A friend of mine gave me a wooden floor puzzle of the United States as a gift for my boys. The puzzle is about 2′ x 3′ so some of the state pieces are fairly large. Upon his first use, my older son had nearly finished the puzzle: he had about 5 states remaining, scattered across the country. He picked up the piece that represented Maine. He looked at the gaps in the puzzle, then at the piece and quickly placed the piece where Maine belonged. After taking a moment to be proud of my genius offspring, I considered what had occurred. He basically compared shapes mathematically: what two-dimensional outline of the five wooden shapes I have is the best match for the open space I am looking at? Here it is…
This document stated earlier that all defining properties of an information vector must be quantifiable. Let’s focus on an important aspect of information vector perception: visual object recognition. Suppose you were shown sheets of paper with these images:
- A flat line (parallel to the top of the bottom of the page) and a slanted line
Two parallel lines; one longer than the other
- Two lines intersecting in an acute (sharp) angle and two lines intersecting in an obtuse (wider) angle
- Two circles; one twice the diameter of the other
- An elliptical arc vs. a circular arc
You would recognize the difference between the shapes on all of these pages and those differences can all be defined pretty simply in mathematical terms:
- The slope of a line is indicated as a coefficient in the algebraic formula that defines a straight line
- The length of any straight line can be determined from the x and y coordinates of its endpoints
- The acuteness of an angle can be measured in terms of degrees
- The radius of a circle is indicated by a coefficient in the equation that defines it
- The definition of an ellipse involves coefficients for both the x and y points that make up the shape. The difference in these coefficients elongates the ellipse (the coefficients are equal for a circle).
Of course our visually-perceived universe is not all straight lines, curves and simple angles. However, in the spirit of compression, these basic constructs go a long way in supporting our perception. Try this exercise — it would really help if you found pen and paper and played along. Look at the image below and make a mental picture of it. Once you think you have a good mental representation of it, look away from this document and draw the image from memory.
Ok, now compare your image with the one in this document. How closely do they match up? Chances are, if you did a good job, the basic shape is similar but you probably did not match every zig and zag in the rightmost vertical line. However, you probably did represent that line as ragged as opposed to straight.
So what is this exercise trying to prove? That although visual information vectors are often very complex in reality, we tend to simplify (compress) their definition in our minds. Think back to the smiley face. The image above is an information vector and your drawing was a representation of the concept of this image.
This gap between the exact dimensional detail and our inability to conceive that level of detail is the basis of the challenge of jigsaw puzzles. If you were a computer minded being that could perceive every puzzle piece as its exact geometric representation visually, you wouldn’t have to “try it out” in the context of the picture.
New exercise. Look at the image below and decide if it brings any renowned person to mind:
Did you guess that it is supposed to be Cameron Diaz? Good, because it’s supposed to be Abraham Lincoln. It’s a pencil sketch that I did myself. Believe it or not, I’m not a professional artist. But when I showed the image to my wife, she immediately recognized Abraham Lincoln. She also mentioned that it didn’t look exactly like Abraham Lincoln. And she is right; I realized this fact when I finished the drawing. There is something subtle but important missing about my information vectors that conjure the concept of Abraham Lincoln: I didn’t capture his defining stare and gauntness or the details of his mouth and nose. All of those attributes would have likely been captured and represented by a talented artist. This point is important to realize: this image is not a photo of Abraham Lincoln. It is a collection of lines, angles, curves that you perceived and caused you to (hopefully) recall the concept of Abraham Lincoln. So why didn’t you think this was a sketch of Cameron Diaz? Because the collection of lines, angles and curves that define the representation of her face is very different. Subtle differences in the lines that shape the eyes and set the pupils and those that define the mouth seem to play a strong part in retrieving the related concepts. But this image was probably “close enough” to Abraham Lincoln’s image to enable the appropriate retrieval.
Concept association and information vector perception work together in a feedback system manner. For example, when I drew the image of Abraham Lincoln, I deliberately included the bow tie. When you were asked to identify the image, the fact that it was some guy in a bow tie might have helped. What if I had included a cowboy hat and sheriff’s badge? Would that have thrown you?
This interaction between concept and information vector in support of our intelligence is applicable to all perceptions. There was a television game called “Name That Tune”. Contestants would be given a verbal clue to some song. Then the two contestants would underbid each other as to how many piano notes would have to be played in order for him or her to identify the secret song. A melody is a composite information vector: a sequence of notes (frequencies), separated by time intervals, each often with its own emphasis. When playing “Name That Tune”, the concepts elicited by the clues in conjunction with the information vector of the melody were often enough for a contestant to guess the song.