This dude knows a thing or two about baked goods.

For my money, the best part of the baking process (aside from the delicious final act) is the careful and precise initial measurement of the ingredients.  Keeping an accurate account of the relative proportions of each piece of the recipe is a hallmark of baking, and reflects the nature of baking itself: one part art, one part science.  Unlike some other culinary arts, the measurements really do matter.  Screw up these proportions and those fudgy brownies you want to make will be too cakey (or vice versa).

But what in the recipe accounts for the qualitative differences we see and taste in the wide assortment of baked goods available at this time of year?  This question has been mulled over by electrical engineer and baking aficionado Michael Ohene, who in this article essentially considers all baked goods as a function of three parameters: Moistness Value, Butter Content, and Egg Content.

How are these values computed?  First, each wet and dry ingredient is assigned a value per cup – these values are used in the calculations to follow.  For examples of wet ingredients, buttermilk has a value of 1 per cup, applesauce has a value of .6 per cup, and so on.  With dry ingredients, things like flour and almond paste have a value of 1 per cup, while peanut butter has a value lightly less (2/3 per cup).  More detailed tables of values can be found at the link given above.  Note that flavorings, leavenings, seasonings, and food pieces (such as whole walnuts) are not included in the calculations.  You can think of these values as a weighing the ingredients according to their relative wetness or dryness.

Once the value of each ingredient is known, that value is multiplied by the volume of the ingredient appearing in the recipe, and these products are totaled up for both wet and dry ingredients – let me call these final values the wetness value and the dryness value, respectively.  Let’s see how this works in practice by analyzing my wife’s spectacular recipe for Pfeffernusse cookies (they are delicious bites of gingerbready goodness).  The ingredients are as follows:

3/4 cup molasses
1/2 cup butter
2 large eggs
4 1/4 cup all purpose flour
1 1/4 teaspoon baking soda
1/2 cup white sugar
1 1/2 teaspoon cinnamon
1/2 teaspoon clove
1/2 teaspoon nutmeg
dash of pepper
confectioners sugar

(In case you’re curious, here’s the recipe, cribbed from my wife’s 8th grade German class:

In a saucepan, combine molasses and butter. Stir until butter melts. Cool to room temperature. Mix in the eggs. In another bowl combine flour, baking soda, sugar, cinnamon, cloves, nutmeg, and a big dash of pepper. Add the flour mixture to the molasses mixture and mix well. Chill for 1-2 hours or overnight. Shape into 1”inch balls. (These can be frozen after this point and baked later if desired) Bake on greased cookie sheet at 350 degrees for 8-9 minutes, they should still be somewhat soft as they will continue to cook after they come out (underdone is better than overdone with these). Cool a bit and role in confectioners sugar. Makes 4 1/2 dozen cookies.)

Like a child who has eaten too much candy, these pfeffernusse cookies are the illest.

According to Mr. Ohene, there are three wet ingredients (butter, large eggs, and molasses) and one dry ingredient (flour).  The rest of the ingredients are not considered in the computations.  Butter has a corresponding value of .5, large eggs have a value of 1/6 per egg, and molasses has a value of 1.  Therefore, the wetness value is .5 times the volume of butter, plus 1/6 times the number of eggs, plus 1 times the volume of molasses, or

.5 x .5 + 1/6 x 2 + 1 x .75 = 1.33.

Similarly, flour has a corresponding value of 1, so the dryness value is simply 4.25, as the recipe calls for 4 1/4 cups of flour.  So the wetness value is 1.33, and the dryness value is 4.25.

The Moistness Value is the ratio of these two numbers, i.e. the ratio of wet to dry.  Meanwhile, the Butter Content is the ratio of the value coming from the butter to the dryness value, and the Egg Content is the ratio of the number of eggs to the dryness value.  In the example of Pfeffernusse, we see that the Moistness Value is 1.33/4.25, or about .3129.  Similarly, the Butter Content is .25/4.25, and the Egg Content is 2/4.25, or .0588 and .4706, respectively.

