Science Fair Projects

Introduction

I have done some knotting investigations that could be turned into science fair projects. Most science fair projects are demonstrations of existing scientific principles or results. The disadvantages of doing small scale research might include 1) safety, 2) time required to do the project, 3) cost of special equipment, and 4) analysis of data requiring concepts and computations that are usually taught only at the graduate level. I believe that students can learn to collect, organize, and present data while avoiding or minimizing these problems. I have thought of a demonstration project but it requires algebra, so I put it in the appendix. As I have no expertise in knotting, the knotting experts on this site might suggest interesting knot comparisons, references for how to tie the knots, and instructions on how to untie the knots. I have not seen instructions on how to untie knots in knot books.

Disclaimer: I offer no guarantees about the safety, feasibility, or suitability of these proposals.

Time

I once timed how long it took to tie, and untie, 16 bends. Different methods of tying a bend were counted as different bends. I did not put any stress on the rope before untying the knots. I used bends, but any type of knot may be used (bends, hitches, loops, shoelace knots, stopper knots, whippings, etc.) To turn this type of investigation into a science fair project, I suggest the following.

Number of knots: 4 to 8.

Number of times each knot is tied: 4.

Omit timing of time to untie.

Data collection: The natural tendency is to tie the first knot four times, then the second knot four times, etc. This procedure is bad because people do not work at a steady pace. Thus, the first knot might appear to be slower to tie than the fourth knot, when all that really happened is that the person speeded up between the first knot and the fourth knot. If there are 8 knots and each is tied 4 times, there are a total of 32 tests. To separate the speed at which the experimenter is working from the time required to tie a knot, the tests should be done in a random order. This randomization of the order of the tests is easily done by using a deck of playing cards. Shuffle the deck of cards. [A mathematician who studied the problem stated that 7 riffle shuffles is enough to randomize a deck of cards adequately. Not sure of my card skills, I do ten riffle shuffles.] Use the shuffled deck to determine the order of the tests as follows. Whenever you come to an ace, tie the first knot; whenever you come to a duce, tie the second knot; . . . ; whenever you come to an eight, tie the eighth knot. Measuring the time to tie a knot is straightforward if you have a stopwatch. If you need to use a plain watch with a second hand, write down the start and stop times. The start times should be ahead of the actual time by 5-10 seconds. If the actual time is 10:23:39 (hours, minutes, and seconds); you might write the start time as 23:45 (minutes and seconds). When the second hand gets to 45 seconds, begin tying the knot.

To organize the data, compute the average time for each knot. Rank the knots by their average tying time. To present the data, give on one line, the knot, its average tying time, the four tying times, and a graph of the four tying times. For example,

Knot A 4.25 (4, 6, 3, 4) XX-X

Knot B 6.25 (9, 6, 5, 5) XX?-X

Knot C 11.75 (15, 13, 10, 12) X-XX-X

I use dashes for blank spaces between the X?s that represent the data to make it easier to see the range of the data for each knot. The recommendation to tie each knot four times is tailored to the comparison of knots by the range of their tying times. It is a coincidence that a deck of playing cards has four suits.

Length

I was using a figure 8 loop in 1/8 inch diameter braided nylon cord for a certain purpose. The length of the cord was barely adequate, so I investigated the length of cord in a figure 8 loop knot using a 3/16 inch diameter braided cotton cord. The cotton cord was 58.75 inches long. I tied a figure 8 loop in it and measured the length of the two ends. To measure the length of the loop, I squeezed the sides of the loop together and measured the length of the resulting bight, and multiplied the length of the bight by two. Subtracting the length of the ends and of the loop from the length of the cord yields the length of the cord in the knot. Thus, the length of cord in the figure 8 loop knot was 58.75 ? 17.375 ? 17.375 ? 2 x 7.875 = 8.25 inches. I found this result surprising. I did it again and got 58.75 ? 3.625 ? 3.5 ? 2 x 22.125 = 7.375 inches. The change in the size of the loop was intentional. I considered the difference between 8.25 inches and 7.375 inches to be replication error---that is, due to another tying of the loop. However, I still thought it seemed too big, so I tied a third loop, but used a different method to estimate the length of cord in the knot. The figure 8 loop knot was 1.125 inches long; I spread it apart and counted six segments of rope traversing the figure 8 loop knot. Therefore the length of cord in the knot is about 6 x 1.125 = 6.75 inches. I tied a fourth figure 8 loop and, using the original method, got 58.75 ? 19.125 ? 19.25 ? 2 x 6.75 = 6.875 inches. I switched my application in 1/8 inch diameter nylon cord from a figure 8 loop to a seized loop and got about 5 inches of additional length.

