Mathematics

The logistic function revisted

Today we revisit one of our most popular articles, on the logistic function:

f(x) = ax(x-1)

In the original article, we demonstrated how we can use this function as a “logistic map” by iterating (i.e.,applying it repeatedly to the previous result). The logistic map produced different sorts of behavior depending on the values of a. For example, for some values of a, iteration settles into a cycle, bouncing among two or more points on the function.

The original article provided more examples and more detail about the mathematics, and those who are interested are encouraged to go back and check it out. One of the main this we discussed was how one can characterize the logistic function over
different values of a using a graph called a bifurcation
diagram. As the values of a increase (a is labelled as “r” in this graphic I shamelessly but legally ripped off from wikipedia), one can observe vertically the period doubling where the logistic map converges on a single value, then bounces between two points, then four, then eight, and so on, until the onset of chaos at approximately 3.57.

When a is greater than or equal to 4, the function “diverges”, i.e., it just gets bigger and bigger (or smaller and smaller because the numbers are negative) when you repeatedly apply it.

The bifurcation diagram shows what happens for real values of a, i.e., all integers, fractions and other numbers that can be expressed as a decimal. But suppose we allow a to be any complex number, or any combination of real and “imaginary” numbers (i.e., square roots of negative numbers). Real numbers can be expressed a line, while complex numbers are expressed on a plane. So we can produce an analog of the bifurcation diagram over a plane instead of a line as above.

In the following diagrams, we are looking at the complex plane of
different values for a. If the logistic map converges to either a single value or a cycle, the location on the plane is colored in black. If it diverges, i.e., gets infinitely farther away from zero, then the location is white. Unlike for real numbers, where the convergent “black” values of a form a simple line segment, for complex numbers the set of convergent values is a lot more “complex”:

Zooming in, we can see the structure of the set, with lots of smaller “bubbles” and “filaments” off of the main circles. The large circles are the complex-number equivalents of the single-line sections of the bifurcation diagrams, with the small bubbles representing cycles and period doubling.

Some readers might recognize similarities between this set and more well-known Mandelbrot set:

The similarity is more than coincidence, as the Mandelbrot set is based on a map similar to the logistic map. But the Mandelbrot set has the pronounced cardioid shape and asymmetry different from the logistic-map set. Zooming in further, we see that filaments and local areas of the two sets have more similarity. Indeed, we see small “Mandelbrot-like sets” at the junctions of the filaments:

It is interesting how these miniature versions have a shape similar to Mandelbrot set rather than the double-circle of the logistic-map set.

Although these sets have a “fractal-like” qualities, neither is a fractal in the strict sense of the word. They are not strictly self-similar, nor do they have fractional dimension. Nonetheless, we are featuring the logistic-map set as a “Friday Fractal” , an event started by our friend Andrée at meeyauw.

I am not sure the logistic-map set has a name like the Mandelbrot set has, so how about calling it the CatSynth set?

Submitted to Carnival of Mathematics #29.







Fun with Pi (Day)

We saw this picture on meeyauw, and thought it was a good way to open our own Pi Day offering. Pi Day, or π day is celebrated on March 14 (3/14) of every year.

π does turn up in some interesting places besides circles and standard trigonometry (and LOLcat photos). There is of course Euler's famous identity:

which unites π with four of the other most famous constants in mathematics: zero, 1, i (the imaginary root of -1) and e. But it also turns up in some more surprising places. Consider the well-known factorial function, where n! or “n factorial” is the product of all the integers between 1 and n. For example:

5! = 5×4×3×2×1 = 120.

Simple enough. But of course some troublemaker is eventually going to ask for the factorial of 1/2. Not so easy. Fortunately, there is a function, called the Gamma function, that provides a solution:

Not really as simple as the original integer-only factorial. Once calculus is involved, might as well forget about it. But if you go through the trouble of plugging in 1/2 to the formula, you get the following intriguing result:

or

So the factorial of one half is one half the square root of π. Who knew?

Mobius Transformations video (A YouTube hit)

From our friend Andrée at meeyauw:

An informative and beautifully produced YouTube mathematics hit. There is a YouTube link, where you will find a link to the original source of the video. You can download the video to your own drive. You really should watch this, even if you don't like math (gasp).

Well, we at CatSynth of course love mathematics, and appreciated this video. It is about Möbius transformations, an important concept involving complex numbers and geometry. You can watch the video below:



Science, light and cats

A few scientific articles that made the popular press this weekend also piqued our interest here at CatSynth. They both involve electromagnetic phenomenon – which in our daily world is most commonly observed as light.

So here's a story about light and cats, or rather a cat engineered to glow in the dark:

South Korean scientists have cloned cats that glow red when exposed to ultraviolet rays, an achievement that could help develop cures for human genetic diseases, the Science and Technology Ministry said.

Three Turkish Angora cats were born in January and February through cloning with a gene that produces a red fluorescent protein that makes them glow in dark.

You can read the whole article here. It is quite interesting, though we at CatSynth are not so sure about genetically modified cat concept.

We now move from glow-in-the-dark cats to the field of quantum computing, in which quantum states of electrons are used to store computational values (much like semiconductors are used in conventional computers). From the folks at ZDNet Australia:

Researchers from the University of Queensland have taken a significant step in the quest to build a quantum computer, creating a light-based quantum circuit capable of basic calculations and moving quantum computing closer to a becoming a reality.

Theoretically, quantum computers leave even today's most powerful conventional supercomputers in the dust. It has also been long known that hypothetical large scale quantum computers could find the prime roots of large composite numbers, allowing them to “crack” modern data encryption.

This additional computing power is a result of the quantum bits, or “qubits”, upon which quantum computing is based. Qubits are special bits that use the quantum properties of subatomic particles to make calculations. Quantum computers take advantage of a special quantum property called “superposition”, allowing one quantum computer bit to act as many.

Pretty hard core, but those interested are encouraged to read the full article, and maybe a bit more about quantum computing.

Cats of course have a famous history in quantum physics as well…



New Podcast: "Bi-fur-cation" demo


Click here to listen or subscribe.

Some might consider tonight's podcast a “rerun” of sorts, as this musical example was featured in the CatSynth article The Logistic Function and its Discontents. This is actually one of our most popular articles of our stats/records are to be trusted, combining mathematics, the work of Antoni Gaudí, and some of my favorite electronic-music techniques. Those who have not read the article are encouraged to do so – I hope to post a follow-up one of these days. Or you can just listen to the podcast as a musical curiousity.

CatSynth 1st Anniversary

Today we celebrate CatSynth's first anniversary.

It's been one year since we posted this photo on July 19, 2006:

The idea came from a friend who said something like “dude, you should do a website about cats and synths.” So I did. Really didn't have too much idea what I would write about. I quickly learned that there was quite an abundance of pictures of cats and synths, and sites like matrixsynth had been collecting such images for a while. Indeed, one of the first external “CatSynth pics” to be posted here was of matrix's own cat JD:

Sadly, we recently learned that JD passed away this month. We offer matrix our sympathies. It's always sad to hear about cat friends passing away (or human friends, for that matter), and we at CatSynth have seen our share this past year.

I expect to post more this evening reviewing the year with photos, not-so-useful stats and the other things we at CatSynth like to think we do well…





CatSynth pic: lissajous (chaos link)

Via matrixsynth:

Originally from gerald:

My cat loves the Lissajous this thing generates

So what is a “Lissajous”? it is actually short for Lissajous curves or Lissajous figures, a class of 2D (and 3D) curves describing complex harmonic functions, or more simply multi-dimensional sine curves. The following equations describe a general Lissajous curve on an x-y coordinate plane:

x = A sin(at + φ)
y = B sin(bt)

Most of the time, one leaves out the A and B, which case all the curves fall on a convenient unit square.

The most commonly described Lissajous curves set the phase term φ to π/2, i.e., a standard cosine function, and have a and b at integer ratios, like 1:2, 6:5, etc. You can think of these as natural harmonics, like in musical sounds. You can see a few of the graphs below, first for a=1 and b=2:

Here are 3:2 (a:b), and 9:8, respectively:

As you can see, the higher the ratio, the more complex and dense the figure. If you add all the figures up together, you should be able to fill the entire unit square.

There are all sorts of interesting special cases. For example, if you set a and b equal, you will get a circle. If you additionally set the φ to zero, you will get a straight line. Finally, you can mess with different values of φ, like 0.3 in the first drawing below, or set a and b to non-integer values, to get all sorts of interesting variations:

It is interesting to think about these sorts of functions by relating them both visually and aurally (i.e., synthesizing the corresponding waveforms), but we will leave that as an exercise for interested readers, perhaps returning to the topic in a future article.






Trying a little experiment. Trackposted to Gone Hollywood, Conservative Cat, The Crazy Rants of Samantha Burns, and The Pet Haven Blog, thanks to Linkfest Haven Deluxe. The links here and in the trackbacks do not necessarily reflect the opinions of this site or its contributors.

Fluxus

I needed some intellectual diversions over the last couple of days, and last night I took another look at concept of software art that has intruiged me recently.


Fluxus is a system for live software art that combines programming with audio, visual and interactive elements. It comes to us from the same people who made Quagmire, in which programs ran inside of monochrome images.

Some interesting statements from the Fluxus website:

act of a flowing; a continuous moving on or passing by, as of a flowing stream; a continuous succession of changes

On a more technical level:

Fluxus reads live audio or OSC network messages which can be used as a source of animation data for realtime performances or installations. Keyboard or mouse input can also be read for simple games development, and a physics engine is included for realtime simulations of rigid body dynamics.

The use of OSC is of particular interest, as such a system becomes an interesting companion to Open Sound World. It also rekindles my idea of providing an OSC-based livecoding environment for OSW.

Unforunately, I have had some difficulty getting it installed (or compiled) for Mac OSX, so I haven't been able to do much myself. Hopefully I will be able to get that working soon…