Who cares about π?
Let me break some news to you that you may not like:
There is no inherent value in the order of digits, 3 1 4 1 5 9 2.. nor is there in the shape of the greek letter π.
That sequence of digits: “three one four one five… ” is really just a representation of the constant in our base 10 number system. In base 12, it would be written as 3.1848… , and in binary (base 2), it would be written as 11.001001… If humans had evolved with 8 fingers, perhaps we’d be calling it 3.11037… Needless to say, pi(e) in any other base would taste as sweet!
Pi is not just valuable for it’s status as symbol of math itself, but more so for the wonderful, useful properties it possesses and the way it always seems to pop up unexpectedly in every situation in the most mysterious way. Since ancient times, humans have understood how vital this constant is, even if they didn’t know the myriad ways it would be used in the future. Even the ancient Egyptians had a approximated value for pi, almost 4,000 years ago.
But let’s get to the main event.
Today I’ll be showing you how to make a Taro-Coconut custard pie with a sugar cookie tessellation inspired by a design from the talented M. C. Escher, my favorite artist, and someone who employed mathematics heavily to much success in his work.
Let’s get started.
Step 1: Design
For this pie, I wanted to try introducing some taro into a creamy custard base. Taro, a vegetable mostly unknown in the western world, is a classic in Asian cuisine. Although this thick tuber may appear intimidating, it can be cooked exactly like a potato, although I assure you it tastes completely unlike a potato. Taro has a nutty, warm vanilla-ey flavor that pairs well with coconut, which together form the backbone of the filling. I enhanced the flavor with several extracts, and enhanced the color with natural purple colorants made from reduced pea flower and hibiscus teas.
For the decoration on top, what better way to honor the dynamic duo of math and art than with a sugar cookie tessellation? This specific design is from Regular Division of the Plane No.99, “Flying Fish”, an ink and watercolor piece by M. C. Escher. To quickly cut out the units, I will make a custom cookie cutter, which is pretty easy to make with some patience and caution. To make the pattern pop, the entire surface will be “inked” with black royal frosting.
Throughout this recipe, π will infiltrate this Instructable, asserting its dominance as a number to be reckoned with. Be on the lookout for these interludes! You can choose to skip them if you just want the food, but those who stay will be rewarded with some interesting nuggets.
So, then, let’s begin!
Step 2: Ingredients
As a general overview, the recipe will be divided into five sections. I will give specific amounts in each section but give sources for those hard to find ingredients here. I will warn you now, this pie is very intensive and takes a few days and lots of effort, but anyways:
- All purpose flour
- 1/2 of a large Taro root - check any Chinese/Asian grocery store such as H-Mart
- Coconut Milk
- Dairy Milk
Cut out Sugar Cookies
- Baking Powder
- Powdered Sugar
- Vanilla Extract
- Coconut Extract
- Taro Flavoring - I bought this from Thailand on Ebay LOL:
Many south-east Asian cuisines use taro, hence why this flavoring is produced for the Thai market. Note! The seller usually sends a free baggie of pea flower tea with the order, perfect considering it's also needed in this recipe!
- Black food coloring - craft stores have this, or Sur La Table / Williams Sonoma (fancy cooking stores)
- Hibiscus Tea (the calyces of the roselle plant) - Wegmans or any specialty tea shop
- Pea Flower Tea (dried flowers) - Look on amazon/ebay
In short: if you have a pantry stocked with the basics, the only "weird" things you'll need to get are one can of coconut milk, coconut extract, black food coloring, hibiscus tea, pea flower tea, taro flavoring, and of course, a large taro root.
Step 3: Make the Pie Crust Dough
I will not re-invent the pie crust here, as this has been done many times before on the internet. Here I will summarize the process for making a pie crust:
The following recipe is from: https://www.livewellbakeoften.com/homemade-pie-cru...
- 1+1/4 c All purpose flour
- 1 tsp Sugar
- 1/2 tsp Salt
- 1/2 c (115 g) Butter
- 1-2 tsp. Water
First, whisk the dry ingredients together.
Then, dice up the butter into small cubes. Add to the dry mix and add a few teaspoons of water. Roughly mix with a fork.
Lastly, put the mix in batches through a food processor until large crumbs form. This could also be done by hand with a fork or pastry cutter.
Now, you can press the crumbs together into a ball of dough. This will make enough for one crust, which is sufficient for this pie.
