Math and Art: Woolly Thoughts

My chair recently forwarded an email from a former department member who’d found a really neat mathematical knitting/crocheting site: Woolly Thoughts by Pat Ashforth and Steve Plummer. They have a slew of designs with a mathematical bent. Here are a few of the examples.

The walls of Troy:

The Curve of Pursuit:

and Penrose:

More afghans can be found here (click on each afghan for a brief mathematical explanation), and more about the mathematical inspiration can be found here. There’s also more on the site about how they were constructed, since some were made as several small pieces and joined and others were created as a whole from the start (I thought I’d be able to tell from the pictures which was which, but I was surprised by a few!)

Pat Ashforth also mentioned (by email) that there is a yahoo group with a mixture of math/science people and knitters at http://groups.yahoo.com/group/woollythoughts/, and a Ravelry group (only available to Ravelry members at the moment), called Woolly Thoughts Fans http://www.ravelry.com/groups/woolly-thoughts-fans.

All photos are posted with permission (thanks!)

Mathematicians of the week: Eduard Cech and Witold Hurewicz

June 29 is the birthday of two topologists:  Eduard Cech [born in 1893] and Witold Hurewicz [born in 1904].  Curiously, in addition to sharing a birthday, they also share credit for the independent discovery of higher homotopy groups, a subject which Cech had spoken on at the 1932 ICM, and which Hurewicz  developed independently in the mid 1930s.

 

Mathematicians with birthdays or death anniversaries during the week of June 29 through July 5:

June 29: Birthdays of Eduard Cech [1893] (algebraic topology; Stone-Cech compactification) and Witold Hurewicz [1904] (higher homotopy groups)

June 30: Death of WIlliam Oughtred [1660] (slide rule)

July 1: Birthday of Gottfried Leibniz [1646]

July 2:  Birthday of William Burnside [1852] (abstract groups); death of Bartholomeo Pitiscus [1613] (coined term “trigonometry”) and Thomas Harriot [1621] (solutions of equations)

July 3: Death of William Jones [1749] (introduced symbol π)

July 4: Deaths of Guido Grandi [1742], Peter Tait [1901], Oscar Zariski [1986], and Marshall Hall [1990]

July 5: Deaths of René Baire [1932] and Oskar Bolza [1942]

Source: MacTutor

Hidden Heptagon

We recently found a heptagon hidden on the underside of a Wegman’s rectangle cake, in a design all around the edges. (I’m no longer sure why we thought to look at the bottom of the cake, especially since I’m pretty sure we hadn’t cut into it at this point.)

The heptagons are subtle, because they’re not regular and they look like they’re masquerading as either hexagons or octagons. But they’re there:

In retrospect, it might not be too surprising. Wegmans is, after all, the store that thought to put heptagons and nonagons on the design of their cookie cakes (from back in the day when I thought heptagons were unusual, instead of all over the place).

Hooray!

Not so simple games

Just sit right back and you’ll hear a tale,
A tale of a fateful trip
That started from a Rubik’s Cube
(Erno gets a hat tip).

The authors were the mighty Igor Kriz,
Paul Siegel brave and sure
With simple groups they tried that day
for a three game tour, a three game tour.

The puzzles started getting tough,
The Monster game was tossed,
If not for the courage of the fearless pair
The new games would be lost, the new games would be lost.

The pair made games from the math of three
Sporadic simple groups
With M12
M24 too,
(From Émile Mathieu, not his wife)
The final star*
From Professor John Co-on-way
Here on Sciam Isle!

* Dotto, based on Conway’s group .0

See this article in Scientific American about how the authors, inspired by the Rubik’s cube, used sporadic simple groups to create new games. The games can be found here.

Monday Math Madness Still Going Strong!

Monday Math Madness #9 is over at Wild About Math this week — solutions are due this coming Monday night/Tuesday morning at midnight. This week’s puzzle is the following:

Consider all of the 6-digit numbers that one can construct using each of the digits between 1 and 6 inclusively exactly one time each. 123456 is such a number as is 346125. 112345 is not such a number since 1 is repeated and 6 is not used.

How many of these 6-digit numbers are divisible by 8?

While you may use a computer program to verify your answer, show how to solve the problem without use of a computer.

