Paradoxes
0Thank you very much to the student in Mathematical Thought at the University of Arkansas that showed this to his class while I was sitting in on the class.
The Internet Maths Aperiodical – dinner and a theorem
3Hello. I’m Christian Perfect and here are some more maths links, after an almost geological gap of one month and one day. I told you this wasn’t going to be a periodical!
There’s going to be some sexy maths at the bottom of this issue of the Aperiodical. I’m just letting you know now so you can prepare yourself by getting in an appropriate state of mind. Maybe some of the links on the way down there will help with that.
Mathematicians like counting. We like counting how many of a thing exist and how many ways a thing can be done and how many kinds of finite simple group there are. In much the same spirit, this happy lady enumerates in a very pleasing video the twenty-five ways she knows of wearing a scarf fashionably. Is this all of them? Are they all different modulo rotation and translation? I think these questions require rigorous mathematical analysis, hopefully in the form of more snappily-edited videos featuring smiling ladies.
OK, so it was a bit of a stretch to claim that video about scarves was also about maths. But this video contains both Leibniz and scarves, so you know it’s maths. It’s also really interesting on its own merits: it’s a presentation by someone who knits stories into scarves using Morse code.
What’s the biggest number of chicken nuggets that you can’t order at McDonald’s without wasting any? In the 1980s, the answer was 43.
McDonald’s is more interesting than that though. In the US they make a thing called the McRib, a sandwich whose appearances seem as transitory and unpredictable as those of the Virgin Mary or the English summer. This article at The Awl makes the case that McDonald’s uses the introduction of the McRib to influence the price of pork. With a graph! So it’s maths!
One final fluff link before we get into the real maths. Here’s a logic quiz that you are guaranteed to fail because of your own PERFIDIOUS BRAIN.
OK, enough weaksauce links and burgers! Time for real maths!
Sphere-packing is a little bit easier now (and here’s the arXiv paper for those who, like me, get nothing out of pop-science writeups and their awful “hey-I-can-relate-to-that-but-you-haven’t-really-made-the-story-any-clearer” way of analogising abstract results). That isn’t the real maths though: it’s imprecise fiddling by a physicist. Real mathematicians only recognise one kind of hard: NP-hard. And circle-packing is hard.
It’s December now, so you probably want to put up some Christmas decorations. Make mathematical ones! Math Craft, which has been aggregating some really good stuff on how to make mathsy objects since it opened recently, has got you covered. They’ve got instructions for six-sided paper snowflakes, an origami christmas tree and they link to a fun article about the fractal patterns you can find in Christmas tree baubles. They’ve got loads of other interesting stuff as well, and they seem to be posting pretty often, so do have a look around. If you’re thinking about making the Christmas tree, be warned: I got roped into an attempt to make it before a seminar on Thursday and it turned out to be way more effort than I thought it would be. I gave up and I feel no shame about that.
A while ago, the people who make episodes of Futurama made an episode of Futurama where everybody’s minds get swapped. One of the writers came up with a group-theoretic justification for the way they resolve the head-swapping. It was pretty fun, and generated quite a few popular maths pieces in the media. Dana Ernst has written a really fantastic set of slides to go with a talk about the episode and its maths. He uses all the trendy web technologies – embedded videos, MathJax, and so on. He even includes an interactive Sage input so you can play about with the groups involved yourself. This is the future of maths exposition!
My last bit of real maths is something I found via Jeff Erickson. It’s an essay on the topic of wooden train-track sets, answering the question of which layouts of left- and right-turning pieces construct a closed curve. It’s a piece of mathematics so beautiful that you will want to kiss your monitor. I’m going to print it out and give it to people I meet. I think I love it.
Finally, now we’re in the mood, turn down the lights because here’s the sexy maths I promised. It’s a short story called “this is math”, by Joey Comeau.
The Internet Maths Aperiodical – temporarily a periodical
1Hello again, here’s another Internet Maths Aperiodical. Sadly, until I write a third edition, this Aperiodical currently has a publication period of thirteen days, so I’ll be as quick as I can with this one and I’ll make sure to post again either before or after another thirteen days have elapsed. I can only apologise for the temporary regularity of service.
Here are some more math links.
Shakespeare wrote loads of plays and sonnets and things. Or maybe he didn’t! Isn’t it fun being a literary scholar?
