Archive for January, 2009

Math and the Fiber Arts

January 10, 2009

hyperbWe’re back from the Joint Mathematics Meetings in DC!  There were lots of great books at the Exhibits, lots of great rocks at the Smithsonian (more on that tomorrow), and lots of great talks at the conference.

One of my favorite sessions was the AMS Special Session on Mathematics and the Fiber Arts.  You can see the complete session (with links to home pages and talk summaries) here, but a few of the speakers also have web pages about their topics.

sara-marie belcastro talked  about braid words that describe the twisting that occurs when making helix stripe pattern.  She has an extensive page on mathematical knitting.

Irena Swanson showed how to make semi-regular tessellations while quilting — not just by sewing the pieces together, but by making use of some intriguing shortcuts in pieces and cutting.  She has a page with some of her mathematical quilts.

Mary Shepherd used cross stitching to illustrate cosets in group theory.  She has a web page about using mathematics to design cross-stitching and symmetry which also has some of the coset information from the talk.  It also shows her using cross stitch to show Frieze Patterns and most of the wallpaper patterns (but not all, because those that have a rotation of 60° can’t be put on the grid pattern).

Good times,  good times.  Now I just have to switch my mind away from knitting and back to classes!

The photo above is of two hyperbolic planes that I knit crocheted last year for my geometry class.

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Hot off the Press! A History of Math resource

January 5, 2009

capitol_domeWe’re hanging out in Washington, DC right now at the Joint Mathematics Meetings, which is lots of fun because the weather is beautiful (no snow on the ground and Actual Sun peeking through!  Except for the sleet predicted for tomorrow, but I’m going to ignore that) and there are a bunch of good talks.

One I heard of today (by Sarah J. Greenwald, with Gregory Rhoads listed as a co-presentor) was of a brand-spanking-new web site, online as of yesterday, put together for people who want to incorporate some history of math into Multivariable Calculus or Differential Geometry.  The intent was to be a general resource of information, but also to have classroom activities.

The link is http://mathhistory.mathsci.appstate.edu

So hey, be one of the first to visit!  And check back if you don’t see what you need, since the site is still in its early stages.

Note 1: I had wondered at first if this site would be like Convergence, but it’s geared towards a more specific topic so the two are complementary.

Note 2: When I saw Sarah J. Greenwald’s name, it was so familiar to me that I figured that I knew her from some math thing or other.  But then when I saw her I didn’t recognize her, and then I was too embarrassed to start a conversation with her even though she was really friendly because clearly I’d forgotten where I knew her from.  And then when I was looking up her name for this post to make sure I had spelled it right I discovered that the reason her name was so  familiar was that she’s the Simpson’s Math person (along with Andrew Nestler).  And she talked to David X. Cohen about Futurama Math in an interview that has been referenced on this blog.    So the mere fact that I knew her name so well that I thought we must be best buds is a testament to how delusional I am much I enjoy her work.

If you can’t trust celebrities, who can you trust?

January 4, 2009

Mariah Carey has been in the science news lately, amazingly enough.  No, not for a discussion of her five-octave vocal range, but rather for a discussion of Einstein’s Theory of Special Relativity.   Her most recent album appears to be named for Einstein’s mass-energy equivalence formula:  E=mc^2, as you can see from the album cover art seen above at this link. [I’m not including the image here, for copyright reasons.]

Apparently Mariah Carey was recently asked about the album title; she has been quoted as having said that it stands for “emancipation equals Mariah Carey times two”.

I suppose “Mariah Carey times two” might be loosely translated as “buy my two most recent albums”, as her previous album specifically referred to emancipation in the title.  But regardless of her intent, the algebraic notation does not refer to multiplying by two, but rather multiplying by a second copy of “c”.  (Perhaps the album should have been a series of duets by Mariah Carey and Charo….)