Varying these three values distinguishes different baked goods from one another.  Want a baked good with relatively high Moistness Value and Butter Content, but relatively low Egg Content?  Perhaps a coffee cake is what you’re after.  How about a moderately wet dough with high Egg Content and low Butter Content?  In this case, maybe a brioche is what you’re after.  A periodic table of sorts that plots baked goods according to these three values can be found here; note that the Pfeffernusse discussed above fall into a square marked “biscotti.”  It’s quite possible that the Pfeffernusse dough would make for good biscotti, though the process of baking is quite different for these two treats, since biscotti are typically twice baked.

You can do other cool things with this analysis.  For example, you can compute these values for a list of ingredients and make an educated guess about what type of baked good the ingredients will produce.  Also, one can automatically create recipes for baked goods based solely on the qualities one would like that baked good to have.  This service is provided here – you entire the baked good you want, the desired level of richness, and the desired level of sweetness, and it creates a recipe for you on the spot!  So if you can’t find just the right recipe for what you want, I’d encourage you to try this recipe creator on for size and see what comes out.

Whatever you or your family decide to bake this holiday, know that adding a little mathematics to the mix never hurt.  And, as evidenced by the work of Mr. Ohene, added analysis can provide some interesting and unexpected insights. Thanks go to him for sending me the links that I’ve shared with you here!

Needless to say, Cameron’s version of the story is much more embellished. In his rendition, their experiment was a success; as he puts it, the pumpkin flew across the field, “goal post to goal post.”

When I first heard him say this, my initial thought was “Is Cameron telling the truth?”  How likely is it that a pumpkin, launched from a slingshot at one end of a football field, could sail through the air to land on the other side?  Note that his story actually ends with the pumpkin falling through the sun roof of someone’s car – this outcome is, of course, highly implausible, and therefore casts a shadow of doubt upon the entire story.  While the sunroof claim would be difficult to verify, one can at least use some basic math and physics to test the plausibility of the first portion of the story.

We would like to be able to find a formula for the distance Cameron’s pumpkin should travel.  Key to our analysis is the conservation of energy principle, and Hooke’s Law.  In particular, we will assume that the slingshot behaves like a spring, i.e. the potential energy it stores is proportional to the square of the displacement (for example, stretching the slingshot two meters gives you four times the potential energy as stretching it one meter).  So, if Cameron and his friends stretched the slingshot a distance of x meters, the slingshot would then store a potential energy of , where k is the spring constant and depends on the physical properties of the slingshot.

Now let us invoke the conservation of energy principle: when the slingshot is released, suppose all that stored potential energy will be converted into the kinetic energy of the pumpkin.  The formula for the kinetic energy of an object is , where m is the object’s mass, and v is the object’s velocity.  So, when the pumpkin is released, by conservation of energy, this kinetic energy should be the same as the potential energy the system had before we chunked the pumpkin.  In other words,

,

which, with a bit of algebra, tells us that the initial velocity of the pumpkin when it comes out of the slingshot must be .

These poor little guys have no idea what's in store for them.

So we have an equation for the velocity of the pumpkin in terms of the spring constant k, the distance x we pull the slingshot, and the mass m of the pumpkin.  Let’s not forget our main goal, though – what we’re ultimately interested in isn’t a formula for the initial velocity of the pumpkin, but rather the distance the pumpkin travels.  With a little more physics, it’s not hard to get from one of these pieces of information to the other.

More specifically, we have a very good understanding of projectile motion.  As I’ve discussed before, if one ignores air resistance (as we will do for now), all the equations of projectile motion can be derived from the fact that the acceleration due to gravity is a constant, meters per second per second.

Using this fact, suppose you throw an object with an initial velocity at an angle to the horizontal.  Then if one sets the initial position of the object to have coordinates (0,0), the x-coordinate of the object at any time t will be given by , and the y-coordinate of the object at any time t will be given by .  Using these formulas and a bit more algebra, one can determine that the distance d an object will travel in the x direction can be written in terms of the initial velocity, the angle , and g, via the formula

Example trajectory of an object shot with an initial velocity v at an angle A. The distance the object travels in the x direction is given by the above formula.

