What Geometry Is Good For

At some point, everybody who's taken geometry in school has said, "When am I ever going to need geometry?"  The surprising answer: baking.  You really do need to know about degrees and angles and circles and triangles and rectangles (and various other geometric ideas) to bake and construct certain baked goods.  Listen up, class - today's lesson is on checkerboard cakes.

The idea of a checkerboard cake is that when you slice a piece, the inside of the cake isn't just one cake, but two different types arranged in alternating squares, like a chess- or checker-board.  The first time I saw a checkerboard cake, I actually thought that somebody had taken two cakes, sliced them up into little cubes, and then glued each cube into place with frosting.  I suppose you could make a checkerboard cake like this, but it's much easier to use geometry.

The key is to use circles to make the squares.  Hang on, what?  It's just like being in math class all over again.  To make a checkerboard pattern, you need to have the cake colors alternate.  You've got two round cakes and when you slice out a wedge, you want to see interlocking squares.  But the squares are really an optical illusion.  Instead, you cut out circles of cake and stack them so that the colors switch back and forth both horizontally and vertically.  (Or you can buy a pan that separates the rings for you.)

This sounds complicated but it's easy to see when you look at a picture.

A special pan for baking checkerboard cakes.  One color batter goes in the inner and outer rings, and the second color goes in the middle ring.  Swap colors in the next pan, and then stack the cakes vertically.  Cutting out the rings after the cakes are baked lets you skip using the special pans.

To make a two-layer checkerboard cake with six squares, you need two layers of cake where the inner and outer ring are the same color horizontally and a different color vertically.  You can achieve that part by simply stacking up two cakes like you normally would.  To make the middle squares be alternating colors, slice out a ring of cake from the middle, then switch the middle ring of one cake with the other one.  When you look at them horizontally, they make a pattern: Color 1, Color 2, Color 1.  The other layer will have the opposite pattern:  Color 2, Color 1, Color 2.  Then stack the two layers, and each ring will be sitting either on top or below of a ring of different color:

Color 2  Color 1  Color 2
Color 1  Color 2  Color 1

This is how you put together rings to make alternating squares when a wedge is cut.

See?  Geometry at work.

This particular example also involves fractions, because I messed around with the cake recipe.  So now we're using both algebra and geometry.

When you make a checkerboard cake, you need to use two cakes with very different colors.  In the first example, I just dyed one of the white cakes blue.  In the more recent example, I made one white and one dark brown cake.

I'm trying to be more organized so last week I cleaned out my cupboard in the kitchen.  There were all kinds of plastic bags with just a small amount of something left in them cluttering up the shelves.  One bag had about half a cup of shredded coconut, which put me in the mood to make a coconut cake.  At that point, I remembered trying the checkerboard cake and thought it would be fun to do that again, with coconut and chocolate.  Of course, I was so organized that I accidentally threw out the bag of coconut and ended up not adding it to the cake batter like I had originally planned.

Finding a recipe for a cake where the coconut flavor is in the batter turned out to be more difficult than I was expecting.  Most coconut cakes are white cakes with coconut frosting.  I finally found a recipe that used coconut milk in the cake and that sounded like it would be good.  It made two 9 inch cakes, to be layered together with coconut frosting.  I decided to make the recipe but leave out the coconut from one half of it and put in cocoa powder instead.  This is where the fractions show up.

I love the taste of coconut and I use coconut milk every morning in my chai.  You can get unsweetened coconut milk at Trader Joe's, next to the rice and soy milk.  Trader Joe's also started carrying coconut cream, which is the thicker part of the liquid inside of a coconut.  Unfortunately it's emulsified, but I decided to use it in the cake recipe anyways to get a stronger coconut taste.

The coconut taste comes from the coconut milk (or cream), which is a liquid.  The only change I needed to make to change the cake from coconut to chocolate was to use a good quality cocoa powder (Ghiradelli) in the solids and heavy cream instead of coconut cream in the liquids.  To avoid measuring all of the solids twice, I mixed the flour, baking powder, and salt together in one big bowl and then split them up into two smaller bowls.

I added three tablespoons of cocoa powder to one bowl for a strong chocolate taste, and then added some extra sugar to taste right before pouring it into the pan.  The cake batter should taste similar to the baked cake's taste, so if the batter isn't sweet enough then the cake won't be either.  I probably added somewhere between 1/8 and 1/4 cup of sugar to the chocolate batter.

The recipe calls for you to mix the ingredients in a certain order.  The reason for this is to have a very smooth batter at the end.  I couldn't follow the directions because of the changes I had made, so I used a strong metal whisk to incorporate all of the clumps before baking.

I also decided to try making my own buttercream frosting instead of buying it pre-made.  Frosting is one of those things that seems like it would be really difficult to make and nobody likes the store-bought version, but I also don't know anybody who makes their own.  I was already using heavy cream in the cake recipe so I figured why not try to make buttercream with the rest of it?

