Awesome. I like how thorough you're being in your investigation of this fractal and of all the implications and extensions it might have for other polygons. One simple organization comment: The material you have currently in "Why It's Interesting" would probably fit better in the "More Mathematical Explanation," especially as you expand on the "magic theory."
Speaking of that theory, at some point, you should move from testing the theory -- which you're doing now by experimenting with different polygons -- and into proving the theory. Proving it may lead you to modify it a bit, but I do think you all are more than equipped with the geometric knowledge to show not just that your theory applies, but why. But please understand that when I say "proof," I don't mean you need to follow any kind of strict format such as the two-column proof layout. Rather, I mean that you should work on constructing a rigorous explanation of how the mathematics behind these shapes makes them behave in this way.
-Diana (21:55, 3/10/12)
This is a terrific start, girls. I agree with Diana that you should eventually apply your knowledge of geometry and proofs to explain the mathematics behind your "Magic Theory." -Mr. Taranta (13:43, 3/12/12)
Looking forward to seeing this with the math syntax for the equations. Also, it would be helpful to have more explanation of why those equations work with your image (beyond "GSP told us so").
Another thing to look for might be real-world objects that resemble your image; are there any?
Also please spell-check!
-Leah and Diana (4/1/12 18:25)
Hi Anea, I'm sorry I didn't get to to your section soon. It looks really fantastic overall, and I love the way you walk the reader through so we can follow your process for finding these equations. A few suggestions:
- Two overall points:
- Choose to express you multiplicative ratio as either or . I'd suggest the former, but it's really consistency that you need, either way. Right now, you switch between the two.
- You tend to refer to the ratio you've found here as the ration between the "sides of the iterations" or the "lengths of the iterations," but this is unclear. Make sure you say "the side lengths of the pentagons in the iteration," or something like that, assuming that's what you mean.
- Minor point: You've written "multiple" in the first paragraph, where you meant "multiply." Also in this paragraph, you have where it should be .
- You refer to your equations as "summation equations," but this isn't actually the right term. ("Summation" means that you're adding, but there's no addition here.) I think what you were going for was "recursive equations" — equations that refer to themselves, so you get xn from xn-1, as you do in the equations you give here.
- You could consolidate your equations, and simultaneously make it clearer that both sets give the same values, by writing them like this:
- Note that I also changed the division-based equations to fraction notation, and that I added vertical dots to indicate that this pattern continues up to any arbitrary value of n. You don't have to use fraction notation if you don't want to; it just seems clearer to me. But I strongly suggest you use the vertical ellipses, since that's standard mathematical notation for this kind of thing. You can just copy and paste the syntax I used here, both for the ellipses and for the fractions, if you want to use them.
- You'll also see that I changed to and to , etc. This makes the math a bit easier to read without taking much away from the explanation, since readers already know that 2-1=1. Also, the important fact (which I think is what you meant to show by writing your subscripts this way) that, to calculate the nth iteration, you're looking at iteration n-1 is clear in the final line of equations.
- Also note that, if you combine your equations like this, the next sentence should start, "In all of these equations," rather than, "In both of these sets of equations." Similarly, two paragraphs down, "...iteration's side length divided by . In the first set of equations it would equal the 282nd iteration's side length when multiplied by ," should be, "...iteration's side length divided by , or multiplied by ."
- You can completely remove the little paragraph, "The subscripts next to ... is equal to ." At this point, this fact is clear to the reader without extra explanation.
- I suggest you move the sentence, "The ratios and ... different iteration lengths," up into your first paragraph, to right before the last sentence, "To get further sides ... achieve your goal."
- You should briefly explain why the equation, is equivalent to your recursive equations. If you need help understanding or explaining why, check out the page, Geometric Sequence.
Again, I really love what you've done here; these are relatively minor changes that I'm suggesting. Overall, great work!
-Diana (23:26, 4/18/12)
First let me start off with: Awesome Job!
Ok next are some nit picky things.
The sentence after” the shapes get larger as the iterations go on” has a few spelling errors. Look over your text again and make sure things look/and sound the way you want them to look and sound.
You don’t need to bold the description of the gif given. Also you citation is in an awkward position could you rearrange it. (this could also just be how my computer is displaying it in a weird fashion though)
Also, like Diana mentioned above explain why your equation, y=1.9^x is equivalent to your recursive equations. Also is there a better way to write this like using summation notation.
By the way I really enjoy your note about possible flaws. (Spoken like true mathematicians).
Lastly, in your theory could you make sure all equations are written in the mathematical syntax that you used in paragraph 2 -Leah
Swu2 15:35, 11 June 2012 (EDT)
Hey girls, my name is Bill Wu (rising sophomore from Beijing, China at Swarthmore College, potential math/physics major) and I've been looking at your fabulous page for possible improvements. To begin with, I confess I had zero knowledge of fractals of any kind before reading your page and that's not true anymore.
I'll list my suggestion (upon one brief read, more to come later):
- In the MME section, I think instead of jumping to the astounding factor "1.9", we could consider listing a table of the length values (which Geometry Sketchpad can nail) of x1,2,3,.., also including a column that calculates x_n/x_n-1, would would let us eventually reach "1.9".
- Furthermore, I think we should work on a more mathematical derivation of x_n/x_n-1, rather than getting it from empirical data. (Which worked out fine for arriving at the number, but is inefficient in terms of actually "explaining".) We should talk about this more.
- As for the "Basic Description" section, I think it's awesome. I would recommend in any current stage, shading the image of the previous step with light colors so the reader will know what we are building upon, thus conveying a stronger sense of continuity. Let me know if this isn't making sense, and I'll visualize my thoughts.
- Minor issue: the first image of the fractal you guys made was beautiful, yet when I first saw it, the three colors you picked for the pentagons, triangles, and decagons seemed to blend together somewhat. So I suggest maybe picking colors that contrast more with each other.
- Your "magic theory" is not only "magical", but also points to a deep-rooted mathematical urge, the urge to generalize special cases to general ones, which makes it more awesome. I'll think about it tomorrow and comment more.
This is Anea. Sorry for the late reply.
- I could most definitely make a list of lengths other than 1.9. Just let me know how you wnat me to go about this (I dont know GSp very well).
- I can shade the previous steps in the Basic Description very quickly too. I work at a hospital for about 6 hours each day so I'm a bit busy, since I do have to bring some work home. If you could give me until at lastest this Sunday for that, then that would be great!
- I could also recolor the first image of the fractal. No problem, that can also be done quickly. Once again just give me until Sunday.
- As for the magic theory. I don't have much of the math on that. My other two partners worked on that teh most. I could look into it but without the original math from them. I feel as though my efforts would be a bit useless. But I will still try to work with and also contact my other two partners.
- As for x_n/x_n-1 math. Tell me what you think we should do. I would love to explain that part even more!
Thanks you so much! And once again sorry for the late reply. I will try to contact Erin and Melina.