## Archive for the ‘Geometry’ Category

### A decagonal snowflake from pentagons

25 June 2017

So far in these posts about fractals, I’ve shown (1) how letting triangles reproduce like this:

generates a bunch of copes of the Koch snowflake at different scales:

Similarly, I’ve shown (2) how letting squares reproduce like this:

generates a bunch of copies of a fractal related to the Koch snowflake, but with 8-fold symmetry:

So what about letting pentagons reproduce?   For pentagons, an analog of the replication rules above is this:

Each of the 10 cute little pentagon children here is a factor of $\frac{1}{2+\varphi}$ smaller than its parent, where $\varphi$ is the golden ratio.

However, something interesting happens here that didn’t happen with the triangle and square rules. While triangles and squares overlap with their ancestors, pentagons overlap with both their ancestors and their cousins.  The trouble is that certain siblings already share faces (I know, the accidental metaphors here are getting troublesome too!), and so siblings’ children have to fight over territory:

In this three-generation pentagon family portrait, you can see that each second generation pentagon has two children that overlap with a cousin.

As we carry this process further, we get additional collisions between second cousins, third cousins, and so on.  At five generations of pentagons, we start seeing some interestingly complex behavior develop from these collisions:

There’s a lot of fascinating structure here, and much of it is directly analogous to the 6-fold and 8-fold cases above, but there are also some differences, stemming from the “cousin rivalry” that goes on in pentagon society.

Let’s zoom in to see some collisions near where the two ‘wreaths’ meet on the right side of the picture:

I find the complicated behavior at the collisions quite pretty, but the ordering issues (i.e. which members of a given generation to draw first when they overlap) annoy me somewhat, since they break the otherwise perfect decagonal symmetry of the picture.

If I were doing this for purely artistic purposes, I’d try resolving the drawing order issues to restore as much symmetry as possible. Of course, I could also cheat and restore symmetry completely by not filling in the pentagons, so that you can’t tell which ones I drew first:

It’s cool seeing all the layers at once in this way, and it shows just how complex the overlaps can start getting after a few generations.

Anyway, because of these collisions, we don’t get seem to get a fractal tiling of the plane—at least, not like we got in the previous cases, where the plane simply keeps getting further subdivided into regions that converge to tiles of the same shape at different scales.

Actually, though, we still might get a fractal tiling of the plane, if the total area of overlap of nth generation pentagons shrinks to zero as n goes to infinity!  That would be cool.  But, I don’t know yet.

In any case, the picture generated by pentagons is in many ways very similar to the pictures generated by triangles and squares. Most importantly, all of the similar-looking octagonal flower shaped regions we see in this picture including the outer perimeter, the inner light-blue region, and tons of smaller ones:

really are converging to the same shape,  my proposal for the 10-fold rotationally symmetric analog of the Koch snowflake:

How do we know that all of these shapes are converging to the same fractal, up to rescaling?  We can get a nice visual proof by starting with two pentagons, one rotated and scaled down from the other, and then setting our replication algorithm loose on both of them:

Proof:

We see that the area between the two fractal curves in the middle shrinks closer to zero with each generation.

Puzzle for Golden Ratio Fans: What is the exact value of the scaling factor relating the two initial pentagons?

Next up in this infinite series of articles: hexagons!  …

I’m joking!  But, it’s fairly clear we can keep ascending this ladder to get analogs of the Koch snowflake generated by n-gons, with (2n)-fold rotational symmetry.  More nice features might be sacrificed as we go up; in the case generated by hexagons, we’d have collisions not only between cousins, but already between siblings.

### More fractal fun

23 June 2017

In the previous article, I explained how the famous Koch snowflake can be built in a different way, using “self-replicating” triangles. This was a revelation for me, because I had always thought of the Koch snowflake as fundamentally different from other kinds of fractals like the Sierpinski triangle, and now I think of them as being basically the same.

In the Sierpinski triangle, each triangle yields three new, scaled-down triangles, attached at the midpoints of sides of the original, like this:

These triangles are usually thought of as “holes” cut out of a big triangle, but all I care about here is the pattern of the triangles.  As I explained last time, the Koch snowflake can be built in a similar way, where each triangle produces six new ones, like this:

You might say this bends the usual rules for making fractals since some of the triangles overlap with their ancestors.  But, it makes me happy because it lets me think of the Sierpinski triangle and the Koch snowflake as essentially the same kind of thing, just with different self-replication rules.

What other fractals can we build in this way?  The Sierpinski carpet is very similar to the Sierpinski triangle, where we now start with squares and a rule for how a square produces 8 smaller ones:

This made me wonder if I could generalize my construction of the Koch snowflake using triangles to generate some other fractal using squares.  In other words, is there some Koch snowflake-like fractal that is analogous to the ordinary Koch snowflake in the same way that the Sierpinski carpet is analogous to the Sierpinki traingle?

There is!  Taken to the 5th generation, it looks like this:

The outline of this fractal is an analog of the Koch snowflake, but with 8-fold symmetry, rather than 6-fold.  Compare the original Koch snowflake with this one:

Just as I explained last time for the Koch snowflake (left), the blue picture above actually provides a proof that the plane can be tiled with copies of tiles like the one on the right, with various sizes—though in this case, you can’t do it with just two sizes of tiles; it takes infinitely many different sizes!  In fact, this tiling of the plane is also given in Aidan Burns’ paper I referenced in the previous post.

But, my construction is built entirely out of self-replicating squares.  What’s the rule for how squares replicate?

Before I tell you, I’ll give two hints:

First, each square produces 8 new squares, just like in the Sierpinski carpet.  (So, we could actually make a cool animation of this fractal morphing into the Sierpinski carpet!)

Second, you can more easily see some of the bigger squares in the picture if you make the colors of the layers more distinct.  While I like the subtle effect of making each successive generation a slightly darker shade of blue, playing with the color scheme on this picture is fun.  And I learned something interesting when my 7-year old (who is more fond of bold color statements) designed this scheme:

The colors here are not all chosen independently; the only choice is the color of each generation of squares.  And this lets us see bits of earlier-generation squares peeking through in places I hadn’t noticed with my more conservative color choices.

For example, surrounding the big light blue flower in the middle, there are 8 small light blue flowers, and 16 even smaller ones (which just look like octagons here, since I’ve only gone to the 5th generation); these are really all part of the same light-blue square that’s hiding behind everything else.

The same thing happens with the pink squares, and so on.  If you stare at this picture, you can start seeing the outlines of the squares.

So what’s the rule?  Here it is:

### 4!-torsor a la George Hart

20 April 2015

As a project with a certain 4-year-old relative of mine, we constructed the proof I described before that the outer vertices of George Hart’s 12-Card Star form a 4!-torsor.  (I guess I didn’t say it that way before, but it’s true!)  Here’s our proof:

Last time I suggested using a deck of 12 cards like this:

But instead, we used four solid colors, three cards of each.  So, our “star” permutes the colors red, white, black, and silver:

You can get any permutation of these colors in our Star by exactly one symmetry taking outer vertices to outer vertices.  The “exactly one” in this isomorphism is what makes the set of outer vertices a 4!-torsor rather than just a 4!-set.

Here’s what it looks like when you put three pieces together, from both sides: