Questioning the Answer

http://www.topchair.cn/en/Question-mark-chair.htm

Have you ever found yourself searching and searching and searching for an answer to a question? You explore all perspecives. You look at it from many point of view. Time drags on – you are still searching – climbing into your mind’s attic for new insights in hopes to find it.

You pause and ask yourself: Uh, what exactly was the question? Now you try to articulate the question, and then you realize that you never really knew what the question was. So then you try to come up with the right question.

Having shifted gears, it doesn’t take you long to find it – it pops out crystal clear. And just as soon as the question comes, the answer comes along right after it. You find yourself in a new place of understanding, and you realize: everything happened in exactly the right order.

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Why is it a Color “Wheel” and Not a Color “Line”?

This blog post was published in May of 2012 on EyeMath. It is being migrated to this blog, with a few minor changes.

I’ve been discussing color algorithms recently with a colleague at Visual Music Systems.

We’ve been talking about the hue-saturation-value model, which represents color in a more intuitive way for artists and designers than the red-green-blue model. The “hue” component is easily explained in terms of a color wheel.

Ever since I learned about the color wheel in art class as a young boy, I had been under the impression that the colors are cyclical; periodic. In other words, as you move through the color series, it repeats itself: red, orange, yellow, green, blue, violet…and then back to red. You may be thinking, yes of course…that’s how colors work. But now I have a question…

Why?

Consider five domains that can be used as the basis for inventing a color theory:

(1) the physics of light, (2) the human retina, (3) the human brain, (4) the nature of pigment and paint, and (5) visual communication and cultural conventions.

(1) In terms of light physics, the electromagnetic spectrum has a band visible to the human eye with violet at one end and red at the other. Beyond violet is ultraviolet, and beyond red is infrared. Once you pass out of the visible spectrum, there aint no comin’ back. There are no wheels in the electromagnetic spectrum.

(2) In terms of the human retina, our eyes can detect various wavelengths of light. It appears that our color vision system incorporates two schemes: (1) trichromatic (red-green-blue), and (2) the opponent process (red vs. green, blue vs. yellow, black vs. white). I don’t see anything that would lead me to believe that the retina “understands” colors in a periodic fashion, as represented in a color wheel. However, it may be that the retina “encourages” this model to be invented in the human brain…

(3) In terms of the brain, our internal representations of color don’t appear to be based on the one-dimensional electromagnetic spectrum. Other factors are more likely to have influence, such as the physiology of the retina, and the way pigments can be physically mixed together (a human activity dating back thousands of years).

(4) Pigment and paint are very physical materials that we manipulate (using subtractive color), thereby constituting a strong influence on how we think about and categorize color.

(5) Finally: visual communication and culture. This is the domain in which the color wheel was invented, with encouragement from the mixing properties of pigment, the physiology of the retina, and the mathematical processes that are formulated in our brains. (I should mention another influence: technology…such as computergraphical displays).

Red-Green-Blue

Consider the red-green-blue model, which defines a 3D color space – often represented as a cube. This is a common form of the additive color model. Within the volume of the cube, one can trace a circle, or a hexagon, or any other cyclical path one wishes to draw. This cyclical path defines a periodic color representation (a color wheel). A volume yields 2D shapes, traced onto planes that slice through the volume. It’s a process of reducing dimensions.

But the electromagnet spectrum is ONE-DIMENSIONAL. The physical basis for colored light cannot yield a higher-dimensional color space. The red-green-blue model (or any multi-dimensional space) therefore could not originate from the physics of light.

DID HUMANS INVENT PURPLE IN ORDER TO GLUE RED AND VIOLET TOGETHER?

An alternate theory as to the origin of the color wheel is this: the color wheel was created by taking the two ends of the visible spectrum and connecting them to form a loop (and adding some purple to form a connective link). I just learned that Purple is NOT a spectral color (although “violet” is :) Purple can only be made by combining red and blue. Here’s an explanation by Deron Meranda, in a piece called…

“PURPLE: THE FAKE COLOR – OR, WHAT REALLY LIES AT THE END OF A RAINBOW?“

And here’s a page about how purple is constructed in the retina: HOW CAN PURPLE EXIST?

Did the human mind and human society impose circularity onto the color spectrum in order to contain it? Was this encouraged by the physiology of our eyes, in which various wavelengths are perceived, and mixed (mapping from a one-dimensional color space to a higher-dimensional color space)? Or might it be more a matter of the influence of pigments, and the age-old technology of mixing paints?

Might the color wheel be a metaphorical blend between the color spectrum and the mixing behavior of pigment?

