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Carnival of Mathematics 85

85
85 by brighterorange

Introduction

Welcome to a new Carnival of Mathematics! Traditionally the Carnival opens with facts about the number, this time 85, but first I have an important point of admin to address.

From Carnival of Mathematics 59 in November 2009 until Carnival of Mathematics 84 in December 2011, the Carnival was coordinated by Mike Croucher. Mike said in a blog post (of course!) that he had “had a lot of fun doing so” but that: “Recently, however, I have struggled to find the time to give the CoM the attention it deserves and so it is time to hand over the baton.” We should all be very grateful to Mike for his effort over these two years. You can still find Mike blogging over at Walking Randomly.

Now, as a result, the Carnival has a new home (including an index of previous Carnivals) at The Aperiodical, a not-quite-yet-formally-announced blogging collaboration between Katie Steckles, Christian Perfect and me. We’ll do our best to look after the series for the time being. If you’ve been following the Carnival for a while you’ll know it only works because of a parade of volunteer hosts. The next few Carnivals are lined up but if you’d like to volunteer to host one on your blog later in the year please contact Katie. The other necessary element is submissions, and we have plenty of these this time, so let’s get to business.

Funky new Carnival logo by Katie Steckles

85

This is Carnival of Mathematics 85. The ever-faithful Number Gossip tells me that 85 has no unique or even rare properties, being merely composite, deficient, evil, odd, square-free and a Smith number. Beyond mathematics, Wikipedia tells me that 85 is the atomic number of astatine, the ISBN Group Identifier for books published in Brazil and the lower bound (due to incomplete research) found by Jorge Stolfi (2004) for The Hollywood Constant, the smallest non-negative integer that has never been used in the title of a movie.

Serious mathematics

(Not that the rest isn’t!)

Brent Yorgey at The Math Less Traveled wrote a series of four posts in response to Depressing Expressions by Patrick Vennebush over at Math Jokes 4 Mathy Folks. In these, he is working to prove why a certain iterative arithmetic algorithm always results in a factorial. Brent says:

In particular I’m proving it using a *combinatorial* proof, a lovely proof technique that (in my opinion) isn’t used or taught as widely as it ought. 

In part four Making Our Equation Count, he says,

I go through the different bits of the equation we’re trying to prove, and explain (with pictures) how to interpret each of them combinatorially.

Rebecka Peterson at Epsilon-Delta writes Extraneous Solutions of Log Equations–A Graphical Explanation. In submitting this post, Rebecka wrote:

Last semester a College Algebra student of mine asked why we sometimes get extraneous solutions when solving log equations. It was such a good question. I tried to do it justice in this post.

Drawing inspiration from the award of the Leroy P. Steele Prize for Mathematical Exposition to Aschbacher, Lyons, Smith, and Solomon for a work about the classification of finite simple groups, Gianluigi Filippelli at Doc Madhattan offers a post about finite simple groups and connections with physics in The classification of finite simple groups.

Frederick Koh from White Group Mathematics submitted Understanding MATTERS (4) saying:

Of late I noticed quite a few students (in online forums) experiencing difficulties in comprehending the concept of calculating distances between 3 dimensional vector planes in space, hence I am sharing an in depth explanation behind how things work.

Rohit Gupta at Kali & the Kaleidoscope investigates a new visualization of Möbius’ Mu, “a notorious function to classify all integers in three different boxes” in The 3 Pills of Möbius.

Art

Thomas Egense from Thomas’ mathematical adventures writes about an attempt to create mathematics-inspired art algorithmically using something called “Fractal flames”. Thomas defines fractal flames, details the algorithmic work involved, and gives several examples of the generated artwork, including the image below (used with permission).

a Fractal flame

Thomas Egense also submitted Dimensions (series of nine videos embedded below) calling this an “impressive graphical visualization of hard to grasp mathematical concepts like higher dimensions and topology”.

Last week Gathering for Gardner 10, the meeting of mathematicians, magicians, puzzlers and others inspired by the life and work of Martin Gardner, took place. Edmund Harriss, from Maxwell’s Demon, previewed his G4G10 talk in a blog post, The 2×1 rectangle and Domes. Edmund begins with the humble 2×1 rectangle, “not one of mathematics most celebrated shapes”, and ends up with structures built as accommodation at the Burning Man event.