Step 4: Roll and Bake the Pie Crust
Preheat the oven to 400F (205C)
Now, you can roll the dough out. Use plenty of flour and a rolling pin until the pie dish appears to just fit inside the sheet.
Place it in the tin and cut off the excess with a knife.
To get these nice crimps, I use this technique where I push the dough with my thumb into the space between my index and thumb of the other hand. See the picture above. This quickly and easily gives an almost machine-perfect edge once you master the technique!
Now, poke many holes into the crust with a fork before baking it in the oven for 18 minutes or until golden and cooked through.
Step 5: Interlude: Calculate the Filling Needed
How am I supposed to know how much filling to make for the pie? Look no further than g e o m e t r y. The inside cavity of a pie is nothing but a slice, or frustum, of a cone. Since I can easily measure the large and small diameters of the solid, as well as the height, I used similar triangles and other basic facts to derive that the volume needed equals πh(r²+rR+R²)/3 where r is the small radius, R is the larger radius, and h is the height.
For the curious few, feel free to look at the full derivation above. It’s a fun little puzzle to try out on your own!
Anyways, plugging in the measurements I took gives that I need 793 mL of filling.
For the filling, I decided to do a mix of 2 parts custard to 1 part taro paste, meaning I needed about 530mL custard and 260mL taro. Previously, I had determined that ⅕ of the custard recipe makes 110 mL of custard, and so 1 full recipe makes 550 mL, surprisingly close to the 530mL I needed. I decided to simply make one full recipe, since it would in fact be sufficient to exactly fill the pie.
Feel free to use a slide rule to do the calculations. :) This sleek little piece of technology is a calculator that works based on logarithms. We as humanity used them from the early 1600s up until electronic calculators replaced them in the 1970s. Even despite being in use for so long, they seem to be a forgotten part of our history. Though they do not give the same precision and accuracy, they're normally good within 0.2% of the exact answer, which is enough for most building purposes. I mean, we built the Brooklyn bridge and went to the moon with nothing but slide rules as calculators!
Step 6: Prep the TARO
Prepare the tuber just as you would any potato.
Peel the thick skin with a knife and dice into large 1 inch cubes.
Make sure to stop for a moment and take a look at the little flecks of purple running through the white flesh. This is where the signature purple color comes from.
Step 7: Cook and Grind the Taro
To prepare the taro paste, boil a large pot of water and simmer the taro cubes for around 15 minutes, until a fork enters the cubes very easily, with little resistance. It should be quite soft.
To make them into a paste, you could use a mortar and pestle, but I would be inclined to recommend investing in a food processor. With a few spoonfuls of water added into the taro as lubricant, the machine makes quick work of this step. Pulse for 30 seconds until a fine paste with no chunks is produced.
I chose to strain it as well, to remove clumps and chunks. Use a spoon and be patient, it takes time to press it all through.
Step 8: Prepare Custard
The custard recipe is a riff on Ray Pajar's Thai Tea Pie (which I HIGHLY recommend you try as well, by the way):
- 2.5 c (600g) coconut milk*
- 5 egg yolks (KEEP the egg whites for later!!)
- 1/2 t salt
- 3 T (23g) flour
- 1/2 c (100g) sugar
- 2 T (28g) butter
*I used one can of coconut milk, but it wasn't quite enough, so I topped it with a little more than 1/2 c of dairy milk until I reached the required 2.5 c of liquid.
To prepare the custard, first whisk together the egg yolks, salt, flour, and sugar thoroughly until no lumps remain.
Heat the milk in a saucepan until just steaming, or almost but not yet boiling. Now, temper the egg mix by slowly adding ladles of the hot milk into the eggs while whisking vigorously. This helps avoid instantly cooking the egg.
Once you've added all the milk, pour the mix back into the pot, through a strainer. (please do use a strainer!! I found many nasty egg bits in the strainer afterwards!)
Step 9: Cook Filling
Into the pot, scoop in 260 mL of taro paste. This happens to be almost exactly 1 cup.
Whisk vigorously while on medium-low heat to try to dissolve any clumps of taro.
Now, cook on medium heat until large gloopy bubbles and steam begin to appear. Let this bubble and thicken for 2-3 minutes. Ideally, it will reach a point where when picked up with a wooden spoon, the mixture slowly drops into the pot leaving a jiggly triangle of the paste on the spoon. It's hard to explain, but you will know when it's there when it appears very thick.
Add in the butter and stir until melted through.
