Introduction: Apple-onion Gasket (pie)

For π day this year, I thought it would be fun to make a fractal design based on circles - an Apollonian Gasket. The flavor choice then seemed obvious - apple and onion, right?! My hope was to make a fractal that was both space filling AND stomach filling as well!  I had never made a savory pie before, but after browsing the interwebs for some ideas I was convinced that this filling had merit (especially with bacon...) and decided to give it a try.

I also like to support my Farmer's Market, so the apple, onion, cheese, and duck prosciutto were all purchased there from local farmers. 

To get started you will need the following ingredients:

Cheese Dough  : (enough for 2 pie dishes)
2 1/2 cups unbleached all-purpose flour
1 teaspoon salt
1 teaspoon sugar
1/2 cup fresh grated parmesan
8 oz (2 sticks) unsalted butter
8-10 tablespoons ice water

Filling :
3 slices thick cut bacon or thin-sliced prosciutto, chopped
1 yellow onion, medium, sliced
1 Granny Smith apple, medium, peeled and sliced
2 eggs
3/4 cup milk
1/8 teaspoon fresh grated nutmeg
1/2 teaspoon ground red pepper
1/2 teaspoon smoked paprika
Fresh cracked black pepper to taste
3/4 cup fresh grated cheese (either parmesan or Gruyere)
2 tablespoons olive oil (if using prosciutto)

Equipment / Tools
I also used a 9.5 inch pie dish, cutting board, cooling rack, rolling pin, cookie sheet, cheese grater, pie weights (or beans), and a large food processor for the pie.

To design the crust I used parchment paper, a compass, an accurate ruler (up to 1/16th inch), a sharp knife, and a spreadsheet for some quick number crunching.

Step 1: Making a Fractal

The crust design is an Apollonian Gasket. Wikipedia has a great article all about Apollonian Gaskets
http://en.wikipedia.org/wiki/Apollonian_gasket

The essential things to know are:
- An Apollonian Gasket is a space-filling fractal. In theory you could make many more circles to continue to fill the top of the pie, but in practice I found that 16 provided a nice design and kept circles at a reasonable size to work with.
- Curvature is how "sharp" a curve is, and is inversely proportional to the radius of curvature . A straight line would have an infinite radius of curvature, the edge of a large circle would curve slowly (low curvature, large radius), while a small circle would curve very quickly (high curvature, small radius). The designs shown on Wikipedia list the relative curvatures of the circles within the fractal (with the first, negative, number listed being the radius of the largest "frame circle"), so we need to do a little number-crunching to figure out the actual radii we want to use.

I decided to use the {-12, 25, 25, 28, 48} pattern that is shown on Wikipedia (I prefer the almost-D3 symmetry). To calculate the radii of the circles you will use, you need to take the radius of your pie dish (my 9.5 in dish has a 4.25 in radius), and multiply that radius by the first number in the pattern (in my case 12, the negative number) then divide by the curvature in question.

The pattern I chose to follow
http://en.wikipedia.org/wiki/File:ApollonianGasket-12_25_25_28-Labels.png

For example, to find the radius of the "25" circle in the pattern, I used my spreadsheet to multiply 4.25 in * 12 / 25 = 2.28 in, or approximately 2 + 4/16 inches. If you are using a different sized pie dish, you can take each number below, multiply it by the diameter of your pie dish, and divide by 9.5

The radii, in inches, for the sixteen circles I used are as follows
Curvature   Radius (decimal)    Radius (fraction)
25                2.280                         2  4/16
25                2.280                         2  4/16
28                2.036                         2  1/16
48                1.188                         1  3/16
57                1.000                         1  0/16
57                1.000                         1  0/16
97                0.588                         0  9/16
97                0.588                         0  9/16
112              0.509                         0  8/16
112              0.509                         0  8/16
121              0.471                         0  8/16
121              0.471                         0  8/16
168              0.339                         0  5/16
208              0.274                         0  4/16
232              0.246                         0  4/16
232              0.246                         0  4/16

Once you calculate the radii of the circles you intend to cut out, use your trusty compass to carefully measure off each radius on the ruler, then draw the circles on a large sheet of parchment paper. Do NOT cut them out yet. I also found it helpful to label each circle by its curvature, for reference later...

Step 2: Making Cheese Dough

You will need enough cheese dough to line the pie dish as well as cover the pie, so if you feel like buying pre-made dough, just get enough for two 9.5 inch dishes.

To make your own dough:

In a large bowl, mix the flour, sugar, salt, and cheese.

Dice the frozen butter into small cubes, then pulse in a food processor with the dry ingredients until just mixed. Alternatively, cut the butter into the flour mix by hand. The dough should still be powdery and there should be small balls of butter throughout. 

Add 6 tablespoons of ice water to the mix to moisten. If the dough still seems too dry, add 2-3 more tablespoons until you can form the dough into a thin disk.

Wrap the disk in plastic and refrigerate for 30 minutes.

Step 3: Shaping the Dough

Once your dough has chilled, separate it into two even portions. One for the dish, one for the design. Make sure to flour your cutting board for easy removal of the dough once it is rolled.

Heat your oven to 400 degrees Fahrenheit.

For the dish, roll the dough out until it is a large circle 1/4 inch thick, then press the dough into a 9.5 inch pie dish. Dock the dough with a fork.

For the crust design, roll the dough out to 1/4 inch thick, and lightly press the parchment paper with the circles on to the rolled dough. With the tip of a sharp knife, carefully cut out each circle, and leave the marked paper pressed on to each circle of dough for easy labeling and transport. Place the circles, parchment side down, on a large cookie sheet.

If you have left over dough, cut it into fun shapes and cook it along side the circles. Cheese dough is quite tasty on its own.

Freeze the dish and the circles for 10 minutes to set the butter. Then cover the dish with foil or parchment, and add pie weights or beans to help the dish hold its shape. Cover the circles with one large sheet of foil as well.

Bake both the dish and sheet of circles for 10 minutes. Remove the weights and bake both for 10 more minutes until the crust is lightly brown.

Let dough chill while you prepare the filling.

Step 4: Prepare the Filling

Prepare your filling in a heavy skillet over medium heat.

If you are using bacon, start by cooking the bacon first, until crisp. Then add the onions and apples and saute them with the bacon until they soften.

If you are using prosciutto, saute the onions and apples first in good olive oil (a couple of tablespoons), then add the prosciutto at the end for just a few minutes to add to the mix.

In a bowl, whisk the milk and eggs with the spices, while the pan filling cools a bit. A little spice goes a long way!

Grate another 3/4 cup cheese to go on the filling.

Step 5: Fill, Arrange, Bake!

Add the sauted filling to the pie dish first.

Next, top the filling with the grated cheese.

Next, pour the egg and milk mixture over everything.

Lastly, carefully arrange the dough circles according to the Apollonian Gasket pattern you are following. The circles should absolutely touch each other, and may still overlap the edge of the pie a little. 

Bake at 400F for 30-35 minutes or until the filling is golden.

Allow to cool for 10 minutes before serving. It tastes great warm or chilled from the refrigerator and keeps for a few days.

Enjoy!

Comments

author
una_amor (author)2011-11-27

OMG I love the ingredients in this gem. Going to make it for sure!

author
iminthebathroom (author)2011-11-27

Love flavours like this together, awesome

author
jessyratfink (author)2011-11-27

Sounds amazing and it's beautiful too! :)

author
phragments (author)2011-03-20

Perfectly delicious for brunch, rounded off with champagne.

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