Pop culture

Matt over at Math Goes Pop! writes in The Probability Games about the process used to select participants to take part in a fight to the death in the book and film The Hunger Games. Matt says “the rules here practically beg for some mathematical analysis” and Katie Steckles, who submitted the post, called this “just the kind of unnecessary mathematical analysis of a situation I like to see”.

Video game mathematics

Drawing inspiration from the recent arXiv preprint Classic Nintendo Games are (NP-)Hard bringing video games “out of nerdy obscurity and into cutting edge computer science”, Sam Alexander from Xamuel.com writes Toward the Mathematics of Video Game Glitches. Sam noticed a minor error in that article based on a glitch in Super Mario Brothers. Running with this theme, he defines video game glitches, game theoretically speaking, and focusing on “glitches which the player can exploit to win the game faster than intended”, defines a theorem (which he describes as “committing horrendous crimes against mathematics”!).

SNES vs. Xbox Triple60
SNES vs. Xbox Triple60 by avail

Performance and puzzles

Ben Nuttall writes about his experience as a maths busker.

“What is Maths Busking?” I hear you ask. Maths Busking is a street performance of mathematics whereby the buskers demonstrate mathematical ideas and engage the public in thinking like a mathematician

As well as a fuller explanation of maths busking, Ben shares some of the ‘busks’ he performed, his thoughts on the experience and some photos.

Birmingham City Centre by Maths Busking

Katie Steckles, writing at The Aperiodical, discusses a variant of a popular mathematics ‘mind reading’ trick. I don’t want to spoil the puzzle in case you want to play along at home, so go over and check out On Disreputable Numbers.

Paul Taylor, also at The Aperiodical, writes about a puzzle he designed for Katie Steckles’ Puzzlebomb. Puzzlebomb is a monthly puzzle sheet featuring all-new types of puzzles. Following the release of the April Puzzlebomb, Paul made the following claim on Twitter:

I guarantee there has never been, and will never be, another puzzle quite like Hilbert’s Space Filling Crossword

In Words to Fill Space he justifies this assertion and describes how he created the puzzle.

Paul also writes, again at The Aperiodical, about a class of puzzle in which a number of prisoners are all given hats and their fate depends on their ability to correctly determine the colour of their own hat. Paul offers “a nice variation on the theme that I heard about at a recent MathsJam” as Another black and white hats puzzle.

Maths Jam is a monthly meeting of maths enthusiasts in pubs worldwide to share stuff they like. “Puzzles, games, problems, or just anything they think is cool or interesting“. A recent development is blogging roundups of what happened at Maths Jam meetings. Recent outings, full of puzzles and mathematical goodies, include: Newcastle (February), Manchester (March), London (February) and Melbourne (January), and a set of photos from various February Maths Jam meetings.

Maths Jam London February 2012

Last month I attended Newcastle Maths Jam. While there we played with a puzzle that Out of the Norm states as:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

Dice and Dissection: a puzzle discusses this puzzle and gives a nice diagrammatic way to view the solution.

History and society

John Cook of The Endeavour writes about a wedding invitation written under the collective pseudonym of that “semi-secret group of French mathematicians”, Nicolas Bourbaki. In Nicolas Bourbaki’s wedding invitation, John explains some of the mathematical references and reveals how the invitation “nearly cost Bourbaki member André Weil his life”.

A post on Pat’sBlog goes to primary sources to highlight an error in Wikipedia in relation to the origins of the four-fours problem. That is,

using four fours and whatever mathematical operations that were allowed to make a number, or a set of numbers. 

The origins of the problem apparently lie in earlier similar problems, including a problem of four threes. Read about it in Before There Were Four-Fours, There Were Four-Threes.

Guillermo Bautista at Mathematics and Multimedia writes A US President’s Proof of the Pythagorean Theorem about the proof of the Pythagorean theorem formulated by James Garfield, the 20th president of the United States of America.

Alexander Bogomolny of CTK Insights offers a little piece of mathematics in the history of chronology given in Florian Diacu’s The Lost Millennium in a post entitled Chinese Remainder Theorem: an Application to Chronology. Submitting this post, Alexander said:

Chinese Remainder Theorem is a staple of early puzzle books, both European and Eastern. This is very satisfying to learn that the theorem finds practical and important applications in the science of chronology. The book by Florian Diacu where I found this application is an exquisitely written compendium of history and mathematics of calendrical calculations. The book deserves every praise and gets my wholehearted recommendation.