No, it’s more fun being a mathematician. So here’s a blog post about a theory that Francis Bacon was the real author of Shakespeare’s œuvre and hid encoded messages attesting to that fact in the Shakespeare folio by encoding letters in binary, and then using two different typefaces to represent 1s and 0s. As a description of historical events it is of course completely wrong, but logic dictates that all but one of the theories about Shakespeare have to be incorrect, so it shouldn’t feel too bad about itself. Anyway, the idea behind the bilateral code was good, and the people with the Bacon obsession played an important part in the mathematisation of cryptanalysis at the start of the 20th century.
Moving onwards, I have another stupid code for you, but this one’s so stupid it took some really clever people to crack it. The Copiale Cipher is an 18th-century manuscript which had evaded comprehension until some dudes with a load of computers and some fresh new ideas about how to use them had a go at it. Using some computer analysis the team first showed that the manuscript contained a real language and not nonsense (which is a very interesting field of study in itself) and then found it could be deciphered using simple frequency analysis and automatic clustering. They’ve written up their methods and thinking in a very accessible paper. It turned out that the book was written by an esoteric society with a predilection for hazing rituals. HOW TOTALLY UNEXPECTED.
I listened to a lovely little programme about rabbits on Radio 4 this week. It’s available on the iPlayer. There’s, apparently, a breed of rabbits called the Old English Spot. So the programme was all about the trials and tribulations of the people trying to breed the perfect English Spot, and the trials and tribulations of the people who have to live with the people trying to breed the perfect English Spot.
And it is hard to breed a perfect English, because a perfect English is one which looks exactly like the rabbit captured in the painting on this page. In particular, the spots down the rabbit’s side must be in exact correspondence with the picture.
What’s this got to do with maths? Good old Alan Turing worked out how spotty patterns come to be decades ago. Spots are the product of an ever-changing dynamical system known as reaction-diffusion, and are much more congenital than hereditary. So there will probably never be a perfect English, unless someone with a very good knowledge of reaction-diffusion systems and a steady hand with a pipette gets their hands on one in utero.
I saw this story on the BBC News site about how summer babies have it tough throughout their entire lives. I didn’t care, I was born in January. What I did care about was the following pair of sentences:
This reflects that these August children can be almost a year younger than their September-born classmates.
This age gap has not been closed by the time youngsters are ready to leave secondary schools.
Abbott and Costello redux. Sadly, the text has been changed to “This achievement gap…” since I first looked at it. The point about August children being nearly a year younger than their September-born classmates is still pretty much tautological but really I just wanted to post that Abott and Costello clip.
Finally, here’s something interesting. A while ago Twitter was abuzz with the startling revelation that the decimal representations of 1/7, 2/7, etc. all contained the same digits, cyclically permuted. A chap called Lawrence Brenton has written an article in the College Mathematics Journal explaining what’s going on. I like it a lot. It’s all to do with some simple group theory, which he explains very clearly. He makes a persuasive case that all teachers should know a bit of group theory because it leads to more convincing explanations of this kind of thing than number theory alone could.
So if anybody ever asks you what group theory is good for, tell them about this. It won’t take long.
SCC 42: Colin T Graham
0
(via http://reformsymposium.com/)
Samuel Hansen is back at the helm of Strongly Connected Components talking to Colin T Graham, the man behind the twitter hastag #mathchat. They talk about #mathcha, the intersection of mathematics and origami, and mathematics and music. Be sure to check out #mathchat, its twitter, its archive, and Colin’s twitter and blog.
Please remember to update your RSS Feed or iTunes Subscription to make sure that you get the new episodes.
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The Internet Maths Aperiodical – “It’s not class war, it’s math”
0Hello. My name is Christian and I spend all day looking at maths links on the web.
At least, that’s what Samuel thinks, so he’s asked me to post some of those links here. This will happen to no fixed schedule, so I’ve decided to call these posts “The Internet Maths Aperiodical.”
Let’s start with an FAQ which surely can’t have been asked frequently enough to merit the title.
“Why are there happy puppies on the cover of this Bayesian textbook?”
“The happy puppies are named Prior, Likelihood, and Posterior. Notice that the Posterior puppy has half-up ears, a compromise between the perky ears of the Prior puppy and the floppy ears of the Likelihood puppy. (The puppy on the back cover is named Evidence. MCMC methods make it unnecessary to explicitly compute the evidence, so that puppy gets sleepy with nothing much to do.) If the puppies bother you, see a solution at this blog entry.”
I can’t believe anyone would disapprove of happy puppies on the cover of a Bayesian textbook. Or any book.
“Systems, networks and strategies” (via Metafilter)
A maths course for arts students at the San Francisco Art Institute, which is now about halfway through. Because dynamical systems are very easy to visualise, it looks like a well-chosen syllabus for arts students. The wiki page I linked to is a fun stream-of-consciousness collection of links to fun stuff on the internet related to each topic in the syllabus.
“The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.”
Summary: Engineer with a gift for graphic design has trouble with algebra; says he was only shown symbolic methods in school/college; creates graphical tools to help get a feel for where solutions to systems of equations lie; writes essay about that; gives it an enormously provocative title. Take from that what you will. I don’t think anybody will dispute that sketching is an important way of getting a grip on a maths problem. Given that his evidence mostly consists of dynamical systems, maybe he should have taken the course at SFAI that I linked to above, and chilled out a bit.
In Soviet Russia, maths problem solves you! (Feel free to delete this one, Sam, it’s in appallingly bad taste)
In order to get into Moscow State University to read maths, applicants needed to pass an oral exam. Apparently, the examiners had a collection of “impossible to solve” questions they would give only to undesirables, in particular Jews. The questions had very simple answers but were worded in such a way as to make them very hard to solve. This paper by the amicable, happy, and not-at-all-odious Tanya Khovanova, along with Alexey Radul, gives both the problems and their solutions.
Baron Munchausen Redeems Himself: Bounds for a Coin-Weighing Puzzle
While looking at that paper I found this one, also by Tanya Khovanova, describing a coin-weighing puzzle. I think these are fairly well-known now, but the Munchausen framing caught my eye because, as is well known, my favourite number is a Munchausen number. I was planning on featuring a paper from my Interesting Esoterica collection each time I make one of these posts, so this might as well be the first!
Benford’s Law and the Decreasing Reliability of Accounting Data for US Firms
Like skateboarding and ska music, every so often the grand wheel of fashion swings around and Benford’s Law enters the public eye again. This is one of those times, and this article claims that accounting data for corporations have been deviating from Benford’s Law more and more since the 1960s. There’s a lot of humming and hahing in the comments about the applicability of Benford’s Law from both people who understand Benford’s Law and people who clearly don’t, which will either excite or frustrate you, depending on your personal policies towards other people’s wrong opinions. A rather good explanation of the idea behind Benford’s Law was posted in the metafilter thread about this analysis.
The Champernowne Constant is a rarity among constants – easy to define, hard to remember the name of. This paper, blogged about by the Math Tourist aka Ivars Peterson, gives a representation of the Champernowne constant derived, as far as I can tell, from a boozy night out. The picture of the author at the end of the paper certainly lends weight to that theory.
And finally, here’s your math masquerading as class war as promised, courtesy of a Mr Obama of the United States.
Combinations and Permutations is Coming Back!
2Here at ACMEScience.com’s new, world headquarters in Niagara, WI we have been bouncing ideas around as to what to do with the Combinations and Permuations brand since the original show has now run its course. I am happy to say that we now know: We are going to turn Combinations and Permtutations into a first person storytelling and collage show. Every couple of weeks I will post a request for stories on some sort of topic, and any one who has a story that relates can call into the skype account name ACMEScience, leave a voicemail, and I will edit together a story collage from those voicemails. Not only that I already have my first request for stories right now.
I want you to call in and tell me about a teacher or a class that got you really worked up about mathematics.
I do want to be clear, this can be a story about getting worked up in a good or bad way, for example it could be a class that made you super angry or the teacher that made you decide to study it as your vocation. Also, and this is important, you do not have to be a mathematician or math student to call in, I want everyone to take part in this project. If you have a story to contribute all you have to do is log into skype, add ACMEScience, and leave me a voicemail with the story on it. If this does not work for you, just record your story and send it to me via samuel@acmescience.com.
I really look forward to your stories, and we are all happy to get Combinations and Permutations back off the ground.