Buttercream is named for two of its ingredients: butter and cream.  The third ingredient is powdered (confectioners) sugar.  Sugar is not the third ingredient if you rank them by amount.  You use a lot more butter and sugar than cream.  I doubled the recipe because you generally need about 3.5 to 4 cups of frosting for a double layered cake, and I used 1 pound of butter and 8 cups of powdered sugar.

If you're thinking that you've become diabetic by just reading that ingredient list, remember that the cake is usually cut up into 10 or 12 pieces so you're not getting that much frosting.  Having a little bit of buttercream frosting occasionally is fine - just make sure that you're not eating it for dinner every night.

Tips:

Creamed butter is not melted butter.  When butter is creamed, it means that it's soft enough to smush with a fork but it is still solid.  You want it to have the consistency of frozen yogurt or soft ice cream.  What you don't want is for the fats to separate.  Melted butter has a light part and a darker part.  You need to avoid this separation.  It's very common to cream butter and sugar together, and the butter must stay together for the recipe to work and have the consistency you want.

If you've forgotten to take the butter out of the freezer and it's hard as a rock, chop it up into 5 or 6 larger pieces.  Then microwave the pieces for about 10 seconds at a time, checking with a fork after each microwave if the butter has softened.  It will melt a little bit, but you can incorporate that back into the rest of the creamed butter.

You must use creamed butter when making the frosting.  If the butter melts, the consistency of the frosting changes and it will be very difficult to use.

I don't own a stand mixer, which is what most buttercream recipes call for.  Instead, I use my 11 cup Cuisinart.  The only problem is that the motor in a stand mixer is on top and separate from the bowl, but the motor is on the bottom in a Cuisinart.  The butter ended up melting a little bit, just from the heat put out by the motor under the bowl, and it was enough to make the frosting difficult to use.  I ended up refrigerating the frosting for a while until it was solid and then microwaving it for 15 seconds to loosen it up slightly.  You can see the difference in the picture that shows the final coat of frosting over the crumb coat.

This page is an excellent reference for how to frost a cake.  You need to apply a crumb coat, which is a light coating of frosting that holds the crumbs that get loose from the cake in the first coat.  Then you let the cake sit in the fridge for 30 minutes to an hour to solidify the frosting.  Once the crumb coat is dry, you can frost the final coat which won't have any crumbs in it.  I strongly suggest that you read the webpage on frosting a cake.  It has easy to follow instructions and useful hints.

To sift flour or sugar (or any powder), you just need a hand held strainer (sieve) with a fine mesh.  Then you bang the strainer against your hand or scrape the inside of the strainer with a spoon or create friction.  The powder will sift through visibly.  If you don't see it, you're not hitting the strainer hard enough.  Once most of the powder is through, there will usually be some larger balls of powder left at the bottom of the strainer.  You can push those through with your fingers.  Be careful to make sure that the strainer is dry when you sift or you will end up with a clumpy mess.

Most recipes say if the measured amount of powder is pre- or post-sifting.  If a recipe doesn't say, I measure out the amount and then sift it.  For both the cake and the frosting recipe, I sifted after I measured.

Buttercream just tastes like sweet butter but it is easy to flavor.  I zested one lemon into the butter when I was halfway through adding the sugar, and people said that it tasted very lemony.  I couldn't taste it at all so opinions may vary.

You can dye either the frosting or the cakes very easily.  Liquid food dye is generally sold in most grocery stores (in the baking aisle), but gel colors are much stronger.  You can buy them either online or in baking stores.  The bright yellow color of the frosting came from about 1/2 teaspoon of yellow gel dye in 8+ cups of frosting.  It is powerful stuff.

To cut out the circles for the checkerboard cakes, I made imprints of circles using two different sized bowls, one glass, and a mason jar.  Since I wanted to make three cake rings, I needed two circle imprints to cut along.  I decided to use the imprints that made three concentric circles of about the same width.  Using pre-shaped objects like this made it easy to cut rings of the same sizes from both of the cakes.  The central rings fit outside the small ring and inside the large ring because all of the cakes were cut from the same bowl and jar.  The chocolate cake was fluffy enough that I just pushed down with the bowl and jar and didn't even need to cut with a knife.  It's difficult to keep the knife straight up and down when cutting, so the rings from the coconut cake were slightly tilted but not enough to make a difference.

Because I used different liquids, the consistency of the two cakes were different.  The chocolate cake was very moist and fluffy, and slightly crumbly.  Most people like this consistency the best.  The coconut cake was also very moist but very dense.  The Trader Joe's coconut cream was solid, so I added some coconut milk to make it more liquid.  This meant that the liquids added to the coconut cake batter were actually less liquid-y than the liquids added to the chocolate cake batter.  The coconut cake had a consistency like mochi because I used the coconut cream.  I really liked it, but if you don't like mochi you will want to use all coconut milk instead.  Or you can add half coconut milk and half heavy cream, and add in shredded coconut for more flavor.