Once done, strain once again to remove lumps and bumps.
The filling won't look appetizing yet, but that's where the next step comes in!
Step 10: Prepare Natural Food Coloring Teas
On it's own this custard looks pretty gray and unexciting.. hence where these enhancements come in!
As far as coloring goes, Taro desserts are traditionally dyed with food coloring to make the purple more apparent. It's just tradition! However, I wanted to try making my own naturally derived colorants from two highly pigmented herbs.
If there's one thing you remember from elementary school art classes, it's probably the color wheel. To make purple, we'll need red and blue coloring.
To make the red, mix 1 tablespoon of dried hibiscus herbal tea with 1/2 cup (120mL) of boiling water. Let sit for 8 minutes, then strain.
To make the blue, mix 1 tablespoon of dried pea flower herbal tea with 1/2 (120mL) cup of boiling water. Let sit for 8 minutes, then strain.
Now, to avoid watering down the filling, we need to reduce the extracts. To do this, gently heat the teas in a wide pot on low heat for around 10 minutes. You should see vapor rise but the tea should not boil or bubble. I was able to reduce each one down to a quarter of the original volume. Not bad!
Since neither the hibiscus or pea flower teas have a significant flavor, they should be undetectable in the finished pie. Now the natural food colorings are ready.
Step 11: Finish and Pour the Filling
To the filling, add a good 1 1/2 tsp (8mL) of coconut extract, as well as 4mL (almost 1 tsp) of taro extract. This taro flavoring is powerful stuff, and I measured the amount in a small syringe for dosing medicine in order to get the precision needed. As a rule of thumb, 0.4mL of the taro flavoring is needed per 100mL of liquid of final product for a good medium strength flavor.
Also pour in the coloring. They truly are pigmented!
Confession Time: I originally made half of the recipes for coloring I gave in the last slide, which proved to be not enough :( I was forced to resort to adding 2-3 drops of red and blue food coloring, but the hibiscus and pea flower teas did do most of the work. It's obvious that If I had made enough, I would not have needed this additional boost of color. So make plenty, just in case!
Mix, and pour into the pie shell. As you can see, the calculations were good and the filling fit perfectly inside the crust. Math works! Who would've thunk it?
Put the pie in the refrigerator. It may appear liquidy now, but with a few hours of cooling, the filling will set to a smooth, gelled, more cuttable consistency.
Step 12: Prepare the Sugar Cookie Dough
The topping will be made of frosted cut out sugar cookies.
This recipe is a standard sugar cookie dough like any other. The quantities from this site make excellent cookies:
Cut Out Sugar Cookies
- 3/4 c (85g) Room temp. Butter
- 3/4 c (150g) sugar
- 1 egg
- 1 1/2 t Vanilla extract
- 1/2 t baking powder
- 1/2 t salt
- 2 c (250g) flour
First, cream together the softened butter and sugar with beaters. Then, beat in the egg and vanilla.
Whisk together the dry ingredients in a separate bowl and add them in slowly. Mix with a wooden spoon when it gets too thick for the hand mixer.
Lastly, form the dough into a ball, wrap, and refrigerate. The dough tends to keep its shape and is easier to roll when the butter in it is solid from being kept cold for at least an hour.
Step 13: Interlude: How Does a Tessellation Work?
Looking closely at Escher’s “Flying Fish” reveals an underlying pattern that is highly precise and calculated in nature. The individual fish appear in groups of six, with rotational symmetry when rotated π/3 radians, together forming hexagons, which then tessellate regularly in three directions. The sketch attached shows the method used to create the units in this particular piece. If you wanted to, you could use this same method to create your own units to tessellate.
By the definition of a tessellation, this pattern of fish has no gaps or overlaps - the entire thing is just a tiling one shape. At the end of the day, all this means is that to create this seemingly complicated pattern, I will only need to make one single cookie cutter!
Step 14: Prepare the Metal for the Cookie Cutter
I wanted to fit four fish across the the width of pie, so I measured the diameter of the pie and divided by four to get the side length of each equilateral triangle unit. I then sketched the design I wanted on a piece of paper, following the rules in the previous step, exactly to scale (1:1). This will serve as a template later on so be precise.
Making custom cookie cutters is not hard but you must be careful. When dealing with sharp edges and knives, maintain complete control and work slowly. As long as you handle the metal carefully, it's perfectly safe.
