Closest packing of Christmas cookies?

I was cutting out some cookies the other day, and noticed that my method wasn’t really that efficient.

So I started using this pattern, and challenge you to find an even more efficient way to pack stars in a plane.

A quick fractal Christmas Tree in Illustrator

Step 1: Make a thing. 

I went with a leaf shape.

Step 2: Effect > Distort & Transform > Transform.  Make sure to preview, then make some copies of that leaf and scale ‘em down.  Add a reflect X, scale dimensions down to 90%, and put some Move Vertical, and with some tweaking you should get a fern-looking thing.

Why does this remind me of the geometric series elves?

Step 3: Object > Expand Appearance so that the effect becomes materialized.  Select all, right-click and group that sucker.

Step 4: Now for the fractal part: iterate!  I rotated my tree-looking thing 90º clockwise so it resembled the initial leaf.  Then I repeated the transformation to make a bunch of these tree things with a scale factor of 90%. 

With 50 copies of a shape made of 50 parts, you’ll notice your computer start to slow down a bit.  I decided to stop while I was ahead… not a perfect fractal, but good enough for your Christmas card.

Geometric series in Adobe Illustrator

There’s a nice feature in Adobe Illustrator that makes it easy to visualize exponential change.  Here’s a little example to show your students how

Start by drawing something… in this case, an elf.  Say he has a height of one unit.  Select your drawing, right-click, and group it.

Now hit up the Effects > Distort & Transform > Transform

Here’s where it gets cool.  First click Preview, then put a number in the Copies box (I used 100 to approximate infinity).  This will create a bunch of copies of the original, but since they’re all the same size, you can’t see them.  Change the horizontal and vertical scales to 50%, and space out the elves by increasing the Move>Horizontal slider.  Drop the vertical slider so that the elves are on the same baseline.

You can now see that each elf is half the height of the previous.  This is a geometric sequence.  To find the value of the series, we need to add the heights of the elves, which we can do by adjusting the horizontal and vertical sliders to stack the elves on top of each other (make sure to have students make a prediction first).

All that’s left is to see how tall that stack of elves on top of the first one is.  Remember that the big elf is one unit, so let’s use him as the measuring stick.  Copy and paste the big elf, then turn off his transformation effect and drag him up to measure the series.

Students should see that the sum of the heights of all the little elves is equal to 1 big elf, which is to say 1/2 + 1/4 + 1/8 + … = 1.