Double-Coconut Cake
Original recipe from myrecipes.com

Ingredients 

• 1 tablespoon cake flour 
• 2 1/4 cups sifted cake flour 
• 2 1/4 teaspoons baking powder 
• 1/2 teaspoon salt 
• 1 2/3 cups sugar 
• 1/3 cup butter, softened  
• 2 large eggs  
• 1 (14-ounce) can light coconut milk 
• 1 tablespoon vanilla extract 

This recipe also has a coconut frosting but I did not make it.  The reviews say that the cake is good but the frosting is not.

Directions
Preheat oven to 350°.

Coat 2 (9-inch) round cake pans with cooking spray; dust with 1 tablespoon flour.

Combine 2 1/4 cups flour, baking powder, and salt, stirring with a whisk. Place sugar and butter in a large bowl; beat with a mixer at medium speed until well blended (for about 5 minutes). Add eggs, 1 at a time, beating well after each addition. Add flour mixture and milk alternately to sugar mixture, beginning and ending with flour mixture. Stir in vanilla.

Pour batter into prepared pans. Sharply tap the pans once on countertop to remove air bubbles. Bake at 350° for 30 minutes or until a wooden pick inserted in the center comes out clean. Cool in pans 10 minutes on wire racks, and remove from pans. Cool completely on wire racks.

My modifications: Combine flour, baking powder, and salt as directed.  Cream butter with sugar by hand - no need to beat for five minutes.  Add eggs to the butter/sugar mixture.  Roughly divide the flour mixture into two bowls.  Add approximately 3 tablespoons of cocoa powder and 1/4 cup sugar to one bowl.  Divide the sugar/butter/eggs mixture between the two bowls.  Add the vanilla extract.  Stir until mixed, then beat with a whisk to incorporate any clumps of flour.


Classic Vanilla Buttercream Frosting

Original recipe from Savory Sweet Life

Ingredients

• 1 cup unsalted butter (2 sticks or ½ pound), softened (but not melted!) Ideal texture should be like ice cream
• 3-4 cups confectioners (powdered) sugar, SIFTED
• ¼ teaspoon table salt
• 1 tablespoon vanilla extract
• up to 4 tablespoons milk or heavy cream

Directions
Beat butter for a few minutes with a mixer with the paddle attachment on medium speed. Add 3 cups of powdered sugar and turn your mixer on the lowest speed (so the sugar doesn’t blow everywhere) until the sugar has been incorporated with the butter. Increase mixer speed to medium and add vanilla extract, salt, and 2 tablespoons of milk/cream and beat for 3 minutes. If your frosting needs a more stiff consistency, add remaining sugar. If your frosting needs to be thinned out, add remaining milk 1 tablespoons at a time.

Makes 2.5 cups.  I doubled the recipe and used about 3/4 of the amount made.

The original try at a checkboard cake.  The frosting wasn't stiff enough.

Prepare the pans.  I like to use parchment paper on the bottom to help the cakes release easier.

Sifting the flour.

The solids split between two bowls, with cocoa powder added.

The butter creamed with the sugar.  I may have eaten a lot of this just to check that it was okay.

The coconut cream was solid so I added some coconut milk to it.

The coconut batter after whisking.  No lumps!

The chocolate batter after whisking.

Ready for baking.

Baked cakes.

Trying different bowls to see what size rings I wanted to cut.

The mason jar made the center ring.

The impressions.  I decided to do three rings instead of four.

Coconut cake cut up and ready for assembly.

While the cakes were baking, I made the frosting.

The creamed butter in the Cuisinart.  Use the dough blade instead of the regular blade.

If you need to move stuff around inside the Cuisinart while it's on, use a chopstick.

The finished frosting used 2 cups of butter and about 9 cups of powdered sugar.  It is quite stiff.  The dent in the left side is from my finger and didn't collapse.

This is the yellow gel color from the tip of a spoon.  I used about twice this amount.

It wasn't quite so neon in real life but it was bright.

I used the cake flour box to make a cardboard cake base.

You want the cake base to be slightly bigger than the cake, so I traced the outside edge of the cake pan.

The cake base was covered in parchment paper instead of tin foil since my roommate used the last of the tin foil but didn't tell me or buy some more.  The frosting will be used to glue the first ring down.

The first ring (coconut) with frosting applied to hold the second ring in place.

The second ring.  The chocolate cake has more crumbs than the coconut cake.

With the third ring in place.

Frosting to glue the bottom and top layers together.

The outer chocolate ring broke when I removed the middle ring.

I glued the outer ring in pieces.  Nobody could tell.

One of the pieces.

Assembled.

The second layer.  The bit of frosting on the right was to hold part of the outer ring in place.

After the crumb cake was applied and the cake sat in the fridge for an hour, I applied a final coat of frosting.  I've done the top but not the sides at this point.

The finished cake.

What each piece looked like.