"That same year an unlikely mathematical pioneer entered the fray: Marjorie Rice, a San Diego housewife in her 50s, who had read about James’ discovery in Scientific American. An amateur mathematician, Rice developed her own notation and method and over the next few years discovered another four types of pentagon that tile the plane. "

https://en.wikipedia.org/wiki/Marjorie_Rice

Rice's pentagons are illustrated here [1]: Type 9, Type 11, Type 12 and Type 13.

[1] http://www.mathpuzzle.com/tilepent.html

>Types 6-8 were found by R. B. Kershner in 1968. >Type 10 was found R. James in 1975. >Types 9, 11-13 were found by M. Rice in 1976-1977.

Dammit, R. James! You can't just skip numbers like that and mess up Rice's streak!!

She has a website for her tesselations (which that Wikipedia article links to), but unfortunately is no longer able to respond to queries: https://sites.google.com/site/intriguingtessellations/home/

There's a nice tiling of butterflies there.

how cool is this lady seriously. the symmetric tiling is really astounding and the roses one has a julia-set quality to its edges.

Wow. I personally cannot do much work without being part of a community to share and refine ideas with. It is amazing that someone can produce important work while working essentially in isolation.

that's awesome. Outsider math and outsider art deserve as much respect as "outsider" programming ( startups / garage hackers ) garner. Creating semi-closed communities of insiders, united by what are essentially arcane handshakes, and dismissive of value emerging outside of themselves, is really just both a narrow-mindedness that doesn't behoove innovation and also an admission of an unnecessary insecurity about the lack of substantive reasons for cohesiveness in that community resulting in the desire to fabricate arbitrary insubstantial reasons for cohesiveness.

At the same time, you could say that narrow-mindedness was focus and restraint, things which __do__ work for innovation.

Another way of looking at the cohesiveness issue, is that if your community is so substantially strong, why do you need to exclude outsiders ? The best democracies include difference, revealing their strength through diversity. The worst extremist fascist states put it to death, revealing their dread of diversity.

Let the ad-hoc groups of intellectual collaborators be like democratic meritocracies not frightened facist states.

IMO the analogy doesn't hold up very well. Outsider art is often very good, and outsider programming seems like a weird idea (do startup hackers have less training than BigCo employees?) But that's beside the point.

If you're looking at the tiny subset of outsider science that's chosen for correctness and publication-worthiness, of course you'll be enthusiastic about it. But these people, like Marjorie Rice, aren't really facing any obstacles from the establishment! If their work has merit, they usually get a warm welcome.

The elephant in the room is that >99% of outsider science is extremely bad, and can't understand why it's bad even after repeated explanations. Pretty much all the people complaining about the scientific establishment are like "fermatists", who used to hover around math institutes to have their proofs of FLT checked. If you're not familiar with such people, I urge you to actually look at a sample of their work and form your own opinion.

After that I think you'll be less eager to throw accusations around. A scientist's life is hard enough already, without people telling her she's an insecure evil fascist for rejecting torsion fields, the EmDrive, or Time Cube.

> their proofs of FLT checked.

What does FLT stand for? Fermat's Last Theorem?

Yes. Maths cranks are extremely common. If you have an academic maths email, you're likely to receive crank emails with some regularity.

The only person throwing accusations around is you. This is a huge blanket blaming statement:

> 99% outsider science is extremely bad, and can't understand why it's bad even after repeated explanations

A challenge dealing with noise in a system is an opportunity for a solution, not a rationalization to censor people. That < .01% may be onto something big.

Part of my point was that the 1% with good ideas aren't actually getting censored. If you can find any recent examples to the contrary, please let me know.

Part of my point was that the 'accusation' you claimed against 1arity was that communities exclude outsiders. That you made rationalizations about why they might be excluding outsiders (because they are bad) serves to illustrate that the community might very well be practicing exclusion, and with good reason. I'm challenging the status quo here, and looking for an opportunity to allow quacks to be quacks while we're still 100% open to finding people with decent ideas. That may be implausible, but I'm still challenging it. :)

For whatever reason, this reminds me of the recent TED talk that got pulled because the speaker was challenging dogma in research. Can't remember the title off the top of my head.

