The following is an exercise in paper folding and an investigation of how the eye and brain do construction work. Follow the paper-folding directions, and describe what your eye sees.

Take a rectangular sheet of paper and mark a point on it somewhere near the bottom and about halfway between the left and right edges - like so:

Now imagine a point on the bottom edge of the paper - anywhere on that bottom edge - mark it lightly - and fold that point onto the previously marked point, crease the paper when you have the one point folded onto the other, and then unfold. If you pick a point like so:

then you can see what the result looks like as you fold by clicking on the diagram above. And, after you crease and unfold, your paper should look like:

That's all there is to it - except that you do this procedure many times, choosing a different point on the bottom edge each time. If you click on the diagram above, you can drag the "bottom edge point" somewhere and observe where the corresponding crease is made.

Predict the total visual effect of all the crease lines if you could make all the infinitely many creases as the folded point traverses the bottom edge. Then

1. Do the folding to see how it matches your prediction, and characterize the shape you see.

2. Use Sketchpad to simulate this folding process, and construct the locus of the "crease" line segments as the folded point traverses the bottom edge.

Click HERE to see the locus (well, just half of it) traced as you drag the "bottom edge point". Click HERE to see the full locus (even off the paper!) as the folded point traverses the bottom edge.

3. Why is this called folding an envelope?

As a preview, we will do later exercises indicated below:

1. Design a procedure analogous to the above, replacing the bottom edge, a line, with a circle. What kind of locus results when the fixed point is inside the circle? Outside the circle?

2. Characterize the visual image formed by your brain by examining the locus constructed.

Go back to Lab 7 for instructions on completing the lab.