Finding the amount of cord in a knot produces a number for each tying of the knot. Thus the data collection, organization, and presentation can be done exactly as given above for the time to tie a knot. In my figure 8 loop knot example, the length of cord in the knot decreased each time I tied the loop knot and used the same method to calculate the length. This steady decrease might be a coincidence or it might be due to something happening to the rope, or something I was unknowingly doing. I was careful to tie the knots neatly so that adjacent lines within the knot did not cross over each other. I might have pulled harder to tighten the loop knots as I became used to the tying procedure. Given the uncertainty about the decrease in length, the randomization procedure given above for timing the tying of knots should also be used for making measurements of length.

Appendix: Math/Algebra Demonstration Project

The equation of a straight line is Y = a + b X. It can be applied to whippings, which are short sections of twine tied around the ends of a rope to keep the end of the rope from untwisting or fraying. Let Y = the length of twine in the whipping, and X = the number of turns of the whipping twine around the rope.

First example: I used cotton knitting yarn (worsted 4-ply, medium size) for whipping twine, and a mixed fiber (polyester and polypropylene) .25 inch (.635 cm) diameter clothesline for the rope. Henceforth, the whipping material will be referred to as twine and the object to which it is applied as the rope. I used Portuguese whipping. I did two whippings. The first had 6 turns (X1 = 6). The twine was 33 cm long. The trimmings from the whipping were 8.5 cm and 11 cm. Therefore the length of the twine in the first whipping was Y1 = 33 ? 8.5 ? 11 = 13.5 cm. The twine for the second whipping was 34 cm long. There were 12 turns (X2 = 12), and the trimmings were 1 cm and 6.3 cm. So Y2 = 34 ? 1 ? 6.3 = 26.7 cm. Now it is just a matter of solving two equations in two unknowns to get a = .3 cm and b = 2.2 cm.

A science fair project could do any of the following.

1. Replicate the process several times to show the experimental error.

For example, you might do three whippings with X1 = 6, thus obtaining three values of Y1. Likewise, you might obtain three values of Y2 at X2 = 12. To solve two equations in two unknowns, average the three values of Y1 and use the average in the first equation. Likewise, average the three values of Y2 and use the average in the second equation. To present this information, plot the three values of Y1 above X = X1 and the three values of Y2 above X = X2. Graph the line Y = a + b X, where a and b are from solving the two equations, on the plot of the data.

2. Apply the method to different whippings. I think one of the whippings should be French whipping (for which X = number of half hitches), as it is quite different from Portuguese whipping, West Country whipping (for which X = number of half knots) or plain whipping. C.L. Day, in the Art of Knotting and Splicing, gives five methods of doing plain whipping. They don?t all produce the same result.

3. Use ropes of different diameters and/or twines of different diameters and show how a and b are changed.

Give a physical interpretation of a and b. I don?t mean an explanation in words, although that is a good start. Both a and b are functions of the diameter of the rope and/or the diameter of the twine. Write equations for these functions. Doing so may require substantial study and thought, but that is what scientists do. Don?t make the equations too complicated by trying to account for everything. Unknown quantities, such as the compressibility of the rope, make the equations into approximations regardless of how detailed your equations are.

To test some of these ideas, I did another example. I used 3/16 inch (.48 cm) diameter cotton cord as the twine and a cardboard tube with an outside diameter of 4.5 cm as the rope. I used French whipping (where X = the number of half hitches). The cord was 147 cm long. The first whipping had X1 = 1, trimmings of 5.3 cm and 86 cm, and Y1 = 147 ? 5.3 ? 86 = 55.7 cm. The second whipping had X2 = 5, Y2 = 147 ? 3.6 ? 13 = 130.4 cm. Note: I used the same piece of cord for the second whipping. It is not necessary to trim the ends, only to measure from the trim point to the end of the twine. Solving the two equations, I got a = 37.025 cm and b = 18.675 cm.

According to the book I used, French whipping ends by making an extra turn around the rope and passing it under the last half hitch. I did not count the extra turn as a half hitch. If I did, the value of X would increase by one. As both X1 and X2 would be increased by one, the value of b = (Y2-Y1)/(X2-X1) would not be changed, but the value of a would change. Another viewpoint on the extra turn under the last half hitch is to say that the extra turn changes the half hitch into a different type of hitch. As it is no longer considered a half hitch, the value of X would be decreased by one. Again, this alternative would not affect the value of b. These alternative methods of defining X don?t require any additional experimentation. You can calculate the different values of a directly from the data I gave above. Examining these alternative methods of counting X should help with understanding the physical interpretations of a and b.