To recap: we have the distance as a function of the initial velocity, the angle the pumpkin is shot from, and the acceleration due to gravity.  We also have the initial velocity in terms of the distance the slingshot is stretched, the mass of the pumpkin, and the spring constant.  Combining these two formulas, we find that

and we now have the desired formula for the distance the pumpkin travels.

Let’s check Cameron’s story against this formula.  If the pumpkin really went from goal post to goal post, the distance it traveled in the x direction must have been at least 120 yards (100 yards for the field of play, plus 10 yards for each end zone).  This is roughly 109.7 meters.  Therefore, we must have .

On the right side of the equation, the term is maximized when equals 45°, and in this case .  So after making this simplification, we see that in order for Cameron’s story to be true, it must be the case that

The acceleration due to gravity is known, but we need to provide estimates for m, k, and x.  The mass m is the easiest to estimate; let’s say the pumpkin is around 10 pounds (roughly 4.5 kilograms).  To be generous, we’ll even round down to 4 kilograms.  What about the distance the slingshot is stretched?  Based on the slingshot used in the episode, it’s unlikely the slingshot in Cameron’s story would have been stretched more than 2 meters or so, but let’s again be generous and say it’s stretched 3 meters.  Plugging in these values, we would have:

where the units on k are Newtons per meter.  Since one pound is approximately 4.45 newtons, this is saying that the spring constant is about 107 pounds per meter – in other words, for each meter you stretch the slingshot, you need to exert 107 pounds of force.  To put it another way, to stretch the slingshot 3 meters or more, you’d need to exert 321 pounds of force.

This seems like quite a lot, though Cameron is a large fellow.  But recall we were being generous, both in our computation of the spring constant and in our estimate of the pumpkin size.  We also neglected air resistance in this model, but air resistance probably has a non-negligible impact here – not only will it slow down the pumpkin’s forward motion, but it also decreases the optimal angle from 45°.  So in a real world situation, I’d have to remain skeptical of Cameron’s story.  On the other hand, these calculations don’t necessarily rule it out entirely (though a more sophisticated analysis might).

For more on the physics of pumpkin chunkin’, here‘s an article from last year courtesy of Wired.  For related trajectory issues, Angry Birds also provides plenty of fodder.

If you are a masochist like me, though, there are plenty of ridiculous articles floating around today to help you get your blood boiling.  One of my favorites comes from today‘s Philadelphia Inquirer.  It’s full of gems like:

• One may be the loneliest number, [La Salle University math teacher Stephen] Andrilli said, but 11 ranks among the most odd – and not just because it isn’t even. He sees 11 as sort of a netherworld number – one more than the familiar 10, one less than an even dozen. Uhh, what?  What does any of this even mean?
• In geometry, Andrilli said, an 11-sided polygon is called an “undecagon” – the shape of the one-dollar coin in Canada, whose people have a particular affinity for 11. Little known-fact: the Canadian people’s love for the number eleven has been the subject of intense research among anthropologists for centuries.  Oh wait – no, of course it hasn’t.  Because that would be stupid.
• Weirdly, the sum of 1,111 multiplied by 1,111 is 1,234,321, another numerical palindrome. Do editors no longer know the difference between the words “sum” and “product”?  As written, this sentence also makes no sense.

To its credit, the article does state that all of the listed coincidences involving 11 only show “the human propensity for seeing patterns where none exist.”
Not all articles may take the day with a healthy grain of salt, though, so make sure not to get caught up in all the 11/11/11 hubbub.  It will be difficult, I know.

Without further ado, let’s begin!