Similar questions can be applied to many mathematical concepts that we take for granted. We understand number and dimensionality because of the ways our bodies, and their senses, map reality to internal representations. And this ultimately influences culture and language, and the ways we discuss things…like color…which influences the algorithms we design.

 

Enough with this Square Root of -1 Business!

Like so many other people, I was kept from appreciating the beauty and utility of mathematics because of the way it was taught to me.

The majority of introductions to complex numbers start with the elusive and mysterious square root of -1, denoted by i.

A number that has an i stuck on to it is called “imaginary” (a convenient differentiator to “real”). Being asked to learn something that is called “imaginary” is not very motivating to young learners who work best starting with concrete metaphors.

The imaginary number i is counterintuitive and confusing. And it’s not the coolest part. Sure, i was an important invention at a critical stage in the history of math when there was no good way to express z² = -1. And yes, it makes a good ending to a long story (which happens to be true): math has advanced through several expansions of the concept of “number” … from the counting numbers to the wholes – to the negatives – to the fractions – to the irrationals – and finally to complex numbers – where i came along and saved the day.

But…does this mean that invoking i is the best way to explain complex numbers to novices – to everyday people? I join many others in saying that there is a better way to learn about the wonderful world of two-dimensional numbers. One voice among those is Kalid Azad.

He speaks in metaphors and freely engages the visual mind to help us grasp math concepts using our whole brain. In his explanation on complex numbers, Azad says this about i: “It doesn’t make sense yet, but hang in there. By the end we’ll hunt down i and put it in a headlock, instead of the reverse.”

…..

When you get an intuitive, aesthetic feeling for why certain mathematical ideas are being taught, you become more motivated to learn the notation. The corollary: learning math notation without understanding why is like learning musical notation before ever being allowed to listen to or play music.

Paul Lockhart, in A Mathematician’s Lament, compares the way math is taught to a nightmare scenario in which music is taught to students using sheet music notation only (no actual music is played or heard) – until the student is advanced enough to start “using” it.

 

What is a Two-Dimensional Number?

When I read that complex numbers are really no more “imaginary” than real numbers, I decided that I would start dismantling my old worldview. Why should I assume that numbers have to be one-dimensional? Over time, I became more accustomed to the notion that a number can occupy a plane (the complex plane) and not just a line (the number line). Learning how to make images of the Mandelbrot Set helped a lot.

Think of Multiplication as Rotation

Instead of trying to wrap your mind around i, and how it magically makes equations come out right, let’s start with geometry. Think of multiplication as rotation and expansion. In the blog Girls Angle, Ken Fan introduces complex number multiplication in a nice visual way… here.

Here’s a video explaining complex numbers in terms of physical metaphors, and eventually explaining why the square root of -1 becomes a necessary part of the notation.

Squaring

Consider the following diagram showing what happens when you square certain complex numbers that lie on the unit circle:

The dot on the right represents the complex number (1+0i). When you square it, it stays the same (no surprise: 1×1=1). The number at the left is (-1+0i). When you square that, it becomes (1+0i). But when you square the number at top (0+1i) it “rotates” by 90 degrees to (-1+0i). Finally, at the bottom, the number (0-1i) rotates…but would it be correct to say that it rotates by 90 degrees clockwise to (-1,0i)? Depends on how you look at it. Rotating by 270 degrees counter-clockwise has the same result. This is the nature of rotation and angular reality: it is periodic – it cycles…it repeats.

What an awesome idea. Multiplication is like doing a whirling dervish jig.

Animated Squaring

Here’s an interactive tool I made that allows you to play with 200 dots (complex numbers) randomly scattered on the complex plane. You can experience what happens when complex numbers are squared. It also allows you to multiply the dots (using a complex number dot that you can drag along the screen).

http://ventrella.com/ComplexSquaring/

This interactive tool might make you feel as if the dots on the screen are obeying some sort of gravitational law of physics. Well, in a way, yes, that’s what’s happening. When you add, multiply, or exponentiate numbers, you get a new number. In the complex plane, the space where that change takes place is two-dimensional. That’s cool! We like images.

Here’s another visual tool: when we multiply two complex numbers, such as (a+bi) and (c+di), we can visualize the operation in this way:

In pseudocode:

realPart      = (a*c) - (b*d);
imaginaryPart = (a*d) + (b*c);


This explanation of multiplication does not require i.

To this day, I STILL do not feel very much music when I think about the square root of -1.

On the other hand, the more I play around with visualizing and animating complex numbers, the more intuitive they become, and the deeper my sense that these numbers are as real as any old one-dimensional number.

They are not imaginary at all.