Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Seeing Is Believing

November 2001

Random Dot Stereograms (RDS) are arrays of apparently random dots that conceal an image. To see the image, the viewer must force the eyes to focus (usually) way behind the image surface (the computer screen for the applet below.) Those who know how to do that are rewarded with seeing a 3D image perceptively floating above the background. The perception is so vivid, especially when one moves one's head, that it's easy to believe it's possible to steal a glance behind the floating image.

The applet below offers a modest selection of simple geometric shapes, as well as the facility to change the pixel size and to choose between square and circular pixels.

Credits

The applet implements an adaptation of the C code from a very comprehensive site maintained by Stuart Inglis.

I got there by following a link from Dror Bar-Natan's (The Hebrew University) home page. Bar-Natan kindly wrote to inform me of an interesting MS thesis of a student of his, Dori Eldar. The thesis employs Java to a great effect to introduce and describe configuration spaces of various linkages.

The mechanics of human vision that underlie the theory of RDS are quite simple. Focusing the eyes behind the screen makes our brain believe there is a depth separation in the image and interpret the visual signals accordingly. The theory of RDS could be found at the above mentioned sites. However, the fact is that, the simplicity of the theory notwithstanding, some people manage to see those 3D images, while others do not.

My wife is of the latter kind. With all my cajoling and explanations she just could not see anything but an insipid flat dot pattern. Not that she did not believe me when I said there was something, but she did appear skeptical. After multiple futile efforts to help her see the image, the famous dictum Seeing Is Believing came to my mind.

As I already was online, I made a web search for that exact phrase. I have a lot of complaints about searching the web. Pressed for time, I'd settle for an encyclopedia. But one thing stands squarely in favor of the web search. On very rare occasions you won't be surprised by the results. If you have time, go for it.

What did I get this time? Very appropriately there was an optical illusion site that in my Netscape 4.61 appeared as a white blank page. (Subsequent use of the Internet Explorer 5.0 convinced me it was a browser induced illusion), three different books at amazon.com, a texture paint program, a graphic design advertisement, a year old announcement of a missed exhibition at New York Public Library, a computer vision article in ACM Crossroads - the first online ACM student magazine, an article describing an experimental device that directly stimulates the brain, an interesting visual perception site by ThinkQuest.org and another Art, Math and Science site from the same organization with an unbelievable amount of bloopers on its math pages. (See examples.)

Surprisingly, there was also a 1995 project from the MathForum. The project offered excerpts from a book Math To Build On by Johnny and Margaret Hamilton. I understand the book has been written for tradesmen, more like a Do It Yourself manual, than a geometry textbook. Seeing is Believing is the last section of the book that is devoted to geometric constructions. The purpose of the section is succinctly expressed in the opening paragraph:

Many people's understanding of math is enhanced by seeing it work. If you are one of those people, the following section is written for you. Working this section is not only helpful in developing your drawing skills, but in confirming calculations as you work in the other sections of this book.

Here's an example of this approach. After giving a step-by-step instruction for dividing a 6'' segment into two equal parts with the standard construction, the authors remark:

You accomplished two things with this action:

One : You divided the line exactly in half. See for yourself. Measure each side from the enter point to the end point.
Each segment should measure exactly three inches.

Two : You created two lines that are perpendicular to each other.

I feel like taking an issue with the "conforming calculations" that, in this example, are reduced to taking measurements and number comparison. It's not about a pipe fitter who verifies that two pieces of pipe are equally long by measuring both and comparing their lengths. (There might be of course a simpler way. By laying the pipes side by side, like they do at, say, Home Depot when cutting pipes or wood, one is immediately able to compare their length without taking any measurements.) At a teachers' and students' site, such as the MathForum, the "conforming calculations" part I believe is out of place for at least two reasons. First, there is a glaring absence of any mathematical content. Except for memorizing a sequence of steps and improving drawing skills, what else a student is supposed to learn or see? Compare this to, say, folding the paper as to make the crease pass through a pair of the constructed points and have the end points of the segment overlap. No more formal, this approach has a potential to convey an abstraction, an explanation as to why the prescription worked.

Second, even in practical matters, errors in measurements may lead to errors in judgement. More importantly, instead of understanding mathematics better, overreliance on things practical may result in devaluation of mathematical knowledge and development of wrong habits of mind.

Here's an example I owe to Roger Wibberley, Goldsmiths College, London. The example arose from his research into drafting a solution to the division of the octave into 12 semitones.

The main structure consists of a right triangle with the vertical side of length 2 and the horizontal side of length 1. Mark a point on the vertical side at the distance 2 from the apex. Through that point draw a horizontal line to the intersection with the hypotenuse. Join the point of intersection to the midpoint and the foot of the vertical side. Divide the base into a number of equal segments (12 in the "real world" situation). The construction involves joining some points and drawing through some of the points of intersection lines parallel to other lines. The applet below presents some possible mesh configurations. Everything's nice and fits tightly.

My main interest lies with the configuration in which the base has been divided into six equal parts.

By construction, AD = BD = BC = 1 and AD3 = 2. The base BC is divided into 6 equal parts. The lines from A to the division points intersect DF and BF at five points each, from which lines are drawn parallel to BF and DF, respectively.

By considering pairs of triangles with bases parallel to DF and the apex at A, we derive a five term proportion:

a/b = b/c = c/d = d/e = e/f.

The six small triangles that line up on BD have all three corresponding sides parallel and are, therefore, similar. The above proportion indicates the constant ratio, say r, in which the corresponding sides of any two adjacent triangles stand to each other. Let DD1 = m. Then D1D2 = mr, D2D3 = mr2, D3D4 = mr3, D4D5 = mr4, and, finally, D5B = mr5.

Summing all six up, we obtain m·(r6 - 1)/(r - 1) = 1. If we only add the first three terms, we see that m·(r3 - 1)/(r - 1) = DD3 = 2 - 1. Now, divide the first identity by the second:

(r6 - 1)/(r3 - 1) = 1/(2 - 1),

or

r3 + 1 = 2 + 1.

In other words, r3 = 2, which is to say r2 = 21/3. The latter of course shows that the famous Delian problem of duplication of the cube is solvable with ruler and compass.

How come?

You'll have to question your beliefs to find out.

What's the point? I think the Seeing is believing dictum is incompatible with the spirit of mathematics. One reason is that mathematics is not about beliefs, but about reasoning, creativity and inquiry. It's possible and reasonable to believe in the power of mathematics of course. Seeing the technological progress and hearing how much mathematics is involved with modern technology helps strengthen and reaffirm that part of our belief system. Almost every one believes in the power of mathematics, relatively few know its source and subject.

It's also possible to believe that a certain sequence of drafting actions will result in division of a segment into two equal parts without understanding why the prescription works. In fact mathematics has originated with practical matters. It is firmly rooted in a multitude of applications. But seen it may only be with the mind's eye.

New technology, the web in particular, should be used to help open the mind's eye, not to create a belief system. It's unthinkable to fault the MathForum — the premier math and math education presence on the web — in this respect. So much so that I would not believe they host the Seeing is believing piece, if I did not see it with my own eyes.

A sample of the bloopers at the ThinkQuest Seeing is Believing pages:

Elementary Limits:

limx«a f(x) = f(x)

This limit represents a horizontal line, which says that as x approaches a,

f(x) = f(x).

and on another page

The tangent line of f at point A is:

limx«a(f(x + h) - f(x))/h.

In the section on Mathematical Numbers we read

Typical examples of irrational numbers are the numbers p and e, as well as the principal roots of rational numbers.

Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at alexb@cut-the-knot.com

Copyright © 1996-2001 Alexander Bogomolny