to new maths that leads to useful problem solving.

Prove me wrong.

>>7467137
>It's basically as pointless as a geometries discovering that it is possible to construct a regular shape with 177 edges using just a compass and a straight edge.
>it won't lead to new maths that leads to useful problem solving.


Except the example I quoted has already done that.

>>7467140
What example?

>>7466950
this is pasta right?

>>7467152
....did you even read it?
It's incredibly (and specifically) relevant to this thread, and would be relevant to VERY few other threads.

>>7467143
How to construct a polygon with plenty-ass sides using only a compass and a straight edge. The work Lagrange and Gauss did to solve that problem inspired Galois, who in turn inspired the theory finite fields, fields extensions and Galoisian correspondance, who in turn help discern the resolubility of polynomial or differential equation, which are equations who actually happen irl.

>>7467113

They do, I just didn't emphasize that in the previous image. Have a look at this re-work to see what's going on. The colors here don't matter, it's just to emphasise that a group of 12 goes left-and-right, as well as up-and-down.

A "sliding cell" is just some portion of a plane tiling that also gets the job done, but without rotation, stretching, reflection etc, just "rigid, non-rotating movement", which in geometry is called "translation".

How was this pentagon discovered?
Does anyone know?
The article says they searched a "large but finite set of possibilities".
How were those possibilities found?

>>7466890
Combinatorial Geometry seems like such a beautiful area, wish I knew more about it.

>>7467158
You can draw some 'moral' analogies to compass and straight edge and solving polynomial equations.

But that does not change that proving that a particular n-agon can be constructed with compass and straight edge would be in itself pointless to the real world abd the rest of maths.

It would never be a useful lemma that you can make a 177-agon with compass and straight edge.

>>7467175

I don't know grad-school tier algebraic details, but a deep study of Escher earlier in life gives me a very good idea of how to start to look for more such pentagons.

The researchers in the present case used a computer to check cases, to your point. But every existing tiling (and here comes a fast-and-loose conjecture) fundamentally resolves to some convex form (explicitly obvious top left), or some NON-CONVEX ANALOGUE of the known 3,4,6 cases, with NON-CONVEX perturbations up and down in the SLIDING CELL. This is Escher's own description of his method for creating his own plane tilings, and the OP image has (one) SLIDING CELL which is fundamentally parallelogram/like, with its obliquity. So, a special case of tiling in the quadrilateral case, except that perturbed, "not-convex" "parallelogram" is really a collection of congruent pentagons.

tl;dr fuck with triangles and rectangles to cook up new pentagons, by pushing their edges up, down, etc. Notice how Escher's horsemen have symmetric "pushes" up-and-down about some rectangle...

>>7467236

got sidetracked, meant to attach this. Both images are good as illustration.

>>7467239

"Push" the lattices up and down, here and there, and perhaps some more pentagons pop out.

Bold (baseless) conjecture: there are infinitely many distinctive pentagon tilings, given what I've just said. And yet we only know 15.

>>7467244

in my haste I don't attach a good website with conditions for the known tilings:

>>7467199
> But that does not change that proving that a particular n-agon

But that's the thing, to prove that a particular n-agon is constructible, people had to experiment with methods who turned out to be particular cases of more general methods that resolves the general question of polygon construction *and* tell you something about the solution of equations.

It's not a "moral analogy" here, it's looking at the actual techniques employed and realizing they rely on a few simple tricks that can be systematized in order to give the general conditions of resolubility. So yeah, it's pretty useful to maths and its applications actually.

>It would never be a useful lemma that you can make a 177-agon with compass and straight edge.

It's not the lemma that's useful, but its proof. By looking at the proof your understand what's needed in general and why the problem ties with more general and interesting question. Fooling around with a compass and a straight edge is essentially constructing solution of polynomials equation of degree one or two, and when you repeat it you quickly run into solving equation of very high degree. The algrebraic manipulations are the same, and realizing it was key in introducing the proper formalism. And remember, one of the first motivation for introducing groups was Galois theory.

And going back at the real life applications, understanding mathematically what it is you're doing when you're using a compass is unsurprisingly useful when you have to build stuff that has geommetrical shapes. Most of a lot of engineers do isn't much different for designing and measuring shapes with simple instruments. The algebraic operation is the mental reflection of the physical gesture like the gesture is concretization of the algebraic operation.

>>7467252

...

http://mathpuzzle.com/tilepent.html

...!!!

OP bumping this on general principles before I go out tonight. I'm still not over this, and you should just plain /care/. Something this elegant is just cool as fuck.

>>7467137

>However this particular kind of tiling will certainly never occur in nature so it has literally no point.

We just drew it. Therefore, the thing which was always a property of nature, was just shown to humans for the first time.

Yes, I understood what you really meant, but I don't think you understood what I really Mean.

<3, Math

>>7466978
This is actually quite alarming, I mean, there could be living cells in suns made of ice

ITT: We claim that math is useless because we cannot understand it.

>>7466942
Go away, retard. You don't belong on /sci/.

>>7467137
>doesn't understand the point to straightedge constructions.
This has to be bait. No one can be this dumb.

>>7467158
It's also related to computability with applications in shit like CAD. There's some very similar stuff in the constructive geometry for origami which has led to lots of big theorems and applications (even medical ones).

