Friday, September 21, 2007

Seventeenth Edition of the Carnival of Mathematics: "At Seventeen, I Learned the Truth"

[This edition of the Carnival is dedicated to our 12-year old Golden retriever, Teddy, who passed away on 9-15-07. He was everyone's 'best friend' and gave far more than he received. When my mom passed away recently, my 4-year old grandson was told that she became a star in the sky. When Teddy died, he comforted us by saying, "Now, Teddy's a star too and Nana will be able to walk him."]

What was the age of the Dancing Queen in an ABBA classic? (ABBA is one of my favorite palindromic groups).
Does anyone remember the title of Janis Ian's classic 60's song about teenage angst and includes the phrase that is part of the title of this post?

As it is becoming a tradition to begin our Carnival with comments on the cardinal number of our edition, I will choose a few more serious facts about 17, one of which I alluded to in the post inviting submissions to this carnival (anyone recall the pdf reference?). The sources here include the Wikipedia article on this number.

From MathWorld, 1/17 = 0.0588235294117647... (repeating). Note that the length of this repeating decimal is 17-1 = 16, which is maximal. I've always been fascinated by such full repetend primes...
17 is the third Fermat prime, as it is of the form 2^{2^2} + 1. Since 17 is a Fermat prime, heptadecagons can be drawn with compass and ruler. This was proved by Carl Friedrich Gauss. Now you know my pdf allusion: Pierre de Fermat...

17 is known as the Feller number, after the famous mathematician William Feller who taught at Princeton University for many years. Feller would say, when discussing an unsolved mathematical problem, that if it could be proved for the case n = 17 then it could be proved for all positive integers n. He would also say in lectures, "Let's try this for an arbitrary value of n, say n=17." In the class I teach on Wednesday nights, I naturally asked a young lady to give me her favorite number. I kid you not!

I could go on , but this takes away from the excellent submissions, so without further ado, I present to you exactly SEVENTEEN wonderful posts, nominations and favorites of my own choosing. By the way, there is no specific structure or bifurcation. Join me now as we take a pseudorandom walk through a slice of the math blogosphere...

(1,2) Denise over at letsplaymath, sends us two interesting posts:
Egyptian geometry and other challenges and Alex’s puzzling papyrus.
Denise described these fascinating articles:
"...two posts about ancient Egyptian math. The target audience for these was middle school to high school teachers and their students, especially homeschool families (who often try to coordinate their topics of study across the curriculum). The mathematical level is high school geometry or lower."
(3) Denise also included Math Quotes VII (too bad it wasn't XVII!).

(4) Vlorbik recommends that we visit Isabel from God Plays Dice who brings us Why g ~ π2.
Don't miss the astute comments, particularly, John Armstrong's concise summary of Isabel's fascinating post:
"What Isabel is pointing out is that at least one definition of "meter" over the years has been designed so that the numerical value of g coincides with the number pi^2."

(5) Vlorbik also links us to a relatively new blog from a brilliant and modest mathematician, William Gowers, who is contributing his insights to mathematics pedagogy. His post, "How should vector spaces be introduced?" reveals his wisdom and humility. Read the comments! Thanks, Vlorbik, for these excellent nominations!

(6) David Eppstein from LiVEJOURNAL brings us into the realm of combinatorics with Not the multinomial coefficients, a 3-dimensional view of binomial coefficients that is not simply an extension of Pascal's triangle. Some combinatorial background is needed here to fully appreciate this but the presentation is clear and engaging.

(7) John Armstrong of the Unapologetic Mathematician brings us a fascinating post on Newton Fractals, a description of Newton's Method and its behavior for f(x)=x3-1, requiring only differential calculus as a background. John develops the root search in the complex plane. By iteration and suitable coloring, John is able to suggest how a fractal is formed. His ability to clearly explain this tells me how very fortunate his students are.

(8) Mathematics in a Jack Reacher Novel is a post in Jeffrey Shallit's blog, Recursivity. Jeffrey discusses extensions of the mathematics suggested in a novel by Lee Child. The main character, Jack Reacher, displays some math-savant characteristics but Jeffrey enriches this with some fascinating problems involving numbers and their digits. Try the exercises and see if you match the results discussed in the comments!

(9) Jonathan from jd2718 offers us Puzzler puzzled. This is Jonathan's specialty, devising variations on famous number puzzles that require considerable thinking outside the box. In this case, one of his students had difficulty with interpreting the language of the original question and suggested another take on the problem. Now we have two great problems to work on!