If you're talking about Graham Hancock or Rupert Sheldrake, I'll stick with the dogma. I don't think you can talk to spirits or make telepathic contact with dogs.

Now I'm curious, how exactly did you come to believe that the scientific community is not welcoming to outsiders with good ideas?

You seem to be arguing against a strawman. What math communities are closed communities of insiders? While it's true that many parts of math are fairly inaccessible to the beginner, that is more due to them relying on foundations built up on huge amounts of detail that you need to internalize before they all make sense, rather than any kind of "arcane handshakes" or "dismissive" attitudes.

Where are you getting that the math establishment is some kind of "frightened fascist state"? I have no idea what you're talking about here.

It's definitely neat that someone with no formal training outside of high school was able to come up with novel solutions to this problem. But I don't see the "fascist" community that you are arguing against trying to exclude anyone.

But mathematics is about the only field where merit is clear-cut enough that there's no outsider mathematics, even if crank mathematics is clearly recognizable by professionals and even by college calculus-educated adults. If, as often is the case, someone draws a sketchy proof for an ambitious conjecture that seems promising enough, entire communities of professionals spend thousands of man-hours fixing the holes.

What's more, mathematics has consistently avoided the "Lakatos trap" of chasing "narrow-minded" research programs that it might be argued that plague science in general. Even physics: Alan Sokal's mock article accidentally zooms in on how we have failed to understand (and persist on failing to working on, at least since the Kolmogorov formalism) the everyday phenomenon of turbulent fluid dynamics even as we've grown to astounding heights in esoteric reductionist physics that are spuriously compared to a new metaphysics. And don't get me started on economics or even econometrics -- just the vast swaths of subjects we've given up on.

Cranks are sometimes hard to recognise. Ramanujan seems like a crank at first glance.

A big part of the problem is scaling. Something I read in the past day or two about "Theories of Everything" made the point that (some|many|most) professional physicists are often inundated by requests from amateurs to review their stabs at a ToE. But if you're a professional, you don't have time to give every such request a thorough review, and still have enough time to do your own work.

I am as uneasy with credentialism as anybody, and I think we need to find better ways around these problems, but it's important to understand that at least some of this community exclusiveness stuff actually serves a purpose.

Science works when it embraces the unconventional and the unknown, anything that steps away from this limits the power of science to shine the light of a theory and an experiment onto something mysterious.

It is positive, constructive and it works for outsiders to be included in science and art. it is also positive and correct to critically look at thing when you hold the highest potential of those things as important. Nothing is sacred beyond critical inquiry, that way lies dogmatic extremism. Being an apologist for a broken culture doesn’t help that culture, nor the context in which it operates.

If you choose to pretend that outsider art is all good and outsider science is all bad, and that the culture of art, science and math ( and tech ) are beyond reproach, and that the points raised have no value as opportunities to create improvements, and that scientists ( or mathematicians, or the establishment ) are poor fake victims who are suffering hardship for considering whether their cultures possess opportunities for improvement, if this is the case then I can understand how pretending that seems to make it easier for you, since you can hold to an unchallenged perspective and not experience the disruption that comes with questioning the lay of the land under your feet. Tho that choice isn't as likely to lead to reflection, as being open to questioning. And without reflection, no improvement.

And without questioning, no science.

Science is inspiring. Science isn't about protecting yourself from reflection, science is brave, science embraces reflection and self critical thinking, and science embraces belief in the face of the unknown, science embraces acknowledging the unknown, and running the experiment. Science embraces the question. Science is the opposite of dogmatic. Anything that claims to be science and shies away from openness, is the mere color of science.

Science is that very ideal : that strength is open vulnerability, and not dogmatic inflexibility. Science is an inspiration.

I believe the very sensitivity of these comments ( and the fact that they created the accusation in their interpreting, fabricating a connection to an accusation that was not stated ) reveals clearly that this issue is on people’s minds, and that in expectation of it being raised, and in expectation of its raising being troubling, they are not only defending against it, they are prematurely defending against it, and creating themselves the accusation that was not present and which they have hallucinated in their fear of it.

To be afraid of something is to suspect it.