>>7466903
I want red for shower tiles

>>7467461

>>7467029
It is the future, the year 2018.

You are a SPACE MASON in charge of shipping pentagonal space bricks to the human outpost on Mars.
Efficiency is TOP PRIORITY.

SPACE MASONRY is advanced beyond the use of rectagonal construction materials. In fact, COSMIC RAYS prevent the use of anything but the STATELY PENTAGONAL PRISM.

Pictured: a standard shipment of STATELY PENTAGONAL SPACE BRICKS.

Not pictured: YOU. Unable to arrange the bricks for MAXIMUM EFFICIENCY.

>>7467629
i'll advance my prism past your rectogonal construction if you know what i mean.

>take grid of rectangles
>divide half the rectangles into triangles
>fuse the rectangles and triangles in a regular fashion
>distort the picture with software
>get Nobel prize in Mathematics
That's right, I discovered a fundamental property of trolling. I didn't invent it, it was always there.

>>7467668

Your last sentence is right bud. Also, the rest ain't science.

>>7467629
10/10 post

>>7467030
>>7467457

please stay on this board, we're facing a huge autism shortage since /pol/ keeps raiding us

>>7467730
>/pol/
>raiding
It's a daily sci tradition to btfo all the cross posters from retarded boards. There isn't really anything unusual about the rate of /pol/esmokers wandering in. Besides that board isn't capable of raiding dick since it's composed of stormniggers who think raiding means posting racist slurs.

>>7466948
>more surprising all are divisible by 5
Hail Eris.

>>7467629
increase the goddamn samples jesus
is that vray or blender

>>7467629
you are single-handedly making /sci/ great today my friend

>>7467730
> blame /pol/
sorry mate, but /sci/ has always been the autism cesspool way before /pol/ was born

>>7467119
n = 1

>>7467029
i assume you've never seen toy packaging

i mean that would make sense

>>7468024
no i never did. please tell me how it relates to this at all

>>7466903
Man, some of them are just plain cheating, they just turn two or three pentagons into a hexagon.

>>7466942
you must be an engineer

>>7467629
ten out of fucking ten, this is going in my great posts folder

>>7468110

At a fundamental level, all of the known (periodic) schemes rely on a triangular/hexagonal/quadrilateral grid. The trick is to start with a base figure and then "push" its edges up and down such that the resulting figure is a non-convex, polygonal sliding cell, such that it can in turn be partitioned into congruent pentagons. Consider that the post >>7467160 is really just sketching a grid of long parallelograms, etc...


>>7467856 I've had Eris on the brain all throughout this thread and refrained from mentioning her name, but you opened the door. Law of Fives bb

>>7467629
Didn't need those sides anyway.

>>7468112
Why ?

>>7466942
What is the point of going to the cinema?

>“We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey. “We were of course very excited and a bit surprised to find the new type of pentagon.

cool how we're getting computers to work for us

also

>Pentagons remain the area of most mathematical interest when it comes to tilings since it is the only of the ‘-gons’ that is not yet totally understood.

What does it mean to totally understand a shape?

Any autist can scribble patterns on a piece of paper with more interesting design then this. It looks like a bunch of overlapping wishbones.

>>7467629
the good posts on /sci/ are few and far between
but god damn if they're not the best posts on 4chan

>>7468243
to enjoy a well written script along with good acting and get a pleasent experience.

>>7468256

In this case, the vague notion of "totally understanding a shape" can be made concrete:

The goal of the research is to completely discover and classify all of the ways in which convex pentagons can tile the plane. That's all. The reason why this is of interest to mathematicians, is because the pentagon is the only convex polygon case for which this process is incomplete.

And I've left multiple comments on the general approach throughout this thread, so if you're really interested, you can just read the thread.

>>7466942
I heard you can exchange new tiles against rare pepes.

>>7467629
Your effort paid off in spades. Well done sir!

>>7468363
oh ok thanks, sometimes context is lost to me. I'm gonna look up how people prove things possible or impossible mathematically now.

>>7467548
Purple master race.

>>7466924
Sod off, mate!

Take a circle of pi radius. It's modular(The end of one side is the beginning of the other side. 180). create vectors. Connect the dots. After that it's just different kinds of counting. Fuck finding one new shape. I just found more than you could ever count.


I'm a robot, and a liar.

>>7468545
There's something about the white one that catches my eye. It seems like the most chaotic of the bunch. I would color it like a weave like pic related.

>>7468594
forgot pic

>>7468240
>scientist makes fundamental discovery about nature
>HURR WHAT THE FUG IS THE POINT OF DIS SHIT, CAN'T END WORLD HUNGER OR CURE CANCER, BENIS BAGINA :DDDDD

>>7466903
Pink one in the middle is fucking with my head

>>7468600
its like the 8th post that can't come up with one single use of this garbage of a discovery. No wonder why you resort to greentext shitposting.

>>7467629
>Not pictured: YOU. Unable to arrange the bricks for MAXIMUM EFFICIENCY.

My sides are arranged for MAXIMUM KEKS

>>7466903
Grey is best, fags.

>>7466890
>>7466924

>>7470336
everything comes around