Fibonacci cake, a golden ratio Battenberg. Challenge accepted!

While honing my Battenberg neatening skills recently, my partner declared that I *must* turn the Battenberg into a golden ratio cake. Indeed, a Battenberg isn't quite fiddly enough ;)

The Fibonacci sequence is where each number is the sum of the previous two numbers, that is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on. Examples of the golden spiral are found in nature and can be incredibly beautiful.

To quote a particularly alliterative Melvyn Bragg from his In Our Time programme on the topic
"The Fibonacci sequence is found to appear time and time again among the structures of the natural world, and even in the products of human culture. From the parthenon to pine cones, from the petals on a sunflower to the paintings of Leonardo da Vinci, the sequence appears to be written into the world around us".

Tasty maths!

There are quite a few cakes online that depict the famous Fibonacci sequence, and it’s quite interesting to see how different people have approached it. There is also this cute Mondrian cake which looks a little like a Battenberg variation, and this pretty amazing Rubik's Battenberg but I’ve yet to find a Fibonacci Battenberg exactly. Hmm would it be a Battenacci, or a Fibonberg..?

I decided I wanted it to be recognisable first and foremost as a Battenberg, and that part of the fun would be revealing the inherent mathematical layer of beauty of the thing through the decoration. 5cm square cross-section is about as big as my cake tins could take me, and it works out nicely as a slightly diminutive slice of Battenberg, where 8cm would start to become a bit of a monster cake. With an inability to work with infinitely smaller planks of cake that the making of a true golden spiral might require, I decided to start with 1cm square units and work up from there.

All right then, are we ready?

Fibonacci Battenberg recipe - yields 2 cakes

Equipment you’ll need:

• A cake cutter, and/or long knife
• A clean ruler or tape measure
• Cocktail sticks/skewers to mark the cake
• 1 or 2 20cm square cake tins that are at least 5cm deep
• Rolling pin
• Pastry brush
• Baking parchment and plastic wrap
• About 3 hours..
• A maths nerd to present the cakes to

Trusty equipment and colouring. I used pink gel, but a little red also works

Ingredients for the taller yellow sponge:

• 350g very soft butter
• 350g caster sugar
• 330g flour
• 20g/5tsp baking powder
• 0.5tsp salt
• 6 eggs
• 0.5 tsp vanilla extract

Ingredients for the pink sponge:

• 175g very soft butter
• 175g caster sugar
• 165g flour
• 10g/2.5tsp baking powder
• 0.25 tsp salt
• 3 eggs
• 0.5 tsp vanilla extract
• 0.5 tsp almond extract
• A little red or pink food colouring

Additional ingredients:

• About 900g marzipan (Make your own marzipan, or see here for tips on how to get ready-made marzipan in Japan)
• About 280g apricot jam

The yellow sponge will come right up to the top of the tin when baked

Method:

1. Grease and line the cake tin(s) with baking parchment. Preheat the oven to 160°C.
2. Make the yellow sponge. - Sift the yellow sponge's amount of flour, baking powder and salt into a very large bowl and mix well.
3. Add the soft butter, sugar, vanilla essence and eggs all into the same bowl and whisk together until just combined. (If you're using a stand mixer however, it's best to sift the dry ingredients on top of the wet ingredients to avoid a layer of flour at the bottom of the mix. Mix on the lowest speed.)
4. Scrape the mixture into a prepared tin, spreading to the corners and bake for about 1 hour, reducing temperature part way if it browns quickly.
5. Repeat steps 2-4 for with the pink sponge ingredients and bake for 25-30 minutes. (If you're reusing the same tin, cool it down with some cold water before using it again.)
6. While the sponges are baking, make the marzipan if you're making your own. 
7. With the two squares of sponge out of their tins and completely cool, cut off the browned tops of the cake and any sloping edge pieces. You want your yellow cake to be 5cm high, and the pink cake to be 3cm high. (Tip – if this is tricky, then chill or partially freeze the sponge and it will be easier to cut straight lines. Also, save the cuttings to crumble up with frosting to make cake pops.)
8. Use a toothpick to make indents at the top and bottom edge of the sponge to mark out the following long rectangular blocks and cut them carefully from the main block. 
• From your yellow sponge: cut two 5cm-wide strips, creating planks of sponge that are 5cm square in cross-section. Also cut 2, 2cm square planks and 2, 1cm square delicate strips. When maneuvering the delicate strips, guard against breakage by rolling them sideways onto something like a long blade rather than picking them up by their ends.
• From your pink sponge: cut 2, 3cm square planks and 2, 1cm square delicate strips. 

Being a bit punctilious helps ;)

The order I used to stack the cake (you'll be doing it with jam)

Assemble the cake:

1. Roll out the marzipan to at least 25cm x 30cm. If it’s sticky, roll it between two sheets of plastic wrap, peeling the top layer off when you’re ready to wrap the cake. 
2. Heat the jam and paint sides of all the sponge strips, glueing them together with the jam as you stack them into a golden ratio rectangle (see the order I used above, but use what works for you). It will be slightly less sticky if you avoid painting the outside faces of the planks until you’re ready to wrap the thing, but it’s not a terrible problem if you paint a side too early as they will all need jamming up in the end. Working on baking parchment will also make it easier to move it to the marzipan later.
3. Position the whole sticky block of planks onto the center of the marzipan, upside down - you want the marzipan seam to end up on the bottom.
4. Paint the top with lots of jam and fold up the sides of the marzipan up to make a join on the top. Trim any large excess and smooth over the join.

Roll out about a 25cm x 30cm sheet of marzipan

Ooh yeah! We're jammin', jammin' (Sorry ;) )

Use the plastic wrap to help if your marzipan is at all thin or sticky

1. To finish, turn the cake the right way up and give the marzipan a final squeeze to make sure it’s all nicely stuck to the jam. With a sharp knife, cut off a thin slice from each end of the log to reveal a neat presentation of the cross-section form each end. 
2. Decorate the marzipan if you wish, by making indentations with a fork, pinching the edges to make a ruffle and/or demarcating portion sizes on top. Sprinkle with icing sugar if you like.
3. Present slices of your masterpiece with mathematically almost-accurate golden ratio swirls of melted chocolate, jam, whipped cream, a rope of marzipan or something else that takes your fancy. Or even better, provide the tools to let your guest create the beautiful spiral themselves.

Otsukaresama deshita :)

Battenberg recipes referenced
http://www.bbcgoodfood.com/recipes/1120657/battenberg-cake
http://www.bbc.co.uk/food/recipes/battenburg_cake_60878

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Options:

- I chose to use an all-in-one mixing method rather than creaming the butter and sugar together first, as it should result in a flatter top and it’s quicker so will save a little time in this fiddly project.

- Another time saver would be to mix up all 9-eggs worth of batter in a large stand mixer and estimate 2 thirds of it for the yellow sponge before colouring the remainder.

- A final option would be to make just one fibonacci cake by making up just one 6-egg batch of sponge, and dividing your baking tin in two with a folded piece of parchment (see here for how) and being very careful when you’re pouring in the batter. Pour half in one side of the tin, then colour the remaining half and pour it in the other half. You'd just need one of each 'plank' size in that case. This is a bit fiddly, and since Battenberg freezes well I decided to make it a bit easier on myself and just made two.

A mathematically inelegant remainder...