You can find directions for submitting the solution at MMM #9. Problems are posted every other Monday either by Sol on Wild About Math or by Quan and Daniel at Blinkdagger; solutions are due a week later, and a winner is announced the following Friday. The problems have been fun to play around with, and isn’t that the point after all? (And as a bonus, they offer free prizes to the winners!)

Sierpinski Cookies

Last week I was inspired by the folk at Evil Mad Scientist Laboratories to make some Sierpinski Cookies. I was pretty sure that they wouldn’t look as good as those (and they didn’t), but they were fun to make.

Naturally, Godzilla did most of the work. He’s good that way.

At the suggestion of EMSL above, he used the recipe from Cook’s Illustrated (posted at instructables, where they use it to make pixel cookies). You know you can trust the folk at Cook’s Illustrated because they’re the ones who revealed (to me, anyway) that the secret ingredient in pie crust is vodka. Seriously — using a mixture of vodka and water lets you use more liquid, and the vodka boils off and the crust stays tender.

Here’s Godzilla mixing the ingredients.

Godzilla likes to take breaks while cooking to watch Chef Gordon Ramsay.

Actually, Godzilla spent so much time watching Hell’s Kitchen and trying to imitate GR yelling, “It’s RAW!” that he forgot to pose for photographs of the next part. He made another batch of chocolate dough (again, following the instructions at Evil Mad Scientist Laboratories (and the chocolate dough is amazing. Plus, no raw eggs, for those who worry about such things when eating raw cookie dough)). Then he formed the dough into long square strips, put them together with a chocolate strip in the middle, chopped them up, and put them together again (with a bigger chocolate piece in the middle) and Lo, Sierpinski cookies!

See? Not nearly as neat as the originators, but he still thought they tasted mighty fine.

The Math of Gas

TwoPi and I are Live On Location out in California right now, at a giant family reunion. Which is fun both for seeing people and for learning all sorts of new stories about relatives, like my cousin who just got back from visiting Afghanistan (for a month, for fun, just because it was interesting. And in fact it did sound really interesting — he said every single person he met was really friendly, and he didn’t get any negative reaction to being from the US.) (Edited to add: Hey, he has a blog! ) Or hearing about how my uncle-by-marriage and his brother tried three times to do a cross-country road trip by picking up a hitchhiker and taking the hitchhiker wherever they wanted to go on the condition that that person drove while the two brothers sat in the backseat “philosophizing” over a couple of cold ones.

And speaking of driving, Holy Toledo the cost of gas out here is even higher than in New York. Which made me wonder about what the benefits are cost-wise of owning a hybrid. Apparently hybrids are popular enough right now that dealers can charge a little more, either directly or indirectly (by requiring people to buy add-on packages). So here’s what I was wondering:

Suppose that getting a hybrid costs an extra $10,000 [which I think I read somewhere as a ballpark figure, though I had a hard time determining if it was accurate]. And suppose that a person drives 12,000 miles per year, which I think is roughly what we put on our car. Different hybrids get different mileage (see this site for example), but it looks like on average they might get 20 mph better than our car (around 46 mpg compared to 26).

How long does it take to break even if gas costs $4.75 per gallon like we’ve seen around here?

Well, in the situation above, we use about 461.5 gallons of gas per year, compared to only 261 on the hybrid. That translates to our spending about $2,192 on gas, while the lucky hybrid owner would only spend $1,239. That’s an extra $953 per year, which is a lot, but if the hybrid really cost $10,000 more it would take 10½ years to break even. That’s getting close to the life of the car.

Which isn’t to say that hybrids are a bad deal: rising gas prices would drop that 10½ years, as would a lower additional cost to buy a new hybrid, and in any case there are other issues at hand (like simply using less gas). But the numbers suggest that in buying a hybrid it’s worth looking at what the additional cost is, so that the better gas mileage can be put into a proper perspective.

Mathematician of the week: Jules Lissajous

Jules Lissajous was born March 4, 1822.  His doctoral studies were on vibrations of bars “using Chladni’s sand pattern method to determine nodal positions”.  This method of viewing vibration patterns entails covering the object with flour or sand, and inducing vibrations, often by stroking with a violin bow (or in a modern lab using amplified sounds at variable frequencies). The vibrations cause the sand or flour to accumulate into a pattern, indicating nodes in the vibrations of the object, locations where the standing waves of the bar have least magnitude.  

Lissajous died on June 24, 1880.

Mathematicians with birthdays or death anniversaries during the week of June 22 through June 28:

June 22: Birthday of Hermann Minkowski [1864] (mathematical foundations of space-time theories); death of Felix Klein [1925] (algebraic geometry)

June 23: Birthday of Alan Turing [1912] (foundations of computation)

June 24: Birthday of Oswald Veblen [1880] (geometry, topology); death of Jules Lissajous [1880] (visual study of vibration and sound)

June 25: Death of Alfred Pringsheim [1941] (analysis)

June 26: Birthday of Leopold Löwenheim [1878] (Löwenheim-Skolem Theorem); death of George Udny Yule [1951] (statistics)

June 27: Birthday of Augustus de Morgan [1806] (mathematical induction); deaths of Sophie Germain [1831] (number theory, elasticity) and Max Dehn [1952] (group theory)

June 28: Birthday of Henri Lebesgue [1875] (measure theory)

Source: MacTutor

Yet Another Heptagon

I recently found a heptagon in a surprise place: Wikipedia. Not on the heptagon page: I was actually looking up copyright info for the blue triangle I used in this post, and I noticed that the copyright statement, explaining that the figure was a simple geometric figure and so couldn’t be copyrighted, actually had a picture of a heptagon on the right to illustrate (click for a more legible version):

Go Wikipedia!

It does occur to me that the copyright notice might actually be copyrighted. Recursion anyone?

Geometers’ Vanity Plates

A car was parked in front of our house recently:

Clearly a special order license plate for someone who fondly recalls the Angle Side Angle Theorem in geometry. And since New York switched from an ABC 123 to an ABC 1234 format a few years ago, there are 10,000 such vanity plates available for this Theorem alone. How cool is that?

Rare math (and science) auction

Christie’s auction house made today’s news, with the results of Tuesday’s auction of the Richard Green library, a private collection of hundreds of rare scientific works.

The headline-grabbing items were one of the first telephone directories, and a first-edition of Copernicus’ De revolutionibus orbium coelestium (1543) that fetched $2.2 million. (A second-edition of this text, from 1566, went for just under $100,000.)

Some of the mathematical highlights from the auction:

• a 1566 Latin translation of Apollonius’s books V – VII on conics sold for c. $12K
• Menebrea’s account of Babbage’s Analytical Engine, translated by Ada Lovelace (published 1843) sold for $170K
• a first edition (1734) of Bishop Berkeley’s The Analyst sold for c. $9K
• a first edition (1713) of Jacob Bernoulli’s (posthumous) Ars conjectandi sold for $20K
• Boole: Investigation of the Laws of Thought ($4400)
• Jean D’Alembert Traite de Dynamique (1743) [$4750]
• Descartes Discours de la Method… (1637) [$116.5K]
• A 3 rotor Enigma machine from c. 1939 sold for just over $100,000
• Two Euler first editions: Methodus inveniendi… (1744) and the two-volume Introductio in analysin infinitorum [1748] sold for $7500 and $8750 respectively
• an original printing of Godel’s 1931 paper on the incompleteness of arithmetic sold for $44K
• Lagrange Mechanique Analitique [$18K]
• Laplace Traite de Mecanique [$20K]
• Leibnitz’s 1684 paper from the Acta Eruditorum on the differential calculus sold for $7500
• An 1834 printing of Lobachevsky’s Algebra or the Calculus of Finite Numbers sold for $17,500
• Oliver Byrne’s 1847 edition of Euclid (the colorized version), containing books 1 through 6, sold for $3500
• A first (1617) edition of Napier’s Rabdologiae (giving an account of Napier’s bones) sold for $80K; a second (1626) edition fetched $5K
• A first (1687) edition of Newton’s Principia Mathematica sold for $195K; a third (1726) edition for $12,500. A first (1729) edition of the english translation sold for $44,000.
• The first printed edition of Ptolemi’s Almagest (edited by Regiomontanus and Purbachius, printed 1496) sold for $50K
• A 1537 first edition of a work on ballistics and engineering by Tartaglia sold for $20K
• A first (1579) edition of Viete’s Canon mathematicus sold for $93K

There’s lots more cool stuff (including the original publication of the Piltdown Man hoax), and tons of significant early works in the physical and biological sciences.

