Category Archives: Beauty

31 Days – 3 Forest Paintings

If you’ve been following these 31 Days of Wonder closely, you may have noticed that I missed a post on Friday. Here’s an extra one today to make up for it.

I’d like to share with you one of my favorite classical guitar pieces — actually a 3-piece suite called 3 Forest Paintings by KostantinVassiliev.

What always amazes me about music is that the same 12 notes of the chromatic scale are available to everyone, yet most of us could not in a thousand years come up with pieces to match those of accomplished composers. What is it about a good composer’s brain that makes his style unique, and so unattainable by people without his gift? Likewise for the performer, who in these videos is the  very talented Roman Viazovskiy.

31 Days – Crop Circles

If you followed the links at the bottom of yesterday’s post, you saw the wonders of the Julia set. Actually, the Julias are a whole family of sets. Here is one of them. In this representation, a white point is a member of the set. The other points are not, with a darker shade of grey corresponding to rapid disqualification.

A Julia Set

The Julia sets inspired some famous crop circles that I thought you might enjoy. As you probably know, a crop circle is a pattern made in a standing crop. The early ones were just circles, but these days they are much more elaborate.

There is considerable controversy over whether crop circles are produced by humans, aliens, or even Gaia herself. Whatever their origin, they are beautiful works of art, well-qualified for August’s 31 Days of Wonder.

This Julia set-inspired formation appeared right next to Stonehenge (!) in 1996.

Julia Set Crop Circle

What could have caused it? Crop circle researcher Lucy Pringle interviewed an eyewitness who claimed it appeared under a mysterious, swirling mist during daylight hours., a Website for human crop-circle artists, presents a more skeptical interview of someone who claims it was made the night before, by people, and was only noticed during the following day.

Also fun to visit is BLT Research‘s Website, which is largely devoted to the strange attributes of crop circles, which have convinced some people that some of the circles could not be of human origin.

If you just want to look at pretty pictures, has plenty of them. Enjoy and wonder!




31 Days – The Mandelbrot Set

Mandelbrot Set

This is an oldie but goodie. I’d like to show you a shape that is infinitely complex and also very beautiful, known as the Mandelbrot set.

Regarded from a distance, it looks like a bug with little bugs budding off of it. So what’s so beautiful? Zoom in anywhere along its edge and breathtaking patterns emerge. Most remarkably, you’ll continue to see repetitions of the original bug, but with variations.

In the pictures, the actual Mandelbrot set is the area in black. The colors at other points represent how close those points came to being members of the set. (I’ll explain what that means in a moment.)

Here’s a bug that is a bud off a bud off a bud off I-don’t-know-how-many other buds. For the full sequence that got us here, visit this site.

Mandelbrot Satellite Bug

The repetition and complexity continue literally forever!

Amazingly, the set is described by a simple equation: Zn+1 = Zn2 + C. Don’t worry if you don’t know what that means for now. The only point is that it’s simple, yet it produces an object of infinite complexity. How beautiful!

Let’s look at some videos.

The first video has some baroque music to match the baroque nature of the set, so turn on your speakers! At the beginning, notice how the bugs repeat. There’s a transition to a seahorse patterns that repeats with variation as the magnification increases, and finally an unexpected bug at the end.

Here’s a video that zooms in on another area. By the end, the magnification is so extreme that the original picture would be larger than the entire universe – and the complexity continues unabated.

So what’s going on?

To introduce the idea, consider what happens if you multiply a number by itself umpteen times. Let’s say the number is 2. You’ll get 2, 4, 16, 256 and so on. If you were plotting those number on the number line, you’d bounce along toward infinity. However, if you were to start with 0.1, you’d get 0.1, 0.01, 0.0001, 0.00000001, trending toward zero.

The Mandelbrot set multiples and iterates with a different sort of number called a complex number. If you’re interested in the details, please visit the links below. For now, suffice it to say that while a normal number is placed on a number line, a complex number is graphed on a two-dimensional plane. If you add or multiply complex numbers, you get to a different point on the plane, instead of a different point on the number line.

If the iterations don’t fly toward the infinite distance on the plane, then the complex number you started with is a member of the Mendelbrot set. If they veer toward infinity, then they are not members. On the videos, the different colors correspond to how quickly they fly off.

Here’s the best series of videos I’ve found that explain it all.

  1. Introduction to Complex Numbers
  2. Complex Plane Dynamics
  3. How Julia Set Images are Generated
  4. Mandelbrot Set: How it is Generated

31 Days – Cattails

You’ve heard of the “butterfly effect” – that a butterfly flapping its wings in China could affect the weather in New York. It’s a poetic window into chaos theory. When a complex system (the weather) depends on a long chain of cause and effect, and the dependencies are non-linear, then a small change in the initial conditions can produce a large change in the result.

Sometimes I wonder about the butterfly effect, but more often I wonder at its opposite.

