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The graphs on the back cover of the book show y = sin n. This is very different from y =sin x. The graph of sin x is one continuous curve. By the time it reaches x = 10,000, the curve has gone up and down 10,000/2π times. Those 1591 oscillations would be so crowded that you couldn't see anything. The graph of sin n has picked 10,000 points from the curve—and for some reason those points seem to lie on more than 40 separate sine curves.

The second graph shows the first 1000 points. They don't seem to lie on sine curves. Most people see hexagons. But they are the same thousand points! It is hard to believe that the graphs are the same, but I have learned what to do. **Tilt the second graph and look from the side at a narrow angle.** Now the first graph appears. You see "diamonds." The narrow angle compresses the x axis—back to the scale of the first graph.

The effect of scale is something we don't think of. We understand it for maps. Computers can zoom in or zoom out—those are changes of scale. What our eyes see depends on what is "close." We think we see sine curves in the 10,000 point graph, and they raise several questions:

- Which points are near (0, 0)?
- How many sine curves are there?
- Where does the middle curve, going upward from (0, 0), come back to zero?

A point near (0, 0) really means that sin n is close to zero. That is certainly not true of sin 1 (1 is one radian!). In fact sin 1 is up the axis at .84, at the start of the seventh sine curve. Similarly sin 2 is .91 and sin 3 is .14. (The numbers 3 and .14 make us think of n. The sine of 3 equals the sine of π − 3. Then sin .14 is near .14.) Similarly sin 4, sin 5, ... , sin 21 are not especially close to zero.

**The first point to come close is sin 22.** This is because 22/7 is near π. Then 22 is close to 7π, whose sine is zero:

sin 22 = sin (7π − 22) ≈ sin (− .01) ≈ &minus .01.

That is the first point to the right of (0, 0) and slightly below. You can see it on graph 1, and more clearly on graph 2. It begins a curve downward.

The next point to come close is sin 44. This is because 44 is just past 14π.

44 ≈ 14π + .02 so sin 44 ≈ sin .02 ≈ .02.

**This point (44, sin 44) starts the middle sine curve.** Next is (88, sin 88).

Now we know something. **There are 44 curves.** They begin near the heights sin 0, sin 1, ... , sin 43. Of these 44 curves, 22 start upward and 22 start downward. I was confused at first, because I could only find 42 curves. The reason is that sin 11 equals −0.99999 and sin 33 equals .9999. Those are so close to the bottom and top that you can't see their curves. The sine of 11 is near −1 because sin 22 is near zero. It is almost impossible to follow a single curve past the top—coming back down it is not the curve you think it is.

The points on the middle curve are at n = 0 and 44 and 88 and every number 44N. Where does that curve come back to zero? In other words, when does 44N come very close to a multiple of π? We know that 44 is 14π + .02. More exactly 44 is 14π + .0177. So we multiply .0177 until we reach π:

if N = π/.0177 then 44N = (14π + .0177)N = 14πN + π.

This gives N = 177.5. At that point 44N = 7810. This is half the period of the sine curve. The sine of 7810 is very near zero.

If you follow the middle sine curve, you will see it come back to zero above 7810. The actual points on that curve have n = 44·177 and n = 44·178, with sines just above and below zero. Halfway between is n = 7810. **The equation for the middle sine curve is y = sin (πx/7810).** Its period is 15,620—beyond our graph.

*Question* The fourth point on that middle curve looks the same as the fourth point coming down from sin 3. What is this "double point?"

Answer 4 times 44 is 176. On the curve going up, the point is (176, sin 176). On the curve coming down it is (179, sin 179). **The sines of 176 and 179 differ only by .00003.**

The second graph spreads out this double point. Look above 176 and 179, at the center of a hexagon. You can follow the sine curve all the way across graph 2.

Only a little question remains. Why does graph 2 have hexagons? I don't know. The problem is with your eyes. To understand the hexagons, Doug Hardin plotted points on straight lines as well as sine curves. Graph 3 shows y = fractional part of n/2π. Then he made a second copy, turned it over, and placed it on top. That produced graph 4—with hexagons. Graphs 3 and 4 are on the next page.

This is called a **Moiré pattern**. If you can get a transparent copy of graph 3, and turn it slowly over the original, you will see fantastic hexagons. They come from interference between periodic patterns—in our case 44/7 and 25/4 and 19/3 are near 2π. This interference is an enemy of printers, when color screens don't line up. It can cause vertical lines on a TV. Also in making cloth, operators get dizzy from seeing Moiré patterns move. There are good applications in engineering and optics—but we have to get back to calculus.