Welcome! Hello there! My name is Gabrielle, and I am currently working on a project with my colleague, Alex. Our project revolves around Pop Up Fractals. These, as you may have guessed, are fractals constructed in a pop-up form, like of a pop-up book. As this is relatively new, we are now focused on brainstorming ideas and experimenting with different fractals and their construction methods.
Visit the following page to view our first pop-up fractal, "Staircases Galore!" Pop-Up Fractals After creating it, we decided to go into the topic further. Not only, we realized, that pop-up fractals are mathematical, they are easy to make and have a somewhat artistic quality to them. We decided it would be cool to recreate Staircases on a larger scale, to see more iterations of it. So, we decided to use poster board and go to the fifth iteration. The result was very interesting, and named "Mama Staircase". We wish we could go further, but the poster is very thick! Pictures shall be uploaded soon.
We have noticed how our staircase fractals are similar to Sierpinski's triangle. They display the same concept, but are not exactly identical. The interesting thing about a pop-up fractal is that the area of the product never changes as iterations are made. This is because it is the same paper that was used, nothing was cut away or added, only slits were made. So in a way the area never changed, but instead rearranged. However, the perimeter does change throughout the iterations. This relates to Sierpinki's triangle. Our next goal is to discover the pattern of the area, using our pictures of each iteration of Mama Staircase.
We have also related this changing-of-perimeter-but-not-area concept to a clever trick that Alex shared. We call it Zigzags Galore, where you cut an index card in a zigzag fashion so it can fit around your body. We plan on using this to help our readers understand the idea that our staircase fractal holds.
- I really like this connection to another paper cutting phenomenon. Maybe this would be a good point to really try to define the focus of your project. Do you want to look at the effects of changing perimeter on the structure of these shapes? Do you want to look at the relationship between the cuts you make and the appearance of changing area? (Note that this illusion of changing area varies depending on the angle from which you see the pop-up.) Or do you want to look at how different types of cuts -- that is, different pop-up shapes -- interact with the fractal process and whether it's possible to construct a pop-up fractal with that shape of cut? You could pursue any of those ideas or plenty of others, but to help you move forward, now is a good time to nail down exactly what mathematics you want to investigate here.
- -Diana (21:31, 3/10/12)
When we emerge from the staircase phase, we plan on exploring other ideas for Pop-Up Fractals. We have recently acquired "Introducing Fractal Geometry" by Nigel Lesmoir-Gordon, Will Rood, and Ralph Edney; and "The Pop-Up Book" by Paul Jackson. We hope that these two books will spark inspiration for new projects. The latter holds simple pop up works that are fractals, so we are most likely starting with those.
We've also considered turning all of our fractals into fun relate-able works of art. The fractals we come across resemble everyday things, such as pyramids, staircases, apartments, even Pacman boards. We are still deciding whether or not to do this, as it can make understanding fractals easier!
Before you get too deep into looking for more fractals to make -- or perhaps after you've made one more for comparison -- you my want to stop and look at what mathematical concepts you see emerging from this image, and which you'd like to pursue. That way, you can have a better idea going forward of what kinds of pop-up fractals you want to investigate, and you won't end up with a lot of images that are related artistically, but hard to relate mathematically.
-Diana (17:43, 3/4/12)
Thanks so much for giving us all these possibilities. What we decided to do is find the relationship of the area of the pop up's "holes" (the holes are the thing that makes it pop-up, what makes it so different from a Sierpinski's triangle) in comparison to the Stage. Since these calculations are based on the highest iteration step made, we need to see if there is a connection of the formulas between our big (stage 5) fractal to our smaller (stage 4) fractal. We want to see if there is a pattern. We think, thanks to the help of your comments, that the pattern lies with triangular numbers. What we plan to do is get more research on these and find a way to apply them as a pattern.
Our goal is now set and ready to go! We've decided to find a pattern between the area of the holes using all Stages. We have taken precise measurements and inserted our data into tables. Instead of trying to find a large pattern for the total areas of all the fractals for all the stages, we've decided to focus mainly on the sections of the fractal. If we can discover equations to find the amount of a particular section, we will be able to calculate the area from there. This, as you can see from our image page, has now been accomplished, thanks to a double math period and inspiration. But now, another idea strikes us. Is there a way to find the total area of the fractal using only the Stage number? Yes, there is a way. Actually two ways. Both use triangular numbers, one simpler, and one more complicated. The first one uses the order of the triangular numbers, and its pattern related to the section numbers. Once you know the next section number, you can easily find the amount by using the equations. From here you can calculate the area. The other way is using the fractal's steps and gaps to calculate the highest possible section for the fractal. Once you know its amount, you can work your way backwards, also leading to discovering the area of the fractal.
Our page is up and ready! This has been an interesting experience for us and we're glad that our we have contributed to the site. :D