Richard Green clearly had an amazing library. Seeing it scatter at auction is slightly sad, although we are rapidly entering an era of ready access to (scans of) first editions of many of these historic materials. Possibly the era of coveting rare vellum is nearing an end.

It’d still be way cool to own an Enigma machine, though.

MathFest Sudoku!

The folk at Brainfreeze Puzzles (creator of the famous Pi Day Sudoku) have created a new puzzle for MathFest 2008 in Madison this August!

The official rules are:

Each row, column, and block must contain each digit 1-9 exactly once. Since the middle block is missing, rows and columns that intersect the middle block will have only six visible numbers. In addition, each “Worm” in the puzzle contains numbers that increase from tail to eyes (although not necessarily consecutively). For example, a worm of length four could contain 2, 5, 7, 9 in that order, from tail to eyes.

Sudoku posted with permission. Thanks!

What happened?

Last night we had dinner out on the porch. The boys chugged their milk almost immediately, so I went and got the bottle out of the fridge, replenished their glasses, and then put the plastic top back on but kept the milk out in case they wanted more.

Maybe fifteen, twenty minutes later all of a sudden the top popped right off and landed a foot away. So either we have a poltergeist or there’s physics (i.e. applied math) involved. There hasn’t been much other evidence of a poltergeist, so physics it is! And I’m thinking that since it was a warm and humid evening, it had to do with either the air inside the bottle warming up and expanding (or was the air already warm from after I poured the milk?) or the milk itself warming up and expanding. Anyone know?

Mathematician of the week: Julius Petersen

Julius Petersen [1839 – 1910] wrote on a wide variety of mathematical topics throughout his career. His dissertation concerned geometric constructability, but he also published work on differential equations, analysis, number theory, algebra, and even such applied topics as mathematical economics and cryptography.

He is best remembered for his pioneering work in graph theory; his 1891 paper on the theory of regular graphs has been cited as the beginning of graph theory as a discipline. His 10 vertex graph, known simply as The Petersen Graph, is “the smallest bridgeless cubic graph with no three-edge-coloring“, and has become one of the standard (counter)examples studied in introductory graph theory courses.

Mathematicians with birthdays or death anniversaries during the week of June 15 through June 21:

June 15: Birthday of Nikolai Chebotaryov [1894] (generalized Dirichlet’s theorem on primes in arithmetic progressions); death of Giovanni Ceva [1734] (Ceva’s Theorem)

June 16: Birthdays of Julius Plücker [1801] (analytic geometry), Julius Petersen [1839] (Petersen graph), and John Tukey [1915] (fast fourier transform); death of Julius Weingarten [1910] (theory of surfaces)

One wonders if John Tukey’s parents had considered the name “Julius”.

June 17: Birthday of Maurits Escher [1898] (artist); death of Frank Yates [1994] (statistics; design of experiments)

June 18: Birthdays of Frieda Nugel [1884] (one of the first German women to earn a PhD in Mathematics [in 1912]) and Alice Schafer [1915] (differential geometry; founding member of AWM); death of Kazimierz Kuratowski [1980] (topology)

June 19: Birthday of Blaise Pascal [1623] (pioneer of probability)

June 20: Birthday of Helena Rasiowa [1917] (interactions of algebra and logic)

June 21: Birthday of Siméon-Denis Poisson [1781] (influential work on definite integrals and trigonometric series); death of Gaston Tarry [1913] (combinatorics; results on magic squares)

Source: MacTutor

Carnival of Mathematics #35!

The 35^th Carnival of Mathematics is up at CatSynth! CatSynth has many pictures and stories about cats, and also touches on a variety of other topics (the tagline is “cats synthesizers music art opinion”); sometimes there’s even overlap, as with the April entry on the book Calculus for Cats.

The Carnival made full use of yesterday’s (in)auspicious date, with facts about both 13 and Friday the Thirteenth, plus some non-13 related posts (including one with our own Godzilla).  More posts might be added over the weekend, so check back more than once!