In a blog post here on Travels in a Mathematical World, I was very taken with an answer given to a question about working outside traditional academia by Neil deGrasse Tyson in an interview with Samuel Hansen for the Strongly Connected Components Podcast episode 45. The whole interview (18 mins) is worth listening to. My reflections can be found as Culturally an academic.

Over at Second-Rate Minds, Samuel Hansen writes about The True Importance of Friends, explaining the origin of a result in social network theory and its implications in epidemiology. 

I really noticed the absence of the Carnival back in February when I thought I might submit a couple of blog posts which got a particularly warm reaction and found the submissions form deactivated. The posts were Apparently Gauss got in this bar fight with Hilbert… and the follow-up Why do we enjoy maths history misconceptions?

Technology

John Chase from Random Walks writes about the surprising features of Microsoft Office Equation Editor, in which he makes the bold claim: “LaTeX lovers will love it”. Find out why at Microsoft Office Equation Editor.

I’ve saved the last word for this revived Carnival to the previous coordinator, Mike Croucher. Mike’s month of math software for March 2012 offers the latest news in the world of mathematical software (a month of math software has been a monthly series since January 2011).

The End

Well, that’s that for this Carnival. You can help spread the word by blogging, tweeting, etc. about the revival of the Carnival and directing people to this edition. If you’re hungry for more mathematics blog posts then there’s the previous Carnival of Mathematics 84, the latest Math Teachers At Play Carnival 48 is over at Math Is Not A Four Letter Word and you can get a weekly selection of blog posts from the Mathblogging.org Weekly Picks.

Future outings for the Carnival of Mathematics are queued over at The Aperiodical. Carnival of Mathematics 86 will be posted in May by Brent at The Math Less Travelled. Submissions for this are now open, so keep your eye out for great posts and get writing!

What sank the Titanic?

RMS Titanic

RMS Titanic, which sank on 15 April 1912 after colliding with an iceberg during her maiden voyage from Southampton to New York City, is the subject of seemingly a million TV programmes this month and a new article in Physics World. The article attempts to answer the obvious question:

When people ask the question “What sank the Titanic?”, at first glance the answer is obvious: she hit an iceberg. But that simplistic answer masks deeper and more substantive questions: why did the Titanic hit the berg in the first place and why did she sink so quickly?

George and Julian

Yesterday, the @mathshistory Twitter feed tells me, was the anniversary of the birth of Julian Schwinger (1918-1994), one of the great physicists of the 20th century. (Technically I queued this tweet up but there are a lot of days and a lot of mathematicians to remember…)

Schwinger is known to me particularly through his connection to the story of George Green. Green was a Nottingham mathematician who did work on electricity and magnetism (among other things) that, largely unrecognised in his lifetime, was discovered and brought after his death to further attention by William Thomson (later Lord Kelvin). The application of Green’s work in 19th century science was impressive but it found a new legacy in the 20th century.

At the 1993 celebration in Nottingham of the bicentenary of Green’s birth, Schwinger spoke about his use of Green’s work (a talk written up as The Greening of Quantum Field Theory: George and I).

Schwinger’s account is worth reading. He describes his use of Green’s work first on microwave radar during World War II, then in the development of the microtron and synchrotron particle accelerators, and finally to solve a problem on quantum electrodynamics, work which earned him a share, with Sin-Itiro Tomonaga and Richard Feynman, of the 1965 Nobel Prize for Physics.

In the preface to his most famous work, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), Green had written:

Should the present Essay tend in any way to facilitate the application of analysis to one of the most interesting of the physical sciences, the author will deem himself amply repaid for any labour he may have bestowed upon it.

Schwinger’s account helps us to understand how Green not only impacted the physics of his age, but how it continued to have impact beyond anything Green could have imagined.

What is mathematics?

This morning on Twitter Tony Mann asked the question: “This morning’s class is “What is Mathematics?” Answers in a tweet please.” Answers were collected via the #MATH1103 hashtag.

Some of the answers were what you might expect: patterns, abstraction, order.