The Four Modes of Media Creation/Distribution
1One of the things that interests me the most is the way that media is created and distributed. I am not the first person that is fascinated by this, nor will I be the last, nor can I bring a “The Medium is the Message” Marshal McLuhan level of genius to the discussion, so instead I will approach the matter using my personal experiences and color everything I say with the brush of my personal beliefs. The reason that I am obsessive about these media questions is mostly a function of the times that we are now living in; I was born just early enough to have lived my first handful of years without internet access, the next few with sporadic on-ramps to the information super highway, and the rest with the always on connectivity that Lobot always asked Lando to pay those extra few credits to get. As my level of connection grew so did my wish to add my voice to the increasingly loud, and increasingly diverse media landscape. Finally three years ago, after a few aborted blog and website projects, I found a foothold for myself in the form of podcasting about mathematics. I currently have 3 mathematical podcasts, another in the works, and an odd podcast about 80s science fiction movies. I am also moderately active on twitter and tumblr, and recently started a mathematical writing blog with one of my podcast partners. While doing all of this I believe that I have isolated four distinct, and rather internet specific, modes of media creation and distribution: creating, aggregating, editing, and curation.
Creation is really the easiest one to speak about because it is exactly what its name implies, the person distributing the content is also the one that is creating. The first time I tried to put this into words I wrote that “Creation is the act of making something from whole cloth”. There are certain ways that creation is the one of these modes that has been the least changed by the new media world order, while the tools and specific mediums may be different a creator still has to make something original. The thing that is new and different as far as I can see are on the distribution side of creation. Gone are the days that required that a person who makes a new work needs someone else to help get that work into the world; now a creator not only crafts, she disseminates. We are in a world that when you go about creating something the project can be fully under your control. I really think that this is the key as, initially at least, the media that people are able to consume is entirely yours, no one has modified/mashed-up/remixed/edited/“enhanced” it. This sort of control over content is rare and fleeting. The engine that drives these modes of creation and distribution are also very important and creation has a very singular engine, the creator themselves. It is their idea both in concept and in completion. Once again it comes down to control, and, for me at least, it is this control that most drives me towards using the creation mode in media. A couple examples of creation are fiction blogs or sketch comedy group YouTube Channels. Those are admittedly obvious, but I never said they were examples you did not expect.
Aggregation on the other hand seems to be a completely new mode that could not have happened in an efficient manner before the advent of the connected world. This is because aggregation is all about the collection and distribution of large amounts of content. When a person is aggregating they scour the world for whatever media they can find on the topic that they have decided to aggregate. Unlike creation where the engine behind the media is the creator themselves, the engine for aggregation is the topic itself. This can become quite clear once one looks at the actual content that is being distributed by an aggregator because the quality is not actually taken into account. I imagine that a lot of people would be happy to tell me that even including aggregation on a list of media CREATION and distribution modes is fallacious as aggregation is not creation in any way; here I disagree because while they are not making a new individual piece of media, the aggregate that they gather has never been in a single place before and that clearly constitutes a new media object. This brings me to the control aspect of aggregation, which is essentially a forfeiture of control. In the end when the topic is king the only real control the aggregator has is the size and fineness of the net that they are able to cast. In the current landscape most aggregation is done not by an individual, but by the crowd itself, and the quality problem is dealt with by allowing the same people that form the net that finds the media to decide which media that is caught in the net should rise to the top. Reddit, Digg, and Metafilter are great examples of this mode.
Editing is where all of these modes start to get sticky. No longer in the world of creating something entirely new or aggregating another’s media, editing takes content that was created by someone else and repurposing it such that it portrays the editors opinion or vision. The act of editing is fundamentally a destructive act, because you are altering the media from the form that its creator had in mind. This does not mean that it is fundamentally a negative act though, edited media just represents the editors opinion and not the original content creator’s. Editing as a distribution mode is rather odd as the editor is distributing not only their edited produce which represents their outlook, but they are also distributing the original content creators, and let me just say hopefully citing who created media and where it can be found, media even though the original content may be found to contradict the message that is embedded in the new media. This is why the engine behind editing is the opinion of the editor, but also the original content. The latter is part of the editorial content as no matter how hard the editor works, there is no way that they can completely remove the original content. At the very least it is very possible the the original media could be found and presented next to the editorial by someone from the aggregator mode. The edit mode shows up in many different forms from the mashups of Girl Talk to the MSPaint enhanced photos of Perez Hilton.