(10) Mark Dominus from the Universe of Discourse offers The Missing Deltahedron , an advanced geometry article describing some research Mark is doing. Marc is following up his analysis of the convex deltahedra, which are the eight polyhedra whose faces are all congruent equilateral triangles. The problem is that there is a break in the pattern in the number of faces and it seems to upset the harmony in the universe!

(11) Meeyauw shows us an application of geometry to knitting. She mathematically builds on another blogger's post on knitting some of Escher's classics and even found a link to 'crocheting the hyperbolic plane'!

(12) Maria at Homeschool Math Blog brings us Number Rainbows, useful for primary math teachers. The idea is that you connect two numbers with an arc if they add up to a particular number, such as 13. Using different colors to connect these pairs makes an attractive rainbow pattern and may help students learn their addition and subtraction facts. Mathmom, in her comment, suggests an extension to Gauss' method of addition.

(13) Speaking of Mathmom, enjoy her insightful and provocative Calculator Rant. Perhaps the AP Calculus Committee set the gold standard a few years ago by developing a 2-part test, one with and one without the graphing calculator. This approach is definitely filtering down through the grades and Mathmom informs us that this has been adopted for some time by MathCounts.

(14) Andreas, from Figuring Out Computer Science, brings us a discussion of the The Most Important Problems in Computer Science. He sent out queries to some experts in the field and the main theme is that of complexity vs. simplicity, including the P=NP problem. Great quotes here but my favorite is "...simplicity is the ultimate sophistication." (da Vinci)

(15) Jacob from Winter's Haven, was one of the first to submit an article for this carnival:
Pure Math, Applied Math and A Priori Proofs. He presents a thoughtful and logical case for why scientists cannot prove natural phenomenon using only a priori, i.e., theoretical, mathematical arguments, in the absence of empirical corroboration. He explains that mathematics only provides possible models of reality which then need experimental verification. Nice...

(16) Marc has published his original research: New Prime Formula Helps Investigate Prime Numbers. After reading my post on Fallout From the Sieve of Eratosthenes, he thought I might be interested in his discovery and he is helping me to work through the details. Since the unsolved problems related to primes have been a motivating force for so many mathematicians, I know you will enjoy reading this significant find.
And speaking of primes... Coincidentally with my discovering Marc's research, I also found the accounts of the new algorithm developed by 3 mathematicians from India, which is considered a significant breakthrough in primality testing. You can find the fascinating details here and here.

(17) And last but certainly not least, Kurt over at Learning Computation threw Greg Muller's 'hat into the ring' by nominating Greg's engaging Hat Guessing Puzzles, The Revenge. Greg's blog is The Everything Seminar and if this is a sample of his writing, I plan on returning. The hat-color-guessing puzzle is itself is a fun exercise in logic for all of us, but Greg shows us how it is related to error-correcting codes in computer science.

Well, there it is. Exactly XVII excellent offerings. Enjoy them!

By way of disclaimers, I sincerely apologize for any omissions, invalid links, or errors of commission I may have made. Just let me know and I'll correct them. If you sent in your submission after 12 AM on Fri 9-21-07, it might not have made it into this Carnival, but I will post an addendum and errata if needed.

BTW, if you're wondering why I've made no mention of the Prof. Steen interview and the ongoing commentary, well, I think I just did...

Correction has been made in the link to #10: The Missing Deltahedron. Other links should be working as well. Sorry for any inconvenience caused by this.


Anonymous said...

Thank you very much for sharing your thoughts. It is always great pleasure to read your posts.

Anonymous said...

The link to the Wikipedia article on 17 is malformed.

Anonymous said...

Dave Munger at the ScienceBlogs blog Cognitive Daily had an interesting post of the preference for 17 as a random number earlier this year.

mathmom said...

Sorry to hear about Teddy. :(

Great looking Carnival. Lots of interesting-looking reading to keep me busy!

Jackie Ballarini said...

"At Seventeen" by Janis - love it! Great carnival, thanks.

Jackie Ballarini said...

The link to #11 Geometry & knitting is not working.

Dave Marain said...

The link to meeyauw's post (#11) seems to be working now. And thanks for appreciating Janis Ian as much as I do!

eric-- I had to delete the link to that wikipedia article.

kurt-- thanks for that reference.

mathmom-- thanks for the kind words about Teddy. The Carnival is running a distance second in my mind right now but I'm glad it is being well-received thus far.

There also seems to be some other script running in the background that is flawed. If you get any error messages, just ignore for now.

Andree said...

I am so sorry about Teddy. That is a huge pain. I hope time eases it.

Anonymous said...

What? No yellow pigs?!

Jackie Ballarini said...

Anon, - I mentioned yellow pigs in my post about the carnival!