Talk:Sierpinski's Triangle

Contents

Max 10/18/2012

I noticed that the images don't seem to be loading in the first section of the page.

Abram 7/10

Really nice page. More images to go with the perimeter, number of edges, etc, sections would be really helpful, but I'm not sure you have time for that. The only other thing that bothers me is:

The perimeter of the triangle increases by a factor of 3/2 as we can see from the images above.

Could you explain this a bit? It's not hard to see, but it's also that obvious without a bit of explanation

Anna 7/6

So it looks like all of the other pages are using the D, N and e notation for the Fractal Dimension, as is that page. You should make this page consistent with that.

And, to address one of chris's questions, no, it does not have to be an equilateral triangle for the chaos construction to work. If I have time tonight, I will create and email you a mathematica notebook that shows that.

Alright, sounds good.

Chris Taranta 7/5

My wife and I both loved this wiki. It is clearly presented and Alan did a great job of explaining each section step by step. The images were very useful. It helped that it the math was more accessible than some of the other images.

Edits:

⁋1: Can it be any triangle or is it only equilateral? Can you get the image to stop moving after a while?

It has to be equilateral. I can't get the image to stop moving, but hopefully the Drexel people can

Chaos Construction I don’t get the connection between the image shown and the concept described. An example image where you take a random point and random vertex would be very helpful in demonstrating this concept.

The chaos construction should be more clear once a animation/interactive applet has been made.

A More Mathematical Explanation/ Number of edges I’d capitalize the E in edges in the section title. ⁋3, Sentence 3: I’d put a semicolon instead of a comma at “appear to contradict this; however, this is a flaw…” ⁋3, Sentence 4: “…the triangle has an infinite number of iterations.”

Perimeter ⁋1, Sentence 2: “…function of the number of iterations.”

Area ⁋1, Sentence 2: I think you are unshading the central triangle, not shading it.

Pascal’s Triangle ⁋1, Sentence 1: I assume the link the reader should click on is in the next sentence. It might be nice to make it clear.

Sierpinski’s Triangle in 3 Dimensions ⁋1, Sentence 1: To what “page’s main image” are you referring? Do you mean the one above? ⁋1, Sentence 2: As in the Chaos Construction section, I find the reference to the random point and random vertex quite puzzling. ⁋2, Sentence 2: Does that mean that the volume is 0?

Again, a applet will make it much more clear than words.

Ideas for the Future I like both ideas a lot.

David 6/30

Overall this page is quite nice. I have a concern about the area section though. When describing how the area decreases you first state "splits each triangle into 4 congruent parts and by shading the central one, removes 1/4 of the area." Later you state "The Sierpinski's triangle has total area of 0 (defining area as the shaded region)." These two shaded areas are are the triangle and the negative space. I would revise the original description of shading the central region.

Also, in Fractal Dimensions you have undefined terms.

Corrections to the area section have been made. The fractal dimension page specifically doesn't have defined variables because fractal dimension is probably over the heads of most readers. If they are curious to learn more, they should click the link for more explanation.

Anna 6/29

Remember to cite each picture that appears in the page if you didn't create it.

It's kind of nit picky, but you don't actually need an initial point inside the triangle for the chaos construction to work. If it's not, you just throw away the first few iterations. I can show you my program that demonstrates this.

Your ideas for the future should be pretty easy for the drexel people to do. It takes several thousand iterations to get the triangle, but the program is super simple (and the one for the tetrahedron really isn't any more complicated).

Hopefully I got through everything.

Abram 6/25

Alan, I really like your overall approach to the page. Your images are well-chosen, and the topics you choose to cover and the order in which you cover them is good. Your use of mathematical language, esp. the way you talk about infinity, is basically great, as well.

Small, but important jobs (these give you a lot of bang for your buck):

• Give the opening caption have a bit more punch. Instead of just saying that S's triangle is a simple fractal, you could add one of those exciting details, like how it's created through an infinite iterative process, or how it exhibits self-similarity (but don't use that term in the opening).
• You write, "Since the number of edges a Sierpinski's triangle has is infinite, the resulting perimeter must be infinite." You are right that the perimeter in this case infinite, but I could easily show you a figure with an infinitude of edges and a finite perimeter. This means that the "if-then" logic in this sentence is not sound logic.
• You could talk about how the perimeter increases by a constant factor at each iteration -- this *does* imply that the perimeter is infinite.
• You write, "Each iteration of the construction process reduces the area by a factor of 3/4." This wording isn't good. The area is reduced by 1/4, or the area is reduced to 3/4 of its value in the previous iteration. Being reduced by 3/4 would mean it lost 75% of its area in each iteration.

Main big job

Look for places where it might not be clear what a phrase or word refers to, exactly. For example:

• In the first section, you say, "We then repeat this process on the 3 newly created smaller triangles." Which three triangles? It looks like you have created four of them. One way to clarify this would be to talk about "removing" the center triangle rather than "lightening" it.
• In the perimeter section you write, "Thus we can relate the number of sides..." What constitutes a side at this point?
• You also write, "we can evaluate this as an infinite limit." Does infinite limit mean that you are taking the limit as x goes to infinity, or that the value of the limit itself is infinity? I know both apply to this circumstance, but which one are you talking about in that sentence? Also, what does the "this" refer to in the phrase just quoted.

I've hopefully incorporated all these suggestions into my page.

Alan, 6/19

Chaos Construction: how about showing some steps? Incidentally, this doesn't seem to require more math, so might go hidden (with title visible) before More Math.

Hopefully there will be a nice interactive chaos creation applet in the future, so I'll hold off on showing steps. Pascal: this is a bit abrupt I think. I couldn't find more information about this online, but if I do I'll come back to it.

3 D: neat illustrations, more refs would be nice. Not much info about it online, maybe something from Abram or Anna to do?

Gene, 6/17

Alan, thanks for asking me to look through this. Good stuff. I've mostly asked you to use your 3 illustrative diagrams in many places below, but not to do much further hairy stuff. Let's talk this over before you undertake anything that's going to take a lot of work.

"Sierpinski's triangle is a simple fractal." Maybe add something like "We'll see what that means below" if we will, otherwise a reference? Or at least, "Basically, that means that no matter how far you zoom in the result is much like the original image" or something.

"The animation on the right depicts a true Sierpinski's triangle." Maybe something like "The animation on the right depicts what would happen if you keep zooming in on a true Sierpinski's triangle." or something. And maybe make that important--in a box or something--a clear caption for the animation.

Please have the captions that you get when you click [show] for the More Math Exp always visible (my idea is to engage folks into looking further!) Also, what math is needed? Be good to put that right at the beginning of the More Math. In your case it might be "More Detailed" rather than more math?

Chaos Construction: how about showing some steps? Incidentally, this doesn't seem to require more math, so might go hidden (with title visible) before More Math.

Number of Edges: Could we maybe have the images again with number of edges to the right of the triangle? Again good stuff with not much math.

Perimeter: Maybe have the same 3 images with the calculated length of perimeters on the right?

"Since the number of edges a Sierpinski's triangle has is infinite, the resulting perimeter must be infinite." Not necessarily, you could have the edges getting small so fast that the perimeter doesn't get infinite. I'd leave that sentence out and just use the limit.

Area: Again having your 3 examples would be real helpful. This is a pretty surprising result so should be celebrated!

Fractal Dimension: You might want to say a little more about what this is? Good ref, however.

Pascal: this is a bit abrupt I think.

3 D: neat illustrations, more refs would be nice.