Autonomous Individuals Network, 23 SECONDS ov TIME,

I am happy to announce the release of 23 SECONDS OV TIME, a project of the Autonomous Individuals Network in which I am participating.

The collection contains 97 individual tracks, each exactly 23 seconds in length, with the total assemblage running for 37 minutes and 14 seconds. You can find my 23-second contribution, entitled “Four ideas in 23 seconds”, at track 80!

Volume One will be released in a limited edition of 123 hand numbered CD Copies.
This CD is planned for release on November 23,2010. Until November 23, you can download or stream the entire collection for free as a single MP3. In either format, I encourage everyone to check it out!

It is interesting to hear the pieces as a single unit, with such short durations they become phrases in a larger whole piece, sometimes with very sharp transitions.

You can also find out more about the Autonomous Individuals Network (and the significance of 23) at the official website.

RIP, Benoit Mandelbrot

Today we return to mathematics, and sadly note the passing of Benoît Mandelbrot. His work was very influential not only within mathematics and science, but also art and music.

Benoit Mandelbrot. (Photograph by Rama via Wikimedia Commons.)

He is credited with coining the term “fractal” (literally, “fractional dimension”) and is often dubbed the “father of fractal geometry” – and he is of course memorialized by the Mandelbrot set (which is technically not a fractal). I had written an article that touched on the Mandlebrot set and fractals for this site back back in 2008.

This quote from his official site at Yale sums up the wide-ranging applications of his work to science and the humanities:

Seeks a measure of order in physical, mathematical or social phenomena that are characterized by abundant data but extreme sample variability. The surprising esthetic value of many of his discoveries and their unexpected usefulness in teaching have made him an eloquent spokesman for the “unity of knowing and feeling.”

I did have the opportunity to take a course at Yale for which he was a regular lecturer (the course was taught by his former colleague at IBM Research, Richard Voss). The course was aimed at an introductory audience, and I think many of the students did not appreciate the opportunity it presented – but that left me with more time to directly ask him deeper questions in both lectures and seminars. At the end of the term, he signed my personal copy of The Fractal Geometry of Nature, which still has a place of honor on my bookshelf.

[click to enlarge]

In reading some of the online articles this morning, I also was reminded that he and his family were part the great exodus of Jewish intellectuals from Poland and other parts of Eastern Europe in the years before World War II. It’s a story that comes up time and time again among thinkers, writers and teachers who have influenced me of the years.

Theano’s Day

We at CatSynth are participating in Theano’s Day, an event to celebrate women in Philosophy. It is named in honor of Theano, the wife of the Greek philosopher and mathematician Pythagoras but also a scholar in her own right. In addition to promoting the work of Pythagoras, she put together her own work on mathematics, art, and beauty, all topics that are a regular part of this site. She is often credited with developing the Golden Mean, one of the most well-known ideas in aesthetic theory in which art, whether spacial (painting, sculpture, architecture) or temporal (music, drama) include structural elements based on the golden ratio φ, which is the ratio a / b such that a / b = (a + b) / a.

The number appears in the Fibonacci series as well, and in the well-known spiral that is used to describe proportions in nature and in architecture:

Theano is also credited with writings about child rearing and gender (although I am having difficulty finding a reference to this). Gender identity seems to permeate the work of many female philosophers over the centuries, indeed it seems to be an inescapable topic. The seventeenth century scholar and artist Anna Maria van Schurman published Whether the Study of Letters Is Fitting for a Christian Woman? in which she argued in favor of women’s education. In the modern era, there is of course Simone de Beauvoir whose writing is considered foundational for contemporary feminism. But I am personally more interested in the treatment of gender as it relates to other philosophical and intellectual topics rather than social, political or biological. How does the concept of “the feminine” relate to existentialism in de Beauvoir’s novels, or to Schurman’s art or Theano’s mathematics? This is not a topic that can easily be covered in a single post, or on a single day, but relates deeply to some of the directions I am exploring in visual art (i.e., photography) and perhaps later on in music as well. So perhaps the best way to see this day is as a beginning…

Pi Day

We at CatSynth once again join others in recognizing Pi Day today.  This time, Google is joining in as well with one of the special “Google Doodles” on their front page:

The image very nicely captures many of the well-known geometric and trigonometric properties of π in an abstract representation of the Google logo.

Of course, I tend to me be more curious about some of the more esoteric properties that interrelate to other parts of mathematics.  For example, consider  seemingly unrelated Gaussian Integral.

This is the area underneath the Gaussian function which is usually associated with normal distributions probability and statistics. The interrelation of π and e, which we have presented in previous years, is at play again here.

