The second day of CAMP this year is off to an excellent start – armed with their new knowledge of logic and truth, the CAMPers are ready to begin their exciting journey into the magical world of fractals.
We started off the day with a whole new set of math puzzles – hypothetical chocolate boxes, tangrams, Rush Hour, linking puzzles, and of course, Set.
“… You take a card, and you put down the cars [that are] on the card, and you have to get the red car out by only sliding the cars this way.”
“So basically, we lay out cards, and they all have different patterns, and you [want to] find the sets of 3.”
“So basically, there are clues that you get in the 4-by-4 box, and you have to figure out based on these clues the color and shape of the chocolate.”
The COS group began by breaking into pairs and working on a sheet of logic problems. After that, they went on to review truth tables before Japheth switched gears to talk about fractals.
- Sierpinski Triangle – Starting with one equilateral triangle, we break the triangle into four smaller triangles and take out the triangle in the middle. Now we have three smaller triangles. Now break each of those into four even smaller triangles – then take the middle triangle out…
- Tree/”Neuron” Fractal – Start with a line that branches out into a “Y” shape. Now branch each of those smaller branches into two “Y” shapes. Now branch each of those…
- Fibonacci Spiral – Believe it or not, the Bard Math Circle logo is actually a fractal! It starts with the smallest rectangle and doesn’t get any smaller – it gets bigger. If you start with a rectangle with side-length ratio 1/2, then make a bigger rectangle by adding a rectangle with ratio 2, then continuing to add on rectangles with a bigger ratio each time (following the numbers in the Fibonacci sequence [1, 1, 2, 3, 5, 8, 13…]), then draw a curve through each of the rectangles… you get a Fibonacci Spiral.
- Cantor Set – Starting with the space between 0 and 1 on the number line, get rid of the interval between 0.333333… (1/3) and 0.666666… (2/3). Now we have two intervals [0, 1/3] and [2/3, 1]. Now take the middle third from those. Now we have two more intervals. Now take the middle third from those…
Meanwhile, SINE continued their work with using logical operators to write out true (or false) statements, which led to several productive debates over what the statements really mean, beginning with contention over compound statements (What happens when you negate a statement that already includes an AND or an OR, e.g. NOT [c OR e] ?).
After that, they segued into an exploration of the perimeter and area of the Sierpinski Triangle at various stages, using it as an example of a fractal with self-similarity – that is, if you take a small copy of the original fractal and expand it, it will look exactly the same as the original, with smaller and smaller and infinitely smaller triangles trapped inside of it.
Next, the SEC group crossed the building to join CS class, while CSC enjoyed a sunny stroll to Hegeman for Art, where they were given business cards to fold into paper cubes.
In CS, the CAMPers were introduced to a new logical operator, XOR (exclusive or) – the output only true if one (not both; only one) of the inputs is true. They then split into two groups to try and create an actual, physical XOR statement with batteries, LEDs, and cables.
Next, they transitioned into an introduction to the Base 2 (binary) system, travelling back in time to the days when place value in Base 10 (e.g., 4,598 = 4 thousands, 5 hundreds, 9 tens, and 8 ones) was heavily emphasized in their 2nd-grade math classes.
After a lunch at DTR (Down the Road), several heaping platefuls of cookies, and a not-so-secret trip to the bookstore, CAMPers split into groups based on the electives they had signed up for at the beginning of the day – a hike to the Blithewood garden, the ever-popular billiards table, and the “do-decoration” of a very mysterious dodecahedron.
No one on Bard campus can remember when the wire sculpture arrived – and even those who remember a time before can’t recall where it came from, or who put it there. Nevertheless, the rusty, dilapidated dodecahedron has been exposed to the elements for decades at least – and it blends in so well that most people who pass by need to squint to see it clearly. CAMP has decided to change that.
At the end of the electives period, CAMPers in CSC and SEC returned to RKC and Hegeman, respectively, for their CS and Art classes. Finally, everyone got together for the end-of-day activities – river-crossing puzzles, Rubik’s cubes, more linking puzzles, and Hex.
Day 2 of CAMP was full of color, laughter, and lots of fractals – we’re excited to see what Day 3 will bring!
Photo Credit: Sonita Alizada (images 7-8, 10-14, 22-32), Kateri Doran (images 6, 9, 15-17, 33), Shiven Dabhi (image 18), Japheth Wood (images 19-21), public domain (images 1-5)