A British nonprofit organization, Sense About Science, recently called attention to Mariah Carey’s misinterpretation of exponent notation and other celebrity gaffes when speaking about science in public, in their “Celeb Audit 2008“.  While Mariah’s quote is the only one that is explicitly mathematical, many of the others could be useful fodder for discussion in a statistics classroom environment, and in many cases (as in the quotes from American politicians by the names of McCain, Obama, and Palin) involve far more weighty matters, and in all honesty far more disturbing misrepresentations.

The number 2009

January 1, 2009

Inspired in part by the legendary Hardy-Ramanujan anecdote as well as the post “What is interesting about the number 2009?” at  Walking Randomly, I offer a few arithmetic and demographic curiosities.

2009 is odd, which in particular guarantees it can be written as a difference of squares, most obviously as 1005^2 - 1004^2, but perhaps more interestingly as 45^2 - 4^2.

Being a difference of squares is equivalent to being a sum of consecutive odd numbers (in the first instance “2009”, in the second instance “9+11+13+…+89”);  some other curious sums that lead to our new friend 2009 include

\frac{1}{15} + \frac{2}{15} + \frac{3}{15} + \cdots + \frac{245}{15}

\frac{1}{305} + \frac{4}{305} + \frac{9}{305} + \cdots + \frac{ 122^2}{305}

\frac{1}{1004} + \frac{2}{1004} + \frac{3}{1004}+\cdots +\frac{2008}{1004}

\frac{1}{1005}+\frac{2}{1005}+\frac{3}{1005}+\cdots+\frac{2009}{1005}

Hmmm.  There are our friends 1004 and 1005 again!  Interesting….  [See below for conceptual continuity]

2009 can be written as 7^2 \cdot 41, which among other things guarantees that all groups of order 2009 are solvable.

Based on data published by Statistics Finland, the 2009th largest city in the world has a population of 207,000 people [a three-way tie between São Leopoldo Brazil,  Ashdod Israel, and Lutsk Ukraine].

Forbes famously publishes lists of the top n individuals, companies, etc… in terms of wealth, corporate size, market share, etc…  Forbes’ list of the world’s billionaires has 1125 entries, so we can’t identify the 2009th wealthiest individual.  However, we can use their data to extrapolate and estimate the net worth of the 2009th richest person.

forbes-billionaires1

The Forbes data has a very good fit to a power law distribution (in the graph, the blue rhombi squares represent individual data points, the black line the power law of best fit).  This model predicts that the 2009th wealthiest individual will have a net worth of 279.27 (2009)^{-0.775} \approx 0.770 billion dollars, or $770 million.

Their list of the world’s largest companies, alas, is the “Forbes Global 2000”, and as one might guess, has only 2000 corporations listed.  Unfortunately, Forbes publishes the top 2000 companies as measured by their own internal (unpublished) index, a weighted average of some sort based on sales, profit, assets, market value, etc….  Without knowing how their index is calculated, one can’t easily extrapolate to determine the size of a hypothetical 2009th largest company in the world.  Presumably it is similar to Mitsubishi Gas Chemical, the company ranked 2000th by Forbes in their list.

If these notes inspire you to find more instances or properties of 2009, add them to the comments at “What is interesting about the number 2009?” at  Walking Randomly.

May 2009 bring all of our readers a wonderful new year!

Added 1/1/09: I knew there had to be a rhyme and reason to the recurrence of 1004 and 1005 in both the difference of squares and the sum of fractions deal.  Here’s the scoop:

If x is odd, thenx = \left( \frac{x+1}{2} \right)^2 - \left( \frac{x-1}{2} \right)^2.

Furthermore, 1 + 2 + \cdots + (x-1) = \frac{ (x-1)x}{2}, so in fact

\frac{1}{(x-1)/2} + \frac{2}{(x-1)/2} + \cdots + \frac{x-1}{(x-1)/2} = x,

and similarly since 1+2+\cdots +x = \frac{x(x+1)}{2}, we also have

\frac{1}{(x+1)/2} + \frac{2}{(x+1)/2} + \cdots + \frac{x}{(x+1)/2} = x.