Curation demands that carefully selection of some set of media and just as careful presentation. The whole idea behind the media that a curator chooses is that they are trying to represent some message through the what and how of their curated media. Curations can range from a selection of what a curator feels is the best media about a single topic to a much more abstracted representation of say JOY through the selection of art to be viewed as specific songs are playing in the background. This idea though has to be presented without the changing of the content of the original media that is presented. The topic trying to be communicated does not have to be the same as the original media, in fact many an effective curator uses contradictory media to great effect. The engine that drives curation is multifold; not only does the curators message drive, the curators taste, and the content of the original all spend equal time pushing down the accelerator. Distribution of curation is less fraught than distribution of editorial content or aggregate as unlike editorial content the original media is not being distorted by the curator and unlike aggregate the taste of the curator will demand a certain class of quality from the media. The curator though does have to be very mindful that the original message embedded in the media, or the channel of distribution do not overpower the new content developed through curation. Curation as a mode shows up both at blogs that curate things that they love, Kottke.org, to the radio show Re:Sound, audio as curated by the minds at Third Coast Audio Festival.
These may seem to be arbitrary classifications, especially as editing just seems to be creation using someone else’s cloth and curation could just be called aggregation with taste, but for me they have become a useful shorthand for the media that I voraciously consume on a daily basis. There is also no need for a person to stay neatly locked into a single mode, so why not become an editorial aggregator by remixing the streams of aggregate from your favorite upvote sites or a editorial creating by mashing up all of the original work on your hard drive. As much as I like classification, come on as a mathematician it actually hurts when I can not successfully classify, I know that these modes are as flawed as they are useful but I do hope that you will humor me and let me know just what mode, or combination there of, you feel that you most often fall under. If you wonder about what I feel about myself, well just wait for the next post where I will elucidate my personal modal position.
Combinations and Permutations Episode 71: Unless You Are In Z1
1Samuel Hansen is joined by Brandon Metz, Juan Mariscal, and Cody Palmer for an era ending episode as Samuel is leaving las Vegas within a week of this recording. So they decided to tackle the four most fundamental of all mathematical operations: addition, subtraction, multiplication, and division.
Topics Discussed:
Addition
Subtraction
Multiplication
Division
Download the Episode
Episode 71: Unless You Are In Z1
Leave us an iTunes review!
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Relatively Prime is Funded
0So after a rather long wait everyone came through at the death and the ACMEScience Kickstarter project Relatively Prime got funded. I want to thank all of you who helped make this dream of mine a reality. Right after the pledge that knocked us over the edge happened I talked with my Math/Maths co-host Peter Rowlett to talk about how it felt. Listen to how happy I sound.
Personal Modes of Creation
0A few days ago I posted a rather long screed on what I consider the four modes of creation and distribution in today’s varied media landscape. Today I am going to follow that up by stating where I wish to fall in such a landscape and why that is the decision that I have made.
The path in life that I have chosen at this point is media, and I therefore spend what would appear to be a rather inordinate amount of time to other people thinking about media’s place in the world and my place in media. My particular niche is the creation and distribution of mathematical stories, a rather specific and tiny niche to be sure. In order to cultivate a bigger audience I am always trying to evolve my products into styles and modes that I think will better suit the content and the audience. After having thought so hard about the four modes I decided that I hope to be a Curatorial Creator.
I love stories so much that I do not want to change the content or message of a story that I am lucky enough to get my hands on, there in lies the part of me that wants to curate. The problem comes from the equally large part of me that wants to tell the stories that I have inside of me, in other words I want to be a creator. For a long time I could not really figure out how to merge these two impulses and have therefore created media that allowed me to do one or the other. I was a creator with Combinations and Permutations and a curator with Strongly Connected Components, but then finally after much thought I finally came up with a project that allows me to integrate both creation and curation in Relatively Prime. The modus operandi of Relatively Prime is specifically molded to satisfy both parts of my media soul, as I will the show will be me using the stories of other people, without changing the messages of their words, to tell the stories in mathematics that I feel are interesting and important today.
I sincerely hope that everyone who reads this takes some time to deeply contemplate where in, I can not say “the” because these modes are just creations of my mind, my modes of creation they fall as there is no way to make a product that is better than one you really want to make. And I really want the world to be full of even more awesome media than it is already.