To begin, cut the ends off of a soda can, then slice it vertically to unravel a sheet. It will be rough, so go back in with scissors and trim the edges clean.
The sheet will probably curl quite a bit, but you can run it over a table edge in the opposite direction of the curl to try to flatten it out.
To make the cookie cutter, cut strips 3 cm wide, then fold them in half to form a piece 1.5 cm wide. Use a chopstick or fork to define the creased edge. By making two layers, not only will the cookie cutter be more sturdy, but the folded edge cuts much more cleanly than the loose cut ends. You'll see.
Step 15: Shape and Join the Cookie Cutter
Now, the metal must be folded and shaped to match the template. Before making each fold permanently, which is hard to do, make a small dent in the metal around where you think it should be and compare with the template. If it matches, go ahead and make the crease.
You'll notice the pattern has many curved sections. In order to make them, hold the section you want to curve with one hand and with the thumb of the other hand, push into the metal with your nail while dragging it along the metal several times until a curve is established.
If your piece of metal is not long enough, join another one by cutting a 45 degree notch into it and taping on with scotch tape. See the pictures above. I tried hot glue and it simply did not hold, so tape is your best bet.
Work your way around the shape until you can finally use the same joining method to close up the shape. You can continue to push around, mold, and redefine the shape now. Most important is that the three tips of the triangle line up with the template, or the cookies will not tessellate well.
Step 16: Roll, Cut, and Bake the Cookies
Take half the chilled dough out and roll to about 1/4 inch (6mm) thick. Brush with plenty of flour to help the cutter not stick.
To cut the cookies, dip the cookie cutter in flour, push the folded edge into the dough all the way down and wiggle to separate from the rest of the dough. Push the cookie out of the cookie cutter gently with your fingers.
The reason we use the folded edge is because the other edge is made of two layers, and dough tends to get stuck in between, leaving a not very clean edge.
Many of you will be wondering: "doesn't that mean you're pushing a sharp edge into your hand?" Well, if you trimmed the sheet as I said, the edge will not be jagged, and, since you push down with your entire hand, the force is distributed over the entire cutter, so no, you will not cut yourself. I cut out more than 50 cookies and my hand wasn't even scratched!
Place the cookies on a baking sheet with parchment paper or a silicone mat or aluminum foil, and bake in a preheated oven at 350F (177C) for exactly 8 minutes. They should be cooked through but not yet turning golden brown. This ensures they don't dry out.
Make plenty. Although I theoretically only needed around 25, I made plenty more as assurance in case I got a few duds.
And let's be honest, also to account for the cookies I would inevitably eat along the way...
Step 17: Mix and Bag the Royal Icing
To make the Royal Icing, you will need:
- 13 oz (360g) of powdered sugar
- 2 egg whites (left over from the custard!)
Mix them with a fork and when you get tired beat them smooth and luscious with the hand mixer.
To make the frosting black to simulate ink, I beat in some black food coloring, around 1-2 tsp. If I were to do it again, I would look for a way of making natural black food coloring. The ingredients on the bottle actually say it's made of yellow, red, and blue food coloring mixed together, and since I already know how to make red and blue, I would only need to find out how to make yellow.
Additionally, the black added a slightly bitter taste, which was virtually unnoticeable unless you ate the frosting on it's own. A naturally derived coloring would mitigate this problem. I encourage the reader to try this out on their own!
Moving on, set up a piping bag with a very small round tip. You could use a regular plastic bag with no tip, but you must put packing tape around the end you will cut to reinforce it and prevent the seams from exploding. Make a very tiny cut, smaller than you think.
Put the bag in a glass to make it easy to fill with a spatula.
Test that the frosting is be able to be piped out and hold its shape, as seen above.
Step 18: Pipe the Design on the Cookies
I know, this design isn't exactly the one seen in Escher's piece, but sometimes, you have to pivot. The design Escher used is very detailed, and when i tried it, the lines were so close together that it looked cluttered, so I made up this new design that looked much better on these cookies.
When you're making something, don't be afraid to modify to better suit your situation and resources.
This design I made in four steps, as seen above.
Piping TIPS: (haha very funny)
- When piping, push slightly down toward the cookie as you squeeze, so that the frosting adheres and doesn't just float on the surface
- When you reach an edge, stop squeezing the bag a few millimeters from the edge, and pull through. If you keep squeezing the bag even all the way through the end, there will be too much frosting deposited and it will look messy
- Make more than you need to save yourself agony later on
After about 10 minutes, the frosting hardens slightly and is no longer tacky, making them easier to handle.