1. Tony Stark

Yes, yes, I know – since Iron Man hit the screens in the summer of 2008, the titular character has become a popular costume idea, joining the ranks of comic book icons like Superman and Spiderman.  I’m not talking about dressing up as Iron Man, though.  Instead, I am recommending a costume based on the man inside the suit – Tony Stark, playboy billionaire and (more importantly) mathematical wünderkind.  All you really need is some delicately coiffed facial hair and a glowing circle on your chest.  Aside from that, the world is really your oyster.  You could go as classy Tony stark, for example:Or, if you’re looking for a more rugged look, you could try prisoner of war Tony Stark:

As with many things in life, the only limit is really your imagination.

2. Rubik’s Cube

I know, this bears a striking similarity to Rubik’s cube head from one of my earlier posts – but this one is even better, because not only does it drape over your body, it also explicitly states you are a Rubik’s cube, for those guests at the event you attend who have been living in a cave for the past thirty years.

RetroCRUSH has the costume on a list of the worst Halloween costumes of all time, an honor with which I must respectfully disagree.  I will concede, though, that if you are looking to score some ladies (or fellas), this may not be your best option.

While we’re on the subject, though, I’d like to send a quick shout out to Parks and Recreation for featuring Rubik’s Cube Head as a costume in their recent Halloween episode.  Here’s a picture of that fantastic party attendee standing next to Rob Lowe, who is sporting a less exciting Sherlock Holmes costume.

Rob Lowe next to Rubik's Cube Head

3. Human Calculator

This one requires some work, but the payoff may be worth it.  First, one must decide what type of calculator to become.  Then one must decide on the size – should it be a full body costume, or centered only on the torso, for example?  No matter what path you choose, however, the most important thing is getting the details right.  Nobody likes a costume made by an inferior craftsman (or craftswoman, for that matter).

Here is an example of a calculator costume gone right.  Note the pride this individual takes in his work.  No doubt he secured many digits on Halloween.

Click to go to the source link.

No matter what you ultimately decide, I hope your Halloween is a good one.  See you next year!

I’ll be back next week with something more substantive.  In the meantime, enjoy your weekend, and if you’re in Los Angeles, Happy Carmageddon!

This year, I thought it would be nice to talk with Dr. Hartl in more detail about his inspirations for Tau Day, and where he envisions it in the future.  He was gracious enough to agree to a brief interview, which I humbly submit to you here.

*

Q: When did you first discover that was “wrong”?  Did you have an intuition that something was amiss before reading Bob Palais’s 2001 article in The Mathematical Intelligencer?

A: I don’t remember how deep my suspicions about ran before I encountered that article, but “ Is Wrong!” definitely opened my eyes, and every section of The Tau Manifesto owes it a debt of gratitude.

A: Support has been overwhelmingly positive. I monitor Twitter mentions of Tau Day, and nearly every commenter has something nice to say.

A: As a social hack of geek culture, the project has already exceeded my expectations. At technical conferences, people often recognize me as “that tau guy”. That said, the problem with pi is real, and I do believe that adding tau to the elementary curriculum would make mathematics more intuitive and more fun.  Since the installed base of pi users is so big, the only hope from my perspective is a grassroots effort, which based on reader feedback does seem to be happening. Someday, perhaps the American Mathematical Society and its foreign equivalents will get their act together and we can have a top-down effort as well, but for now it’s bottom-up all the way.

A: Fixing the sign of electric charge (in short, electrons, not protons, should be positive) is virtually impossible, since all the old textbooks would have to be rewritten. In contrast, switching from pi to tau can happen incrementally. There are some other anachronisms, but I’m not sure they’re worth fixing. (The temperature scale, for instance, is subtly broken, but what we have is probably good enough.)

Students can learn from this subject that notation matters, and that even geniuses (e.g., Euler) sometimes make mistakes. They can also learn that just because (nearly) everyone believes something, that doesn’t make it true.

A: Tau Day happens during the summer, so perhaps we could add a distinctive outdoor component. Tau Day at the beach? I’m certainly open to suggestions!

