Let’s learn about the third most famous number in history – the Golden Ratio (aka “golden mean”, “phi”, “golden rectangle”, “Fibonacci Sequence”) Name your program goldenRatio. . First add a “pointer” to your robot that points directly down to the ground, is about ⅛” off the ground, and is on the left side of your robot. Then program your robot to demonstrate the Golden Ratio by performing the following sequence of steps. At each “pause” measure the distance from the pointer’s starting location directly to the pointer’s current location – label this column “pointer distance”.
forward 1 rotation
pause
point turn right 90 degrees
forward 1 rotation
pause
point turn right 90 degrees
forward 2 rotations
pause
point turn right 90 degrees
forward 3 rotations
pause
point turn right 90 degrees
forward 5 rotations
pause
point turn right 90 degrees
forward 8 rotations
pause
point turn right 90 degrees
forward 13 rotations
stop
Questions:
- What is the next number of rotations in this sequence? Try it by adding the lines to your computer program.
- Make a graph of your data, “rotations” versus “pointer distance”. How is shape of this graph different from the shape of the graph of your data for the Stem Activity – distance?
- ADD two consecutive numbers together from your number of rotations and put the answer in a new column on your data table, and in the same row as your first addend. Do this for all of your rotations column. Compare that to your original number of rotations. What do you notice?
- Now, ADD two consecutive numbers together from your “pointer distance” column, and likewise enter the result in a new column on your data table, and for all consecutive pairs. Again – do you see any similarities to your original data?
- Now, for each consecutive pair of numbers in the rotations column DIVIDE the second number by the first number and enter into a new column. If you kept doing this for 20 consecutive pairs of numbers – about what number do you expect to get? ( Note … if this were a high school pre-calculus class, we would call this number the “limit of the Fibonacci Sequence” … you and your robot are together doing calculus! )
- And, for each consecutive pair of numbers in the “pointer distance” column DIVIDE the second number by the first number and enter into a new column. If you kept doing this for 20 consecutive pairs of numbers – about what number do you expect to get?
- Finally, look up online “Golden Ratio” and compare your results to the theoretical predictions of mathematics. How close does your experimental robot system come to the theoretical demonstration of the Golden Ratio? Write a one-sentence conclusion about your robot system and the Golden Ratio.
- (VERY ADVANCED PROGRAMMING) the program as written above is very wasteful. Can you write a program using the variables in the EV3 programming language and a loop, that is less than ten blocks long and not only does the above list of steps, but also does the next 50 steps?