So the very ideas I have raised have triggered this apparition to be conjured in these minds giving life and proof to the very questions I raised, and expanding the scope of relevance of the critique. It's not just about maths, clearly. It's about science as a whole. The argument was never "could this occur?" for we all know it could, and we have seen it, as have we all seen the kind of reflexive apologist rhetoric get triggered that was triggered here in denial of anything wrong with these culture’s held beyond question. The argument instead was a warning against the very defensiveness seen triggered in response to it, and a warning put now in clearer relief against the background of comments that display that very inflexibility and refusal of critical questioning that can lead to the exclusionist frightened culture warned about.

These comments are then nothing except my loyal supporters, they stand beside my warning as additional support for its importance, and in them we can see how the beginnings of dogmatic inflexibility take hold : through defensive denial of divergent ideas and refusal to question the conventional.

Let the ad-hoc groups of intellectual collaborators be like democratic meritocracies not frightened fascist states.

What's the point group for the first tiling on that webpage? It's a periodic tiling (unlike a Penrose tiling, which is only quasiperiodic), so the crystallographic restriction for two dimensions says the rotation subgroup of the point group must be one of C_2, C_3, C_4 (not C_5!) or C_6:

http://mathworld.wolfram.com/CrystallographyRestriction.html

I can't tell by eye-balling it what the symmetry is for the first one, but its periodicity says it must be one of those. Quasicrystals with 5-fold symmetry are not exactly periodic.

There are only 17 wallpaper groups. Since this is a wallpaper, what is its group?

https://en.wikipedia.org/wiki/Wallpaper_group

From the table on the wikipedia page, it looks like it's p1

It's interesting how your eye naturally groups the pentagons into the larger primitive unit, like the pinwheels of type 5.

Wolfram Alpha also has some things about tiling: http://www.wolframalpha.com/input/?i=pentagon+tiling http://www.wolframalpha.com/input/?i=pentagon+type+5+tiling

Meta question: where can I find list of simple unsolved/undiscovered problems like these in math?

It does not appear in https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...

Not to answer your question, but here's a simple, unsolved problem:

    Dissect the circle into congruent pieces
    such that the center point is in the
    interior of one of the pieces.

It is widely believed to be impossible, but neither proof of impossibility nor construction of an example have been found.

Huh. Is it possible to dissect a circle into congruent pieces that are not, uh, pie shapes? Is that the question, or are there known other ways to dissect a circle into congruent pieces, it's just those pieces also have the center point as an edge point?

Non pie shapes are definitely possible - look at the taijitu (yin/yang) symbol for example - . But of course, again, the center is on the divide between both halves.

Yes.

Summarising, it is possible to dissect a circle into finitely many congruent pieces that do not all touch the center point.

Slightly longer, there are at least two infinite families of solutions:

(a) For every natural number n>1 there are f(n)>0 solutions, with f growing exponentially quickly.

(b) For every natural number n>1 there is an uncountable infinite family.

Thus we have a countable family of solutions, and a countable family of continuous solutions.

So yes, there are solutions that are not all just "slicing a pizza" type solutions.

There's a nice collection on Math Overflow: Not especially famous, long-open problems which anyone can understand

http://mathoverflow.net/q/100265/8217

I don't think that tiling a plane is exactly a "simple" problem.

I mean something simple enough to understand, and could be played by average programmer in a IPython Notebook.

I didn't mean simple to solve, just lower barrier of entry.

I like this one:

http://www2.stetson.edu/~efriedma/squinsqu/

Can be easily generalized to other shapes and more dimensions too.

Does the lack of n = 16 mean that there's no current proof?

> For the n not pictured, the trivial packing (with no tilted squares) is the best known packing.

Why not? It seems simple to me.

Simple doesn't mean easy. Factoring a 2048 bit number is very simple and very hard.

Because then anything that can be proven in a given axiom system is considered simple because it can be done by enumerating each possible sequence of transformations.

It might be easier to read the list of all discovered problems. Then everything else is in play.

Wait a second. Aren't there actually two different pentagon shapes in use here?