*

Though I’m usually a curmudgeon when it comes to mathematical holidays, Tau Day does present a somewhat unique educational opportunity, and since it is still new to the scene, there is ample opportunity for people to contribute to future traditions.  It is in this spirit that I offer the following suggestion for today (and future Tau days!):

1. Embrace the season.

I agree with Dr. Hartl here.  Kids are out of school, and this might seem to put Tau Day at a distinct disadvantage.  On the other hand, a summertime holiday naturally lends itself to outdoor activities (at least in this hemisphere). Since tau is all about relating the circumference of a circle to its radius, there are many ways to explore this relationship in an outdoor setting.  If you’re celebrating at the beach, you could have a circle drawing contest, where each contestant is given a line in the sand and tries to draw a perfect circle with the given line as its radius.  The circle for which the ratio of circumference to radius is closest to tau would be declared the winner.  Or, if you are celebrating by a lake, you could attempt to measure the circumference of the lake, and use it to determine the size of a circle with equal circumference.  Planned carefully enough, one could hint at the isoperimetric inequality (though perhaps not too explicitly, depending on how excited your kids are to do math during the summer).  Any activity involving some kind of perimeter measurement could work here.

2. Cut the memorization.

As my readers know, I am no big fan of the recitation contests that have somehow become a Pi Day tradition, in which people compete to see how many digits of pi they can recite.  Reasons for my objection can be found here.  Given that pi and tau are so closely related, it might be tempting to introduce a similar contest for Tau Day.  But these contests offer little in the way of actual mathematical learning, and are terrible PR for mathematics in general.  In order to help Tau Day mature into its own independent entity, I would advocate for removal of any recitation contests.  If the focus is on a mathematical constant, let’s focus on some real mathematical insights – this would be more educational, and could be more fun too.

What would these activities look like?  There’s plenty of freedom here.  If you have kids interested in computers, one of my readers wrote up some Tau Day activities related to formal proof writing and machine automated proof verification.  There is some cool stuff here, though sitting for too long in front of the computer may run counter to the first suggestion.  Whatever you decide, the purpose should be to emphasize mathematics as a creative pursuit full of ideas, not one that relies solely on blind memorization.

3. Take the food to the next level.

Non-math students who enjoy Pi Day probably enjoy it for the food.  If we are to hook people on Tau Day, food will probably remain an important component.  But if you advocate that tau should take the throne from pi, then it seems only natural that the food on Tau Day needs to be cranked up to 11.

As tau is nothing more than two times pi, pie still remains a natural food choice – simply make twice as much.  I think we can do better, though.  One idea: the Tau Day Pumpple.

(Images courtesy of Foodaphilia via Gawker.)

The Pumpple consists of two pies – one pumpkin, and one apple.  This takes care of the pun.  To really take it to the next level, though, the two pies are then baked inside of a cake.  I can think of no better way to celebrate.

Now, given that it is the summertime, perhaps a pumpkin pie isn’t entirely appropriate.  With so much fruit in season, one has tremendous choice in selecting a dessert appropriate for today’s festivities.  One could bake a Chapple, perhaps (cherry and apple), or maybe even a Bleach (blackberry and peach).  As long as two pies are baked inside of a cake, the spirit of the holiday will be honored.

Any other suggestions for Tau Day festivities?  This has the potential to be the only math holiday I’d willingly support, so I hope some truly exceptional traditions take root.

(Thanks to Michael Hartl for taking the time to answer some questions, and to Jim for the Wikiproofs link!)

I’ve written before about my mixed feelings regarding math t-shirts.  Today, though, I’d like to tackle a different type of gift: the math clock.  This is inspired, in part, by a gift I received from my grandmother (bless her heart) over the holiday.  The gift, pictured below, was an analog clock in which the numbers have been replaced by (what one would hope to be) mathematically equivalent expressions.

Figure 1: Clock with a black background.

Don’t tell her, but we haven’t yet put this clock up in our apartment.  In my own defense, this is mostly because we already have a math clock displayed prominently in the kitchen.  My future wife says that she gave me our first clock, and this is a claim I have no reason to doubt.

Figure 2: Clock with a white background.

Sadly, our apartment is simply not big enough for two mathematically themed clocks.  The question becomes, then, which one should be kept in-house, and which one should be relegated to the dungeon of an office I hold at the math department?  It seems that the most natural way to answer this question is to compare the mathematics of each clock side by side.

