carnival of mathematics

Carnival of Mathematics #38: 8-8-8

We at CatSynth are happy to be hosting the 38th Carnival of Mathematics, a bi-weekly round-up of mathematical posts from around the blogosphere. This is our second time hosting, and it falls on an auspicious day, August 8, 2008 or 8-8-8!

Eight is associated with good fortune and prosperity in Chinese culture, and so a trio of eights is especially fortunate. It is no accident that the Olympics in Beijing opened today.

Mathematically, 8 is of course a power of two, the third power of two in fact. 8 * 8 * 8 is 512. 8 raised to the 8th power raised the 8th power is…well…a very big number:

6277101735386680763835789423207666416102355444464034512896

Coincidentally, “888” is the name of a popular former product by my current employer.

So, with so much coincidence and good fortune surrounding the three eights, one would think this would be a great day for us at CatSynth. And they would be wrong, it is in fact rather bad day, capping a not-so-great week. So does that make us uniquely star-crossed? Probably not, but it does call into question the idea of using numerical patterns to divine good fortune.

And that brings us to our first entry, entitled Don’t Listen to Numbers from epsilonica. People have found significance in the patterns of digits of ? and the square root of 2 and countless other numbers for centuries, but as this post suggests, such patterns are inevitable. And very profound, when you think about the fact that this blog post, or any great work of literature or digitized music file can be represented by a unique number in binary, or English text in base 27.

So numerical patterns should be viewed with caution. So should the well-known statistical tool, the histogram, as described the Parable of the Histogram at ecthathy. The parable shows how the partitioning of data in to histograms can lead to very different interpretations, as a slight offset produces very different results for a group of students.

A numerical reality today is the explosion of online math videos, as report by the Teaching College Math Technology Blob. Videos on mathematical topics can be quite interesting and entertaining, especially for people who read blog posts about math. But they can be a source for media-oriented students to get answers to math problems without fully understanding the solution. And since correct solutions should be identical, it is more difficult to track down such sources than it is for essays.

Some of us learned mathematics not just to pass a class in school, but for the enjoyment, and perhaps for the realization that it is valuable to many aspects of our lives. You can read more about why we learn math at the blog It’s the Thought That Counts. This site has a lot of articles about science, religion and politics, and Barack Obama as well, so worth a detour…

Ok, back to mathematics. Sam Shah offers us a mind-boggling maximization problem, it is deceptively simple, but ends up brining in some rather difficult numbers (which I am sure have some strange and mystical patterns hidden inside of them). He also explains the Richter Scale and Logarithms, a topic close to many of us here in California.

Larry Ferlazzo presents a math word game, in which players have to chose the words that correctly describe the displayed numbers. It is geared towards beginning English Language students. I wonder if they can find the solutions to these problems on YouTube…

One of my favorite disciplines within mathematics is number theory, and our friends at Walking Randomly offer a method to generate Fibonacci numbers from matrix determinants. In particular, they can be generated from the determinants of a tridiagonal matrix with the imaginary number i in the “side bands.” It is always fascinating to see interesting mathematical concepts unified, in this case imaginary numbers, linear algebra and the Fibonacci sequence.

More interesting ideas with matrices and sequences can be found in this discussion of Phylogenetics and Algebraic Geometry at Rigorous Trivialities. Again, this crosses over into the complex numbers, and vector spaces.

The order of operations is a fundamental concept in arithmetic and computation. Apparently, it does not apply to the world of reality television, where producers have about as much mathematical literacy as a $1 calculator. Our friends at 360 analyze a recent mathematical puzzle given to contestants on the popular British TV program Big Brother.

I wonder how those contestants would do at coloring a plane, a interesting problem posed by Skeptics Play and explored in detail by Yoo at Stochastic Scribbles

Jon Ingram presents the game of Nim at Lessons Taught; Lessons Learnt. He starts with volunteers at conference to introduce this simple but fascinating game, and then plots the losing positions using Autograph. The result is quite surprising.

More observations on innumeracy in Why Can’t Johnny Add? at Staring at Empty Pages. The problems illustrated are not so much basic arithmetic as basic algebra, and how students can lack these basic skills in both 1983 and 2008.

From basic algebra, we move back to probability statistics, and normal errors for binomial and Poisson distributions. Textbooks say the normal distribution is good when “n is large”, but how large is large enough?

Next, The Endeavor presents an article on connecting probability and number theory. Prime numbers, and some properties of integers like the number of distinct prime divisors, can behave like random variables. This is an area of interest for me right now, not only with prime numbers, but also the other problems such as the Collatz Conjecture.

The above link to the Collatz Conjecture and related problems is from our friend Andrée of meeyauw, who this week presents aFriday Fractal: Pink Dragon Spewing Pink Flame.

That concludes the carnival for now. We will continue to post submissions through the weekend (including any we have missed due to 8-8-8 contrary fortunate), so please let us know if we missed you.