Look at the yellow and blue in the OP. They are actually mirror images of each other. Maybe a mathematician would say they are the same, but certainly not someone cutting tile for a bathroom floor. And if these were proteins trying to form a cell wall, that mirroring would be a serious hurdle.

All depends how you define "the same". Mathematicians use "congruency" which allows mirror images.

Ah. You learn something every day. I guess the guardian was therefore incorrect in their definition and should have said "congruent copies" ... but "copies" is also probably wrong. Congruent congruencies?

>"If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to tile the plane."

"Identical copies" is informal, as is "copies" for that matter. You can think of the information that is defining the shape as what is being copied, hence mirroring and other variations are allowed. A possibly related term to look into would be "isomorphism".

Is mirroring also allowed when 'filling' a 3d space?

Mirroring is allowed because in a higher dimension space, it is simply rotating the lower dimensional object by 180 degrees upon its own axis :-)

Ie. an ant can't flip the hexagon you lay flat, but you can. Similarly, you cannot "flip" the 3d object you have in terms of chirality, but a 4d being can :-)

I wish I had a bathroom to tile - I reckon this could be considered "in vogue" for the next 30 years or so, until they find a newer pentagon. Does anyone know if this can be coloured with 3 colours? Obviously 4 is possible due to the 4 colour theorem and 2 will not work due to to three faces sharing a corner.

> Obviously 4 is possible due to the 4 colour theorem...

Unless your bathroom includes a loop, such as all four walls (even with holes for windows and doors) or over the ceiling. Then the coloured area is no longer a plane, and so the 4 colour theorem does not apply.

Wouldn't all four walls still function as a plane, topologically? Think about looking into a cube (box) with one side open, so that you see five faces. The projection of those faces onto your retina or a photograph is a direct mapping to a plane.

Four walls plus the ceiling are equivalent to that open box. Four walls minus the ceiling are equivalent to a plane with a hole.

You have to include both the ceiling and the floor to break out of plane topology. And what you get is a sphere.

A sphere is also four-colorable, functioning equivalently to a plane for such purposes; consider stereographically projecting the sphere (minus a point) onto a plane.

I think it's kinda funny that most of the tessellations are just using pentagons to make other shapes that tessellate naturally. I suppose the same could be said of most tessellations though, but it's still interesting.

That's because there are only 17 wallpaper groups (https://en.m.wikipedia.org/wiki/Wallpaper_group). Any _repeating_ pattern must match one of them (non-repeating patterns by definition do not)

Can you NOT link to mobile version? Mobile phones switch to mobile automatically, desktop version does not.

The mobile version is very readable on a desktop browser, and it's probably easier for you to edit the URL on your PC than for the poster to edit it using their phone.

...all effort being equal. And assuming, of course, that comments are read (on average) once.

Assuming that the total effort expended by the few who really dislike the mobile interface is less than fiddling with a URL on a phone, plus gratitude toward the person for taking the effort to make the post at all.

Why not just be a bro or cool girl and do the thing you wish others would do? I'm always so grateful when others post links through paywalls, so:

https://en.wikipedia.org/wiki/Wallpaper_group

Now trip the f*ck out to some badass patterns :)

Many of these types are basically combining two pentagons into an octagon (or even hexagon) then tiling it across the plane. For some reason, intuitively those seem more easy to me (2 * 3, 2 * 4), so that you could just generate a bunch of them, and split them in two to create tessellating pentagons?

Even the example in the article can be viewed as a regularly tessellating nonagon. I don't see what's "irregular" about it? The article doesn't mention that word, but the HN title does.

"then tiling it across the plane": see https://news.ycombinator.com/item?id=10046349.

"just generate a bunch of them, and split them in two"

Try splitting a 'random' polygon into a set of identical convex (regular or irregular) pentagons.

"what's "irregular" about it?": this has been answered, but for completeness: the sides and angles of the pentagons aren't all identical.

irregular - with sides of different sizes

https://en.wikipedia.org/wiki/Regular_polygon

Sides and/or angles. It's possible to have all the sides the same length, and yet still not be regular. Similarly, it's possible to have all the angles the same, and yet still not be regular. It is instructive to construct examples of both types of failure.