Let’s start at the top and work our way around.

12: vs.

The black clock has a simpler expression, but perhaps it’s a little too simple.  At least the white clock asks you to work a little for it.  Point: white clock.

1: 102,413 – 102,412 vs.

The white clock’s expression for 1 is a little to esoteric for my taste.  The notation is meant to symbolize Legendre’s constant – this number is related to the asymptotic behavior of prime numbers, and historically it was believed to be greater than 1 for some time, based on numerical evidence.  But if you didn’t know all that, there’s no way you could make the connection between the notation and the number 1.  At least the expression in the black clock makes the connection to 1 clear.  With a little more information so that the casual time-teller could have learned something about primes, the white clock may have had the upper hand.  As it stands, though, I must side with the black clock here.

2: vs.

Here I must tip my hat to the infinite sum.  The square root is nice, but the sum is nicer, and if you’re trying to impress a date who doesn’t know about geometric series, this will provide you with an excellent opening.  Point: white clock.

3: vs. some XML garbage.

Come on, white clock.  That isn’t even math!  Point: black clock.

4: vs.

I prefer the poetry of including “clock arithmetic” on the face of a clock.  Plus, modular arithmetic (as it is more professionally known) is a topic that the general population is not always exposed to, even though it’s not hard to explain.  I’ll take any opportunity I can get for a clock to educate the masses.  Point: white clock.

5: vs.

Given my stance on 1, this may seem a little hypocritical, but I’m going to give the edge to the white clock here.  Part of the reason is that the black clock loses steam pretty quickly – out of the 12 numbers, 3 are expressed in terms of long division.  Come on, guys.

Besides, since we know that the expression on the white clock equals 5, this allows us to solve for and obtain the golden ratio.  This is something someone could discover on his or her own, perhaps with the aid of something like Wolfram Alpha.  So the comparison to the 1 o’clock entry isn’t quite apples to apples.  Or at least, that’s what I’ll keep telling myself.

6: vs. 3!

The factorial is a little less conventional, but every student should encounter it at some point.  Here I’m giving the edge to the white clock again.

7: vs.

Presumably, the black clock wants us to solve for x, using the quadratic formula or something.  I don’t get it, though – if they’re going to express 7 as an unknown in a quadratic function of x, why would they write an equation that has two solutions, one of which isn’t 7? Since is the same as , here’s a graph of , so you can see the two roots:

I realize I’m being a little pedantic (after all, there isn’t any negative 6 o’clock), but it would’ve been just as easy to write a quadratic that had only 7 as its root.  Here’s one:

Besides, the white clock’s entry for 7 is good in its own right.  No contest here, white gets the point.

8: vs. some dots.

I’ll give it to the black clock here.  The white clock is expressing 8 in base 2, but I don’t know why they don’t do it using digits.  Probably because they also play around with the base in the next number, where they write 9 in base 4 as .

9: vs. .

If I could give negative points, I would give them to the black clock here.  Their expression doesn’t evaluate to 9; instead, the clock only perpetuates common misunderstandings about the number .  Admittedly, the black clock does give a fairly good approximation, but I’ve never heard of 9.004778… o’clock.

10: vs.

These are both worthy contenders.  For the sake of fairness, since I gave the white clock the point for 3!, I’ll choose the black clock here in favor of the white clock’s binomial coefficient.

11: vs. some hexadecimal representation of 11.

The long division is redundant, but in a sense, so is the white clock’s entry – we’ve already seen two other cases of representing a number in a different base.  In this case, I’ll defer to the one that’s clearer.  Point: black clock.

By my count, the final score is 7 points for the white clock, 5 for the black.  It was a close match, but it looks like a decision has been made.  Regardless of the outcome, though, both clocks have their share of problems.

I should point out that, somewhat surprisingly, these are not the only math clocks on the market.  Here are even more examples.  The one which speaks to me the most, though, is probably the last one.