Tweaking the Koch Snowflake with Apollonian Gasket
Oh wow this is so awesome! My own page....I like the sound of that. Well, I was planning on creating a Apollonian gasket within a Koch Snowflake. I'd start with an equilateral triangle and inscribe the largest circle possible. By applying the Koch curve to each of the sides, I get three more equilateral triangles. Then, a circle would be inscribed in each of those circles. And so on and so on. The result will be a fractal within a fractal. The difficult part is finding a way to first draw the Koch Snowflake. Doing that by hand is a burden and frankly, I'm lazy because that sounds like a lot of work. It is possible to use GSP (Geometers Sketchpad) for this task but I need to work out the kinks (I'm not very good with GSP). The iterate function seems to be useful. Once I have the snowflake, then I'll worry about the circles.
Hi Victor! A couple of ideas:
- There's an existing MI page on the Koch snowflake, but it looks like the math section is incomplete. Maybe you could work on fleshing it out?
- Here's a page that explains a bit of how to make the Koch Curve (closely related to the snowflake) in GSP: GSP Iteration.
- Could you explain how you came up with this method of constructing an Apollonian gasket within the Koch snowflake? How does the combination of the two change them?
-Leah & Diana (18:13 2/19/12)
If you're still looking for ways to construct the circles in your image as part of your GSP iterations, I suggest you play with the idea of making a rule for how the center point of each circle translates to the center point of a circle in the next iteration. This might involve finding rules for how the angle bisectors of larger triangles relate to bisectors of smaller ones. Once you know this, you have the information to construct new iterations of circles using the intersections of the bisectors of each triangle as the center point for its internal circle, and using the intersection of one angle bisector with the opposite side of the triangle to mark the radius distance of the circle.
-Diana (20:48, 3/4/12)
So I tried your idea of make an iteration tool with the circles and the triangles combined. So when I click the next iteration, both the circles and triangles will show up. This was what happened when I tried this. Only the small circles show up and not the large circles. The picture on the left is the iteration prior to the image on the right. Any ideas on how I can relate the circles so they show up together?
What I did: Draw a line. Mark one endpoint as the center. Dilate the other endpoint to the ratio 1/3. Mark the other endpoint as the center. Dilate the other endpoint (the point you first marked as the center) to ratio 1/3. Click one of the inner points along the line segment and mark as center. Click the other inner point and rotate 60 degrees. Find the midpoint of the segment formed by the two inner points. Mark that as the center. Click the point that was rotated and dilate it to 1/3. Hide original line segment. Connect the points. (The midpoint and the point directly above it will form the circle. Then I highlighted everything and used the Iterate function.
- I was able to get it to work with roughly the same method. That could be for a couple of reasons, but first, this could be an easy fix: when you're in the "Iterate" window, before you tell GSP to iterate, go into the "Display" drop down menu in that same window, and make sure you've got a check mark next to "Full Orbit (All Iterations)," and not next to "Final Iteration Only."
- Hope that fixes it; if not -- or if you've already done that -- we'll figure out what's going on. In general, the method you've come up with works.
- -Diana (11:37, 3/7/12)
- Okay that makes sense. I've always made sure to put my iterations in "Final Iteration Only" because thats how most of the GSP labs are done. I'll try that.