Ah true. A lozenge has all sides with the same size but not the same angles

"Attack on the pentagon" ...aaaand the guardian has lowered themselves to buzzfeed's standards.

Not at all. Witty headline have a long history which was unfortunately cut short by google which, at least for a while, only rewarded the most boring, literal drivel. It's good to see the tradition making a comeback.

This headline is also completely unlike the buzzfeed-clickbait headlines. Buzzfeed tries to exploit psychological weaknesses (Mathematicians attack the pentagon – you won't believe what happened next).

Headlines like the Guardian's are much better in that they're still entertaining after you've read the story. Sometimes I think they're written more for the amusement of the editor than anything else.

Other guardian headline I saved: "Blight in Italy leaves pine nut nuts pining for more"

Its a joke, I found it pretty funny.

Isn't that every newspaper that has ever existed?

I think it's more that they have raised themselves to The Economist standards.

As you can see, it's some kind of column.

very well explained article. even I (with my little grasp on math) can see why its a big thing. Cudos to the author

"Every triangle can tile the plane. Every four-sided shape can also tile the plane."

Can someone point me to a proof of this?

The triangle is pretty easy: take two of the (same size) triangles, with vertices ABC and A'B'C'. Rotate and translate the second triangle to fit the matching side of the first triangle, e.g. AB to B'A'. You now have a quadrilateral with two pairs of equal sides (sides (AB')C and (A'B)C'). The angle on the corners comprised of the two triangles (e.g., C(AB')C') will add together to be 180 degrees minus the angle of the adjacent corners, due to the three interior angles of every triangle summing to 180 degrees. Duplicate that quadrangle and fit the second to a matching side. The angles put together will form 180 degrees, i.e., a straight line. Now you have indefinitely extensible strip. Place the strips next to each other and you've tiled the plane.

Not a formal proof, but close enough I suspect. See comments as well:

http://www.quora.com/Why-do-all-quadrilaterals-tessellate

Besides tiling your bathroom is there any use case for this? Or is this pure for fun and gaining knowledge?

Edit: The article is talking about building structures but isn't a triangle the most rigid form? And triangles are already used in building.

Sure there are. There are many processes where you want to "tile" a surface with identical smaller parts. I'd assume it's something that happens with solar panels. Possibly also on the molecular level with coatings etc. Having more options for the shape of the constituent parts may be an advantage.

see also https://en.wikipedia.org/wiki/Buckminsterfullerene, a molecule inspired by architecture.

Can I actually tile my bathroom with tiles like this? Do any tile manufacturers make them that shape?

EDIT: I found these, at least http://www.ballerhouse.com/2010/08/19/precision-concrete-til...

They probably will.

Should it be discounted if it were done for pure fun?

It's a question, not an opinion. He/she might just be interested in the practicalities.

Of course not. I'm just curious about the possibilities.

I am buying a house and the bathrooms need updating, so tiling your bathroom is actually the first thing that came to mind.

With only five points and < 180 degrees at each intersection, couldn't this problem have been solved by a computer via brute force in seconds?

From the article:

“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.

I would expect it to take longer than seconds, since there are many ways that these shapes can fit together, and there are many possible edge lengths.

It kind of was:

>“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.

This is cool, but I'm not sure why it's a hard problem to solve for a computer running through a ton of 5-sided polygons with some basic rules and attempting to tessellate them? Is there a reason that approach doesn't work, other than being rather un-romantic as far as discovering cool new things like this goes?

Just going to leave this here https://xkcd.com/793/

Yeah, great reasons why it doesn't work—what 5-sided polygons are you going to run through? The hard bit is the side lengths; there are good reasons to think that the angles cannot be too weird. But for example, this pentagon has one side of side length 1 / (sqrt(2) (sqrt(3) - 1)). How long is your exhaustive enumeration going to go until it finds that one?

Anyway, it sounds like clever enumeration is exactly what these authors did—but you do have to be clever to find something at all.

> 1 / (sqrt(2) (sqrt(3) - 1))

I don't mean in any way to demean their research, but I think your exhaustive search could build up formulas in exactly that form and iterate on them. It starts with a valid pentagon. Then it permutes that pentagon with an evolutionary method by modifying the formula.