The approximation of the square root of π is 1.77245385…, and like π itself, is a transcendental number. In addition to its appearance in the Gaussian Integral and the Gamma Function (which we also presented on a past Pi Day), it plays an important part of the ancient mathematical problem of Squaring the Circle, that is constructing a square with the same area as a circle using only a compass and a straight edge. Because the square root of π is transcendental, it suggests that Squaring the Circle is in fact impossible. But that probably won’t stop some people from continuing to try.

Weekend Cat Blogging and Photo Hunt: Spiral

The theme of this week’s Photo Hunt is spiral, a shape that appears frequently in art, mathematics, nature, and of course in Luna’s expressive tail.

This is an old photo with a much younger Luna in our former home in Santa Cruz.  It is one of the best examples of the tight curl she often forms in her tail.  Here is a more recent example:

The spirals in Luna’s tail are all about motion, and often difficult to capture in a still photograph. However, you can see them very clearly in the recent videos from our performance.

With respect to music, a spiral was also one of the symbols I used in my recent piece for conduct your own orchestra night:

This is the classic Archimedean spiral, defined by the formula r = aθ. Essentially the distance from the center increases proportionally to the angle as one “spirals out” from the center. There are other formal mathematical variations on the spiral, such as Fermat’s spiral, which I find visually interesting.

Is mathematics, music and cats not a fine way to start the weekend?


Weekend Cat Blogging #249 will be hosted by our friends LB and breadchick at The Sour Dough.

The Photo Hunt is hosted by tnchick. This week’s theme is spiral.

The Carnival of the Cats will be hosted this Sunday by Kashim, Othello and Salome.

And the friday ark is at the modulator.

CatSynth video: dronecat

From sduck409 on YouTube, via matrixsynth:

“”Bigfoot the cat gets minimal on a Roland JD-800”

Bigfoot should really join in one of our Droneshift performances!

Lambert W Function

We at CatSynth return to the topic of mathematics for the first time in a while. In particular, we visit an obscure topic of personal significance. One day in high school I wrote down a seemingly simple equation:

2x = 1 / x

And set about trying to solve it. It certainly has a solution, as one can graph the functions 2x and 1/x and note their intersection:

In the graph above, the green curve is 2x and the black curve is 1/x. They intersect at an x coordinate equal to about 0.64. I actually moved to a variation:

ex = 1 / x

(somehow thinking that using e would make it simpler), and quickly approximated the solution as:

0.56714329…

While computing this number was relatively simple pressing buttons on a handheld calculator, describing it in a closed form proved elusive. Every so often, I would return to the equation, try to manipulate it algebraically or using calculus, but I was never able to do so.

Years later, in college, I found out that it was in fact impossible to solve algebraically, but that did not prevent mathematicians from naming both the constant 0.56714329… and the function necessary to compute it. Consider a function w(x)) such that:

w(x)ew(x) = x

The function w(x) is known as the Lambert W function, or “omega function”, and is named after 18th century mathematician Johann Heinrich Lambert. It is a non-analytical function, in that it cannot be expressed in a closed algebraic form, hence the difficulty I had attempting to solve my equation). However, one can see that w(1) is a solution for it. And w(1) [=] 0.56714329… is often called Lambert’s constant.

Although Lambert’s function does not have a closed-form expression, one can approximate it with a small computer program, such as this python program:

from math import e

def lambertW(x, prec=1E-12, maxiters=100):
    w = 0    

    for i in range(maxiters):
        we = w * e**w
        w1e = (w + 1) * e**w

        if prec > abs((x - we) / w1e):
            return w

        w -= (we - x) / (w1e - (w + 2) * (we - x) / (2*w + 2))

    raise ValueError("W doesn't converge fast enough for abs(z) = %f"

It was somewhat disappointing in the end to find out both that there was no closed form for the solution, and that the constant associated with the solution already had a name. But it was still interesting to learn about it, and to then apply it to other problems.

On that note, we conclude by showing that w(x) can also be used to solve the original equation:

2x = 1/x

can be rewritten as:

(ln2)xe(ln2)x = ln2

We can now use w(x) to solve the equation:

x = w(ln2) / ln2

which is approximately .6411857…

One thing I never tried in my youthful experimentations with this function was evaluate it with non-real complex numbers. While there are examples plotting w(z) on the complex plane, I would rather take some time to explore this myself.

I also have yet to find any applications to music or the visual arts, outside of literal usage in conceptual art pieces.

36

Today we explore some properties of the number 36. It is of course a perfect square, 6 x 6. But it is also a so-called “triangle number”, the sum of consecutive integers from 1 to 8. It is highly composite, having 9 factors, all 2s and 3s. Composites of 2 and 3 have a particular appeal for humans, and are very common in music (where most rhythms are subdivisions of 2 and 3), and in organization (e.g., dozens, etc.).