Stuart Ravn sent a series of tweets giving his views:

Math is everything you can do with the abilities to count and deduce. There’s literally no end to the fun you can have. No joke.
Math is the only thing which is truly universal; it underlies and makes it possible to understand and communicate with everything.
Everything, immediately or ultimately, is mathematical and arises from mathematics.
Ask yourself what isn’t mathematics, and try to prove yourself right.
Noam Chomsky said of love, “I can’t tell you what it is, but life’s empty without it.” The same is viscerally true of mathematics.
I don’t just have enthusiasm for maths. I love it. It’s the closest thing to my heart after my family. I’m emotional about it.

Noel-Ann Bradshaw noted that “What is mathematics?” is the name of “an excellent book by Courant & Robbins revised by Stewart“. I have the tenth printing from 1960. Although this has a lot to say on the subject, it opens a discussion of historical development with:

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasise different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

I was interested in a related question: When does mathematics become something else? At some point some topic is clearly applied maths and at some point it is physics, astronomy, engineering, economics, computer science, biology, and so on.

We struggle with this a little on the Math/Maths Podcast, where we try to report news from mathematics and its applications. On Twitter I said that I think I tend to stray a little further from that which is unambiguously mathematics than does Samuel Hansen. We both report applications but I think mine are often more tangential than Samuel’s. This was quite noticeable on episode 80 this week when Samuel picked me up on an astrophysics story I was defending as involving statistical models. He said:

This is ‘Math/Maths’, not ‘Stat/Stats’, and it’s definitely not ‘Astronomy/Astronomies’. I’m assuming you put ‘s’ at the end of every single science – you have ‘Chemistries’ and ‘Physicses’, right?

and later

Now you’ve turned us into ‘Geology/Geologies’.

Answering my question on Twitter, Samuel said: “if an application has been around long enough to have own name, Physics astronomy or thermodynamics. It’s not math“. I don’t fundamentally disagree with this and some disciplines, notably computer science, were born this way. However, Sharon Evans made a very practical (if teasing) counter-point: “so it’s only maths if it hasn’t got a name? You’re not leaving much to report on in [the podcast]“.

Clarissa Wornack replied to say “Well, when you start writing code; it’s IT/software eng/comp sci; if you create something that is a material object; it’s eng” and Charles Brain said “Applied maths becomes Engineering when it hits the real world and money becomes part of the equation!” I don’t particularly agree with these. I know people who use high end computing to do mathematics and just because they are using computing as a tool (and writing bespoke code) this doesn’t mean they are doing computer science research. I also don’t agree that it stops being applied maths when it creates a material object. Defining mathematics as that which doesn’t involve the real world or money seems very self-defeating.

Felipe Pait offered this definition: “Applied math interests mathematicians and non mathematicians. Otherwise it’s pure math, or pure engineering. Math stops being applied math and becomes pure physics when it doesn’t interest mathematicians. An operational definition.

There’s something in this. When I think about “what is mathematics?” I am really thinking “what can mathematicians do?” I am particularly interested in what university mathematics graduates might become and would like this to be a broad as possible. I meet a lot of mathematics researchers working in different application areas. For example, back when I was doing the Travels in a Mathematical World podcast for the IMA I spoke to Paul Shepherd. I am much more naturally inclined to consider Paul a mathematician working in architecture than an architecture researcher who once did a maths degree. By extension, I am happy to include Paul’s use of geometry in architecture as part of mathematics than to exclude it.

Daniel Colquitt suggested “a lot depends on the user” and “in many cases, the distinction is arbitrary“. I think this may be the wisest view on the subject I have heard. From my point of view I am biased towards including topics on the edge within mathematics rather than excluding them, and maybe even collecting a little of the host subject along with them. I would rather cast the arbitrary net as wide as possible.

Nobel prize for mathematics

There’s no Nobel Prize for Mathematics

This is a common statement. I’ve certainly used it myself. Recently it occurred to me to be annoyed with this.

Nobel Prizes are awarded in physics, chemistry, medicine, literature, peace and economics, but not mathematics.

On the other hand, mathematics is widely applicable and I think I could convince you it is certainly used in physics (career), chemistry (career), biology (career), medicine (career) and economics (career). (Links to the excellent Plus Magazine and Maths Careers.) The case for literature and peace might be a bit harder to sell. But even without these two we still have a majority.

So perhaps from now on I will try to remember to say:

Most of the Nobel Prizes are for Mathematics1

[1. there is a fallacy here: for example, saying that some mathematics can be applied to economics does not mean that all economics involves mathematics. But, shh!]