For example, maybe the lengths it tries are:

1 / 2

1 / sqrt(2)

1 / sqrt(2) * 3

1 / sqrt(2) - 1

1 / sqrt(2) * (3)

1 / sqrt(2) * sqrt(3)

It would try both these and many others along the way.

1 / exp(2) ...

2 / sqrt(2) ...

Again, not to trivialize, but there are only so formulas made up of a fixed number of terms and operators, and as long as it's easy to check whether a shape is a valid pentagon, and whether it tiles, then I think you could check a considerable number of them. It looks like a number of the pentagon formulas have a few sides with complex lengths, while the others are simple or equal to each other, so you could bias the algorithm to search for those.

I'm sure there's way more complexity I'm overlooking, but that's how one might get started.

"I'm sure there's way more complexity I'm overlooking, but that's how one might get started."

That's how I'd start. There's probably a way to make the search a lot smarter. Here's the part you're overlooking, though:

"... there are only so [many] formulas made up of a fixed number of terms and operators..."

"So many" = "countably infinite." Paring it down to finitely many would require understanding tantamount to having solved the problem in the first place.

[Edit: I can words.]

In this case, (with the hindsight of knowing the answer), it would probably be easier to enumerate the angles, as they were all rational multiples of pi (with the biggest denominator being 12). Still, that gives a naive 5^144 combinations of angles to test. If you reduce this to proper, reduced fractions this comes down to 5^47 [1]. You can probably improve on this further by requiring that the angles form a (convex) polygon, but I suspect that still leaves you with to many to be practical.

[1] http://oeis.org/A005728

This could work for approximations. But if you want the exact value, this might not be super effective, as there are values without closed forms (that is to say, that are not expressible simply with the *-+/ and sqrt/log and other usual operations).

So your iteration procedure will probably get to a good approximation, but might not find the "core" expression.

I'd be even more curious about the computation used to determine if a given set of sides and angles tessellates, and how long that computation takes. Approaches to enumerate possible expressions don't seem that far-fetched, if the resulting possibilities can be evaluated quickly.

mind elaborating on why the angles can't be too weird? I thought it was odd that the angles given were all integers.

The one thing that the unit of "degrees" has going for it is that it tends to give integer answers (because 360 is so divisible). If you convert them into rotations around a circle, they angles become (1/6, 3/8, 7/24, 1/4, 5/12). (Multyply those by τ=2π to get the angle in radians).

EDIT:

To your question as to why angles can't be to weird, consider vertices, where the corners of the pentagons meet. Each vertex is composed of three angles, one from each of the three neighboring vertices (although it is not a-priori obvious that a vertex contains only three angles). The sum of these angles (in rotations) must be 1. This means that, heuritstically, you would want as many permutations of the angles as possible to add up to one.

The angles don't seem so weird though

That approach does work. From the article:

"We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey.

Right? I love when people ask "Why don't they do X?" without even bothering to read the damn article. Does the parent think that people just randomly stumbled upon this shape?

For those interested in the subject here is an interesting video about using penrose tiling for street tiling in Helsinki, Finland: https://www.youtube.com/watch?v=yxlEojkVJ0c

There's a lovely building at North Greenwich which is Penrose'd.

http://wordpress.mrreid.org/2011/02/26/penrose-tiling/ http://www.bdonline.co.uk/foa%E2%80%99s-peninsula-patterns-f...

I'd never heard that Penrose sued Kleenex. Not his finest hour.

I don't know why, but this was the first question that popped into my head after seeing the diagrams:

Does anyone make bricks in these shapes? Those would make an awesome paver pattern.

The first of the image of other tilings at the end of the article is almost obvious: it's just a hexagonal tiling where the hexagons are bisected.

Does anyone have a link to the actual paper?

I'm not certain there is an actual paper yet; I suspect from some of the phrasings that it was merely an announcement that their program succeeded, but a formal paper will take a while.

I did find this reddit post by Dr. Mann [1] where he says:

> We were just in the process of debugging and optimizing the code when our new example was found. Because we are in the early stages of the computational experiments, we were surprised to find this example so quickly. We are hopeful of finding more new examples as we proceed.