We will continue to post properties and facts throughout the day, but feel free to suggest your own in the comments.

Pi Day 3.14159…

[For Weekend Cat Blogging, please scroll down or click here.]

We at CatSynth once again, celebrate Pi Day on its three-digit approximation, March 14 (3-14).

We start with some interesting facts about the digits of pi. We presented statistics about the distribution in our 2007 Pi Day post. From super-computing.org, we present some interesting patterns:

01234567890 first occurs at the 53,217,681,704-th digit of pi.
09876543210 first occurs at the 42,321,758,803-th digit of pi.
777777777777 first occurs at the 368,299,898,266-th digit of pi.
666666666666 first occurs at the 1,221,587,715,177-th digit of pi.
271828182845 first occurs at the 1,016,065,419,627-th of digit pi. (that’s e for those who haven’t memorized it)
314159265358 first occurs at the 1,142,905,318,634-th digit of pi.

Last year, we showed the relationship to the Gamma function, and of course to Euler’s identity, which links pi surprisingly closely to the imaginary constant i and the number e. But it is also surprisingly easy to generate pi from simple sequences of integers. Consider the Madhava-Leibniz formula for pi:

Thus one can generate pi from odd integers and simple arithmetic. Another formula only involving perfect squares of integers comes from the Basel problem (named for the town of Basel in Switzerland):

In recognition of Pi Day, the U.S. House of Representatives passed a resolution this week:

And thus the sad history of pi in politics as exemplified by the Indiana Pi Bill of 1897 is put to rest. Now onto erasing the sad history of science and politics in general of the past eight years…

Knot Theory

Today we explore the topic of Knot Theory. Most of us have a conventional idea of what a “knot” is; and those who were once Boy Scouts may have a more formal idea. But in mathematics, a knot has a very formal defintion: an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). An example of a mathematical knot, the “figure eight knot”, is illustrated below:

Basically, it is a continuous curve in three-dimensional space that loops back on itself, crossing an arbitrary number of times but never cut or spliced. It relates to the conventional notion of knot as a piece of string connected at the ends.

Knots relate to other topics that have been explored here at CatSynth, such as Lissajous curves.

Many knots, when projected onto a two-dimensional plane form Lissajous curves.

It also relates to our interest here at CatSynth in highway interchanges, such as this the intersection of I-105 and I-110 in Los Angeles:

Indeed, our friend whaleshaman of Jelly Pizza suggested the link between highway interchanges and knots, although mathematically such interchanges are really tangles.

Knots (and tangles) can be arbitrarily complex with twists and crossings. But there is order in this twisty world, and indeed knots have properties analogous to numbers, such as equivalency and prime decomposition.

Two knots are considered equivalent if one can be converted to another by simple scaling (stretching or rotating), or performing one of several Reidemeister moves, twisting or untwisting in either direction, moving one loop (or segment) completely over another, or move a string completely over or under another crossing. Basically, this is any operation you can do on a closed string without cutting or splicing it.

Here at CatSynth, we are quite familiar with Reidemeister moves, as they seem to occur spontaneously on our audio cables:

Even more interesting is how knots can be decomposed into prime knots. Just as any integer can be expressed as the product of prime numbers, any knot can be expressed as the “sum” of prime knots.

Here is a chart of the first 15 prime knots:


[Click image for original and more info]

Here, the prime knots are grouped by the number of crossings. For example, the trifoil knot (second from the left on the top) has three crossings. The circle is a degenerate case, known as the “unknot”, with zero crossings. As the number of crossings increases, the number of possible prime knots also increases. For example, there are seven unique prime knots with seven crossings.

For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table.

n number of prime knots
1 0
2 0
3 1
4 1
5 2
6 3
7 7
8 21
9 49
10 165
11 552
12 2176
13 9988
14 46972
15 253293
16 1388705

This sequence (formally listed as A002863), appears to grow exponentially. Indeed, results by Welsh show a lower bound of 2.68 for the exponential base, and an upper bound of 10.40 due to Stoimenow. However, as far as I can tell, there is know known analytical formula for this sequence, and the values for n=17 and above are not known.

I find such sequences of numbers fascinating. Where to they come from, and how does one figure out the next value? In the case of prime knots, these appear to be open problems.

For more information, Giovanni de Santi has an excellent introduction to the theory of knots. Another paper by [url=http://algo.inria.fr/bsolve/constant/constant.html]Steven R. Finch is a resource for advanced analysis of knots and tangles, including more on counting prime knots.