The Fourier Transform of a Cat

We at CatSynth never miss an opportunity to combine mathematics and cats. Recently, our friends at Walking Randomly posted this image:

It was created by Andrew J. Bennieston, and inspired by this comic from xkcd:

Unlike the number theory and other mathematics we like to post here at CatSynth, the Fourier Transform is part of our stock and trade. There are many variants, including the Discrete Time Fourier Transform which one of the basic tools of signal processing and sound synthesis:

Basically, what a Fourier Transform does is decompose a signal (or any time-varying mathematical function) into separate frequencies. If you have a spectrum representation of a sound, this is output of a Fourier Transform. Similarly, if you have a graphic equalizer on your stereo, it can be seen as operating on a very low-resolution Fourier transform, as it allows one to raise and lower different frequency ranges of sound.

For images, “frequency” corresponds to detail. Highly detailed areas of image that change from pixel to pixel are high-frequency, while areas of constant color or intensity are lower frequency. Another variant of the Fourier Transform, the Discrete Cosine Transform, or DCT, is more often used with images because it tends to put more information in lower frequencies.

Theoritcally, one should not lose any information when taking a Fourier Transform of a signal (or image) and taking the inverse to retrieve the original. However, in Bennieston’s image, which applies two DCTs to the original image, results in the “ghost” that loses a bit of the original detail. Certainly, some is due to the rounding error when doing any calculations on the computer, but it seems like more than that. Most, likely, the DCT is more sensitive the boundaries, i.e., what happens at the beginning or end of a signal.

DCTs are often used in “lossy” image and audio compression, such as JPEG for images. However, I have rarely seen them used in music applications, where one tends to see more general Fourier Transformations, which correspond more closely to an intuitive understanding of musical frequency.

As such, it would be interesting to work with DCTs in a musical context and see what transpires. If we ever get around to this project, we will certainly post it here on CatSynth.

This post is part of the Carnival of Mathematics which is being hosted this weekend by Logic Nest.


Carnival of Mathematics #35

We at CatSynth are delighted to be hosting the 35th Carnival of Mathematics.

The date “Friday the 13th” of course evokes many superstitions. And perhaps that is why we have so few entries so far this weekend. That having been said, we at CatSynth take a dim view of superstitions. Many of the traditions that led to the irrational fear of the number 13 (including skipping over of 13 in building floor numbers, etc) are the same ones that promote the unfair treatment of black cats. Neither has a place in a modern, rational, society.

Consider that 13th of the month is no more likely to fall on a Friday than any other day of the week. Over a period of 28 years, that is 48 “Fridays the 13th,” as described here.

Let us turn away from superstition and myth, so some of the more mathematical properties of the number 13. It is a prime number (indeed, the sixth smallest prime). It is also a well-known twin prime, and it is one of only three known Wilson primes. For those less interested that I am in prime numbers, it is also a Fibnoacci number, and the number of Archimedean solids.

You can find more about the mathematics and mythology of 13 at “13 the Unlucky”.

Last week’s host, the blog 360, has a post about 13 and other “unlucky” numbers. But they also contribute a demonstration, using Godzilla, of how to calculate the speed of a
dinosaur by its footprints.

Recursivity introduces us to the Open Problem Garden, a site collecting open problems in mathematics and theoretical computer science. He presents an open problem about discrete iterations that is easy to state yet frustratingly hard to solve, as are many of the great problems of mathematics.

Looking for a more modest challenge? Math and Logic Play presents a bicycle-racing puzzle. Practice your math skills while staying healthy and reducing your carbon footprint…

Larry Ferlazzo describes a site for math games and education in Tutpup Math & Spelling Games.

Andrée from meeyauw treats us to Sierpinski cookies and earings this week. Those cookies look pretty good, but challenging to make. Fractals are by nature very detail oriented, and I am by nature very not detail oriented. But I’m sure they would taste great in either case.

The state of math education (in the United States) in a frequent topic on Carnival of Mathematics. This week, female math professor critiques an elementary school math program at Life on the Tenure-Track. It seems that at least in some classes, teachers aren’t able to make it through the full syllabus.

More topics in math education can be found at Teach College Math, where we are introduced to the Math Girl videos. “Who says there’s nothing good on YouTube?”

For the more discriminating reader, we have Discriminating Determinants, third in a series from Matt-a-matical Thinking.

We get back to our theme of the number 13. The fashionablemathematician presents the sum of prime factors. For a prime number like 13, the sum is of course the same: 13. But are there other numbers whose prime factors sum to 13? Well, the number 22 has prime factors 2 and 11, which add up to 13.

Sometimes a mathematician can end up being an astrophysicist, especially on Friday the 13th, as is the case with this article from Math Trek.

That is all the posts we currently have. We will continue to add sites throughout the weekend, so please leave a comment or contact us if you would like to participate.



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