Archive for the ‘Uncategorized’ Category

Cutting a triangle into infinitely many Koch snowflakes

8 June 2018

I want to explain the construction behind a piece that I posted recently without explanation.  Starting from a single triangle, I’ll show you how to cut it up into pieces, rearrange those pieces, and get this:

triangles

The construction

Here’s how it works. Start with an equilateral triangle:

triangles-gen0

Mark off each edge into thirds, and connect those points with lines using this pattern:

triangles-gen1a

This divides up the triangle into into seven smaller triangles, three equilateral and three isosceles.  Remove the three isosceles triangles to get this:

triangles-gen1b

But now, notice that the three triangles you removed can each be cut in half and then taped back together along an edge, so that you get three equilateral triangles. Do that, and place the new equilateral triangles so that they stick out from the sides of the original triangle, and you will get this:

triangles-gen1

That’s it for step 1!

Now, for step 2, repeat this whole process for each of the 7 equilateral triangles obtained in step 1.  Step 3: Do the same for each of the 49 triangles obtained in step 2.  And so on. My original picture, at the top of this post, is what you get after step 5.

Notice that each step is area-preserving, so in particular, the total area of all of the black triangles in my original picture is the same as the are of the triangle I started with.

Here’s an animation showing the first five steps in the sequence, and then those same steps backwards, going back to the original triangle:

snowflake-animation

The reason the picture seems to get darker in the latter steps is that the triangles are drawn with black edges, and eventually there are a lot of edges.  Since there’s a limit to how thin the edges can be drawn, eventually, the picture is practically all edges.

How many snowflakes do you see?

The outline of the entire picture is clearly a Koch curve, so we have generated a Koch curve from a triangle.  But, what I really love about this construction is that every triangle that occurs at any step in the recursive process also spawns a Koch curve!  That’s a lot of Koch curves.

To make this precise, we can assume that triangles at each step are closed subsets of the plane.  Admittedly, the “cutting” analogy falls apart slightly here, since two pieces resulting from a “cut” each contain a copy of the edge the cut was made along, but that’s OK.   With this closure assumption, each of the Koch curves, one for each triangle formed at any stage in the process, is a subset of the intersection over all steps.

The Koch Snowflake from triangles

11 May 2018

Here’s a new variation in my series of my Koch snowflake-like fractals drawn entirely from regular polygons.  This picture consists entirely of black equilateral triangles.


triangles

I’ll avoid explaining this one for now, except to say that I generated it starting from a single triangle, and iteratively replacing each triangle by seven new triangles.  This is the sixth generation. The construction differs only slightly from my previous Koch snowflake fractal, in which each triangle had six descendants.  I really like this new version, because you can see Koch snowflakes showing up in even more (infinitely more!) places than before.

There are also analogs of this for squares and pentagons!

3d Fractal models

29 March 2018

My student Colton Baumler has been printing 3d versions of some of the fractal designs I posted about a few months ago:

IMG_9818

That’s his three-dimensional interpretation of the first few iterations of this design of mine:


octagon-snowflake

What’s fun about Colton’s version is that each new layer of squares is printed a bit taller than the previous layer.  I had really only imagined these as two-dimensional objects, so for me it’s really fun to have 3-dimensional models of them to hold and play with!  Colton’s idea of adding some depth really adds another … er … dimension to the overall effect:

IMG_8738

His work also gives a nice way to illustrate some of the features of these fractals.  For example, visually proving that the “inside” and “outside” in my fractals converge to the same shape can be done by printing the same model at different scales.  Here are three copies of the same fractal at different scales, each printed with the same number of iterations:

IMG_9830

Not only do these nest tightly inside each other, the thickness is also scaled down by the same ratio, so that the upper surfaces of each layer are completely flush.

Colton has been doing this work partly because designing fractals is a great way to learn 3d printing, and he’s now getting some impressively accurate prints.  But, I also like some of his earlier rough drafts.  For example, in his first attempt with this fractal based on triangles:


koch6

there were small gaps between the triangles, which Colton hadn’t intended.  But, this gave the piece a sort of rough, edgy look that I like, and it casts shadows like castle battlements:

IMG_9839

Colton is still doing nice new work, and we’ll eventually post some more pictures here. But I couldn’t wait to post a preview of some of his stuff!

IMG_9850

(Designs and photos © 2018 Colton Baumler and Derek Wise)

Observer Space: new paper and ILQGS talk

3 October 2012

Steffen Gielen and I just put our new paper on “observer space” on the arxiv:

S. Gielen and D. Wise, Lifting General Relativity to Observer Space

Then, today I gave the International Loop Quantum Gravity Seminar on the same topic. This a seminar between various institutions, mainly in North America and Europe, where people work on loop quantum gravity and related topics. It’s run the old-fashioned way, as a conference call.

I was a bit uneasy about volunteering for such a talk. I don’t like phones. I’m happy to speak in front of any audience I can see — but an audience I can’t see is a little intimidating, even if I do probably know most of them. Besides, on the phone, you never know whether someone might be recording your conversation, hoping to use it against you later. And in this case, they were! Here’s the audio to my talk in aiff or wav format. If you decide to listen to that, you might also want to look at the slides to my talk.

Seriously, I think the talk turned out rather well — except for the part where my Skype connection to the phone bridge cut out, and I didn’t even know it. Fortunately, though, as I only found out after the talk, Steffen took over, explaining to the audience the same stuff that I was simultaneously, unwittingly, explaining into a black hole. Steffen had never seen the slides, and described this as his first experience with live Powerpoint karaoke. I think he did an excellent job of filling in.

I’ll have to explain a bit more about “observer space” on this blog sometime later…