Josef Albers and Interaction of Color

My first post on art is about Josef Albers and his use of color. An idea central to his work is that we do not see “individual” colors — but colors are always perceived in relationship to the surrounding colors.

Day002Albers2

Look at this image for a moment — what do you notice about the smaller rectangles? If you look carefully, you’ll notice that they are all the same color. It may look like some are darker than others — but that’s only because you’re seeing them against different background colors. Albers explored this idea in depth in his famous book, Interaction of Color.

So how can you create this image? Begin with a color specified as RGB — in this case, (0.7,0.6,0.3) — to use as the color of the smaller rectangles. Remember that (0,0,0) is black and (1,1,1) is white; using values for R, G, and B between 0 and 1 will allow you to produce millions of different color combinations.

Day002Square

Now use your computer’s random number generating ability to create three random numbers between -0.3 and 0.3. For this example, I’ll use -0.2, -0.1, and 0.3. Subtract these values from (0.7,0.6,0.3) to get the RGB values for the left larger rectangle — you get (0.9,0.7,0.0). Then add these random numbers to (0.7,0.6,0.3) to get (0.5,0.5,0.6) — these are the RGB values for the right larger rectangle. Notice that this procedure assigns RGB values to the smaller rectangles which are the numerical averages of the RGB values of the colors of the surrounding rectangles.

Keep in mind that this is an arithmetic mixing of colors. If you actually had paints which were the colors of the larger rectangles, mixing them would likely not give you the color of the smaller rectangles. Also keep in mind that Interaction of Color was published in 1963, so that Albers’ ideas were developed with paper and pigment. What I’ve done is adapted Albers’ ideas for use on a computer — and interpreted his ideas as you see here. Color theory is a very complex field, and is widely written about on the internet — so if these ideas intrigue you, start searching!

Day002Albers6Web3Yin Yang IV, 8” x 8”

As a final example, here is a piece incorporating these ideas about color with an abstract yin/yang motif. Visit my art website if you’d like to see some additional images with commentary.

Now we’ll get to the specifics of actually implementing the ideas mentioned above. I’ve decided to use Python for these examples since it is a language growing in popularity and is open source. Also, I’m using Python in a Sage environment because it’s open source, too — and you don’t need to download or install anything. You can just open a Sage worksheet in your browser. Sometimes it’s a little slow, so be patient. You can download Sage onto your own computer if you’d like to speed things up.

Here is the link to the interactive color demo. It’s fairly self-explanatory — and you don’t need to know any Python to use the sliders. Just hit shift+enter as explained in the instructions.

One thing to be careful about, though. If you choose a red value of 0.8, and then choose to vary this value by 0.4 — you’ll get red values of 0.4 and 1.2. But since RGB values are between 0 and 1, the 1.2 is “truncated” to 1.0, so you’re really working with 0.4 and 1.0. This means that 0.8 is no longer the average of the two red values — so the “Albers” effect won’t be so pronounced, and may be absent if your values are too far off. Select your values carefully!

I’ve tried to make the code fairly straightforward, so if you know a little about Python or programming in general, you should be able to make some of your own changes. You’ll have to make a Sage account and copy my code to one of your own projects in order to make changes.

This blog is designed to give you ideas to think about, not be a tutorial. So I won’t be teaching you Python. If you need to understand the basics of the RGB color space — well, just look it up. There’s also plenty online about Josef Albers. Go in any direction you like.

But if you are going to play with the Python code to make more complex images, here’s a suggestion. Whenever you are about to type in RGB values, pause for a moment and ask why you’re choosing those particular values. Use color deliberately. This will make all the difference in the world — and may well be the difference between making a digital image, and creating digital art.

What Is Mathematics?

Mathematics is creative.

Unfortunately, this is lost upon many — if not most — students of mathematics, in large part because their teachers may not understand mathematical creativity, either.  One way to address this issue is to have students write and solve their own original mathematics problems.  This seems daunting at first, until students realize they are more creative than they were led to believe.  (I’ll discuss this more in a later post.)

The difficulty is that the creative dimension of mathematics is a bit elusive.  Give a child crayons and ask her to draw a picture, sure — but give a student some ideas and ask him to create a new one?  To appreciate mathematical creativity, you need some understanding of the abstract nature of mathematics itself.  To create mathematics, you need imagination much like you do in any of the arts — or other sciences, for that matter.

Over the years, I’ve created my fair share of mathematics.  How much of it is really new is hard to determine — how do you know if any of the billions of other people in the world already created something you did?  (Proof by internet search notwithstanding.)

This blog is about sharing some of my ideas, problems, and puzzles.  Some were created years ago, some are new — and I will consider myself lucky if some are entirely original.  I truly did have fun creating them, and I enjoy writing about them now.

I’m hoping to convey an enthusiasm for mathematics and its related fields — in other words, all human knowledge — and to share something of the creative process as well.  The creation of mathematics is not a mystical process, and needs no explanation to a mathematician.  But we can surely do more to make this enlivening process accessible to all in a time when it is certainly necessary.

As you follow, you’ll notice a heavy emphasis on programming.  Every student should learn to program — and in more than one language.  Perhaps this should be an axiom in the 21st century, but we’re not even close.  So many of the tools I use are virtual — the ability to write code to perform various tasks is essential to my creative process, as you’ll see.  In fact, many posts will have links to Python programs in the Sage platform (don’t worry if you don’t know what these are yet).  These tools are all open source, and available to anyone with internet access.

Finally, blog posts will usually have a “Continue reading…” section.  Some posts (like this one) will be essays on teaching, creativity, or a related topic.  Since not everyone may be so philosophically minded, the “Continue reading…” sections of these essays will be a puzzle or game.  Enjoy!

Continue reading What Is Mathematics?