## Wednesday, April 23, 2008

### Power of Two Cake

I know you've been long awaiting a post on my powers-of-two themed cake.  Now that I'm less swamped by work, I can oblige.

But first, a little math.  (I know you all come here for the math!)  Astute commenter Rico mentioned his fascination with the sum of the series 1/2n (where n = 0, 1, 2, ...).  If you add 1 + 1/2 + 1/4 + 1/8 +..., the sum approaches but never reaches 2.  For a given n, the sum of the series equals 2-1/2n.  So for n=4, the sum equals 2 - 1/16 = 1+15/16.

I decided to represent this sum in cake.  In order to do so, I needed to bake two square cakes.  I had one 8" square cake pan, but I needed another, so I did as The Joy of Cooking suggested and squared off a rectangular cake with aluminum foil, putting dried beans on the other side:

The cake was my favorite cake recipe: cocoa devil's food cake from The Joy of Cooking.  You've seen me make this cake twice already, so I won't duplicate it here.

Here's how it turned out: the foil edge is a little uneven, but not bad.  I decided to use it as the base of the cake, to hide the unevenness.

My dad and Marvis were visiting, and Dad helped cut the cake into the proper pieces.  You can see him measuring the fractions with a measuring tape.
Here are the pieces after he was done.  So, if the other square represents one, then we have a 1/2, a 1/4, 1/8, and two 1/16s (one of which will not be used), concretely (or maybe, more aptly, chocolately) demonstrating the sum I showed above.
The frosting is a simple buttercream: butter + milk + powdered sugar + vanilla.  Here's the 1 + 1/2 + 1/4.
And here is the completed cake.
Next I decorated it with chocolate chips, to illustrate positive powers of two.  I put four chocolate chips on the smallest piece, four more on the part of the next sized layer that's showing, eight on the part of the next sized layer that's showing, etc.
So that ideally, if you look straight down on the cake, it just looks like it has 64 evenly-spaced chocolate chips on it.
Here's the cake from its most photogenic angle, complete with tissue paper flowers in the background:
I ate the leftover 1/16th piece while I was decorating the cake.  Then, when I served the cake, I had the 1/16th piece, Jeff had the 1/8th piece, and Dad and Marvis split the 1/4 piece.  We all had equal portions of cake and equal numbers of chocolate chips.  How cool is that?!?

I took the leftover cake to work with me the following Monday, so that Jeff and I wouldn't eat it.  I told everyone it was made in celebration of powers of two.  One of my colleagues emailed me with a message of "Great cake -- terrible reason!"  I had to laugh.

I hope you all enjoyed this mathematically-inspired cake as much as I did!