Redwing Blackbird and CattailsFrom my office, I can see cattails bowing rhythmically in the summer breeze. We don’t think much about the breeze, but it is incredibly chaotic – a turbulent boil of molecules bouncing off each other and only sort of going in one direction. What made the breeze?

True to chaos theory, the causes are simple: the steady fusion of innumerable atoms in the Sun; the inexorable rotation of the Earth; land shaped by ancient, Moon-pulled tides; butterflies in China. The simplicity iterates over time and distance, confusion increasing at every step. Eventually, all predictability is lost and total chaos arrives outside my window.

The butterfly effect then goes into reverse.

The cattails, possessed of a certain shape and flexibility, absorb the chaos. They rock as calmly as cradles.

A single redwing blackbird is stirred and takes flight.

I see the bird’s epaulets and rise slightly in my chair, drawing in my breath.

31 Days – Marek Pasieczny

I promised something less geeky (and less Greeky) today, so here’s some classical guitar music by Marek Pasieczny.

I thought the intensity of his performance in the first video earned it a place among 31 Days of Wonder. My favorite movements are the opening fugue and the nocturne that comes next. He plays the fugue with a conviction that will make your hair stand on end, as one of the YouTube commenters put it. In the nocturne (4:08), he grabs you with a simple idea and will not let go.

Now here’s a young man playing something Pasieczny wrote when he was young – only 19, in fact. If everyone on Earth were to either play or compose music like this for an hour a day, we would all be at peace.

Pasieczny, who is Polish, loves to explore the musical idioms of other cultures. If you’re up for some musical adventure, here are his variations on a traditional Japanese tune. There is an introduction, and then you’ll recognize the tune once it appears at 0:47 (and is stated even more clearly at 1:30). Pasieczny exploits the many tone colors of the guitar to produce an authentic, Japanese esthetic.

Marek Pasieczny is a wonder, is he not?

31 Days – Euler’s Identity

When something is true by coincidence, we can just enjoy it…

HE: What brings you to a grocery store in the middle of the night?

SHE: I’m looking for a pie.

HE: Wow! Me too!

When the truth is serendipitous, we start to wonder whether Something More might be going on:

SHE: What kind?

HE: Strawberry rhubarb with macaroons on top.

SHE: That’s incredible! Strawberry rhubarb with macaroons on top is my favorite! And why are you up in the middle of the night?

When it’s true by definition, it’s no big deal…

HE: I can’t sleep. I’m a mathematician and there’s a problem I can’t solve. It has to do with pi.

SHE: Strawberry rhubarb pie?

HE: No, pi — defined as the ratio of a circle’s circumference to its diameter.

…but when it must be true and nobody knows why, our hearts start to pound.

SHE: Oh! Pi! As in Euler’s Identity, which <batting her eyelashes> mysteriously relates the five most fundamental numbers in mathematics: pi, e, i, 1 and 0, using only multiplication, addition, exponentiation and equality <bat, bat> exactly once each?

HE: I love you.

And here it is: Euler’s Identity. No list of 31 Wonders would be complete without it.

Euler's Formula

In case you’re not as familiar with math as our two protagonists, I’ll explain the three letters there. The wonder is how they can possibly be related so simply.

You can think of “e” as pertaining to continuously compounded interest. If you were to invest a dollar at a 50% interest rate, you’d have (1+1/2)2 dollars after two years, which is $2.25. At a 33% interest rate for three years, you’d have (1+1/3)3, or $2.37. Invest a dollar at an infinitely tiny interest rate for an infinite number of years, and you’d end up with (1 + 1/∞), or “e“, dollars. That’s about $2.72. (How amazing is it that we can even compute that answer!?) More on “e” here.

As an aside, “e” also happens to be expressible as an infinite series:


…which is pretty cool.

i” is the square root of -1. Can you think of a number that, squared, equals negative 1? Neither can anyone else, which is why “i” is called an imaginary number. In fact, in math it is the prototypical imaginary number. All other imaginary numbers are multiples of it.

You already know what pi () is, but here’s an infinite series for it:

Pi Series

…which is maybe even cooler than the one for “e“.

Now what five numbers could be less related to each other than e, i, pi, 0 and 1? How amazing is it that they do, in fact, combine into such a compact equation as Euler’s Identity? Many mathematicians consider Euler’s Identity to be the most beautiful result in all of mathematics.

If 1 + 1 = 2 is for kids and a2 + b2 = c2 is for grown-ups, then Euler’s Identity is for the gods.

And Euler’s Identity is only a special case of the more general Euler’s Formula, which brings trigonometry into the picture.

Tomorrow, I promise I’ll bring you something less geeky. For today, I hope you’ve enjoyed this excursion into the mysterious and beautiful world of math.

31 Days – Tango

We’ve spent most of our 31 Days of Wonder in geekdom. Let’s cut loose and enjoy some tango!

Argentine Tango beautifully captures the essence of romance — maybe life’s greatest wonder. It is usually improvised but even when choreographed you can see the give-and-take, mystery and sensuality of the dance between the sexes.


I think these last two are improvised, which is pretty amazing.