Step 19: Attach Topping to Pie
First, set out the pattern on a paper towel. Switch out cookies until they fit the best you can get them.
While Escher's pattern has fish of two colors, I thought it would be too cluttered to then frost the purple surface of the cake too, so I decided to only add the cookies, so that the purple underneath would naturally form the shape of the fish.
To stick the cookies on, make a simple syrup by microwaving or simmering 2T (25g) of sugar and 2T (60mL) of water for 1 minute.
Brush the syrup over the pie and place each cookie down own the surface. The "glue" doesn't dry for a while, so you can still adjust throughout the process.
For the edges, use a knife to cut the cookies to fit. You can use tweezers to put the small pieces in accurately. If the frosting falls off, simply pick out the frosting that has fallen and re frost the cookie while its on the pie.
Work your way around until it's done. After about 30 minutes, once the syrup has mostly evaporated, the cookies should be more or less affixed in place and won't move if you shake the pie.
The syrup has an added benefit of keeping the top surface moist.
Step 20: Finished!
Behold! The pie is done!
I think visually, it's a decent rendition of Escher's art. The spaces between the fish do in fact form the shape of the exact same fish, meaning it's a true tessellation.
As far as taste, it's not bad, the sweetness and taro flavor is just right. And it turns out, cookies are actually a really great topping for a pie! While taro may be an acquired taste, I think those who have had exposure to it and enjoy it will enjoy this pie. It reminds me of taro bubble teas I've had before.
I think this pie would be best served with a helping of fresh sliced strawberries.
Step 21: Interlude: Calculate Pi Experimentally With a Pie - Part 1: Theory
We're not done yet!!
This project gave me a chance to try out an experiment I’ve always wanted to try. On his Youtube channel, the fantastically clever Matt Parker swung a pie on a pendulum to calculate pi experimentally. I think it would be appropriate to give it a shot too, in the spirit of pi(e)!
Essentially, a swinging pendulum experiences an acceleration which is proportional to the position of the mass, x. Assuming we only swing the pendulum at small angles (<10 degrees), where sin(x) is very close to x, we get a differential equation of the form x’’ = -kx, whose solution is that the function that models x is a sinusoid, or sines and cosines. Skipping several steps, it can be shown that the period of a pendulum (T) equals 2π√(l/g), where l is the length to the center of mass, and g is the acceleration due to gravity on the earth’s surface, 9.80 m/s².
If you want, a slightly more detailed but still abridged explanation is attached. All of my engineers and physicists out there should enjoy it ;)
Too Long Didn’t Read? All you need to know is that rearranging the formula for the period of a pendulum shows that π=T/2√(g/l), where T is the period of oscillation, and l is the length of the string, both of which I will measure. Lastly, g is a constant 9.80.
Step 22: Interlude: Calculate Pi Experimentally With a Pie - Part 2: Results!
To make the pendulum, I clamped a board to a tall cabinet, and tied a long string on it. I tied a quick little net to place the pie inside, then hung the pie in it. I taped a straight line on a piece of paper to wall as a reference for position zero.
In order to measure the period of oscillation, I decided to film the pie swinging ten times, with a timer running in the background. Then, the time for 10 periods was simply the difference between the end time and the start time.
I could very accurately get these times down to the hundredths of a second by individually looking for the frame in the video where the string aligns with the line I marked on the cabinet in a video editing software afterwards.
For my first trial, with ten oscillations, the start time was 4.81, and the end time was 33.25, giving a difference of 28.44, so T for one swing was 2.844 sec.
I measured the string from the top to the estimated center of mass 2 cm down into the pie, with a precision of millimeters. Funny enough, it came out to exactly l = 2.001 meters! I swear I didn’t plan this, but anyways:
Plugging these values into π = T/2√(g/l) gives that pi is…. 3.148!
While it may seem off, it’s actually only 0.2% off of the exact value of pi, which is incredibly good given I literally just swung a pie from a cabinet with some yarn I had. I’m satisfied :)
Step 23: Conclusion
Now, it is time for your taste buds to enjoy the fruits of your labor.
Go ahead and serve 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 +... of a pie, which believe it or not, as proven in 1734 by Leonhard Euler, everyone’s favorite mathematical rockstar, happens to be equal to exactly π²/6.
Judges Prize in the