[1] https://www.reddit.com/r/math/comments/3fe347/15th_pentagon/...

we can shape building blocks like those, now

Perhaps the more impressive number is that they found 7 quintillion new irregular pentagons that can't tile the plane.

I've read your comment a few different ways. I'm concluding that you find it impressive that it took 7 quintillion tries to find a new pentagon that can tile the plane.

I think he just meant that the number 7 quintillion is an impressive number, more than 15.

For the record, I submitted the article with the original article title, as I always attempt to do. (Some articles have more than one "original" title. In doubtful cases, I read the source code of an article, to distinguish top-level headings from sharing titles from HTML titles, and choose the title that I think is most informative for a Hacker News readership.) In this case, I submitted the title from The Guardian with the "Attack on pentagon" title, and some member of the moderation team here changed it after a while. I'm gratified that this submission has received so many thoughtful comments both from mathematically trained onlookers and from amateurs at this level of mathematics like me. (The author of the original article has deep training and experience in mathematics, and is a regular mathematics columnist for The Guardian who likes word play in his column titles.)

I just read your HN bio. You're a very interesting guy! I'm an undergraduate student and one of my favorite conversation starters when I meet new friends majoring in STEM-fields is how they would change primary and secondary math education. Because we're just 18-22 year olds, the concepts that we struggled with and the methods we liked are still fresh in our mind... Though as each year goes by and I learn more advanced maths I find myself going back on how I felt before.

I hope that this bickering over his choice of a title and whether or not it's a pun doesn't rub you the wrong way. I think it's a great article otherwise and it's an incredibly interesting problem that made me think. Thank you for posting it!

For what it's worth, I don't really fault the author for coming up with this title. Ultimately the responsibility falls on his editor and I think his editor should have stopped it. Others are free to disagree with me (and by the downvotes, I'd say most are.)

It's a pun, not clickbait.

This is not the death of "journalistic integrity".

No, the original title is clearly clickbait – meant to be misinterpreted as something other than what the article is about. Here is what a real pun in a newspaper headline looks like:

http://threepanelsoul.com/2010/03/22/on-astronomy-minors/

It's a pun

PUN: Attack on the pentagon results in discovery of new mathematical tile

CB: Breaking News: Pentagon attack

The title, being a pun, is only meant to be misinterpreted for the first half of the sentence. It can hardly be clickbait if the reveal is made by the end of the anchor text.

The address changed:

http://threepanelsoul.com/comic/on-astronomy-minors/

The method was apparently a brute force attack, so the title is accurate.

Something can be accurate and still be misleading.

It would only have been misleading with a capitalised "Pentagon".

Well now we're arguing semantics. But okay.

I think that in the context of a newspaper when you read, "Attack on the pentagon" the first thought that most people have is that there has been an attack on The Pentagon. That's the whole reason it's a pun, right? Because you're lead to believe it refers to The Pentagon when it really refers to a 5 sided polygon?

That's an inappropriate thing for a newspaper to do. Go ahead and make puns, just not about attacks on government buildings.

The SPJ ethics code clearly states,

> "make certain that headlines, news teases and promotional material, photos, video, audio, graphics, sound bites and quotations do not misrepresent. They should not oversimplify or highlight incidents out of context."

At the very least you must admit that it's pushing the envelope, no? The fact that the HN mods changed the title from what OP submitted is evidence enough that I'm not alone in thinking it's a misleading headline.

> most people have is that there has been an attack on The Pentagon

Most people on this planet do not give a crap about some tasteless Northern American building. It is unlikely the first association with the word "pentagon" for anyone outside of the US.

I agree that most people don't think of The Pentagon when they hear the word "pentagon" alone. But the title says, "Attack on the pentagon". Furthermore, given that the Guardian is a UK newspaper, I would argue that most people in the UK know what The Pentagon is and associate the phrase with the building.

But let's assume I'm wrong and most readers don't know what The Pentagon is or wouldn't have associated the title with the US government building. If that's the case, then what is the point of the pun at all?

Just asked a number of people around how do they read this title. Nobody had "The Pentagon" in mind. London, UK, geeky population, so there might be some bias.