Kiselev's geometry ex 7,8

Question

I need some help on two exercises from Kiselev's geometry, about straight lines.

Ex 7: Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight. Hint: Flip the straightedge upside down.

I would draw the first line, then flip the straightedge and draw the second line over the first. The two lines should coincide nicely iff the straightedge is straight. Because, this shows that there is no "unevenness" or "bumps" on the edge of the straightedge. There would be gaps between the two lines if there are "unevenness/bumps" on the edge of the straightedge.

Ex 8: Fold a sheet of paper and, using ex 7, check that the edge is straight. Can you explain why the edge of a folded paper is straight?

Ex 8 is marked as more difficult by the author. I'm completely clueless about this exercise.

Please provide insights and help me with these two exercises. I'd appreciate if they are more of an "experimental approach" than theoretical because exercises 7 and 8 are arranged in between the introduction and first chapter of the book.

Thank you. :)

Answer 1

While these questions are handwavy and cannot be precisely answered, I think that the author is looking for something like the following for exercise 8:

When you fold the paper, you can do it in two ways, corresponding to which side you fold across the line. Notice that the results of these two ways are reflections of one another across the line. But the line is the same no matter which side you fold across it, so by exercise 7 the line is straight.

Now, I want to emphasize again that this is nowhere near a mathematical proof, but it is indeed interesting to think about it this way, as it shows that there are some inherent properties that we assume about our ambient space whenever we perform operations (typically rigid motions) within it.

Answer 2

I’m late to the game but here is my argument for #8. (Incidentally, I don’t think Kiselev is suggesting to use #7 to prove #8. He says use the technique to check that the line is straight, and, separately, asks us to explain why it is true.)

When we fold the paper, we identify each point on the bottom 'half' with a unique point on the top 'half'. (You may imagine the paper to expand as much as necessary to make this a reality! Or, you can imagine we are folding a plane.)

Lay the paper on top of a plane and fold the paper. The edge creates a curve in the plane. Our goal is to convince ourselves that this curve is straight.

Unfold the paper. Now the paper has a crease, which of course is the same curve as the edge. The crease divides the paper into two halves — call them 'left' and 'right'.

Now, imagine any two points on this crease, and further imagine them connected by some curve which does not lie on the crease — say, a curve in either the left or right half of the paper. What can we say about such a curve?

A curve like this cannot be straight. The reason is, folding the paper would give us a congruent but different curve on the other side of the page, and we cannot have two different straight curves connecting a pair of points.

So we have shown that if two points on the crease are connected by a curve which travels off the crease, that curve is not straight. But we know that any two points are connected by a straight line, and hence that straight line lies on the crease. Applying this argument to the endpoints of the curve, we see that the whole crease/edge itself is straight.

Obviously I've left off some details and assumptions (for example, a line congruent to a straight line is straight, etc) but I think I've managed to find the thrust of a good argument.

It’s a wonderful problem that I’ve thought about for years.