Conclusion to Half-Tau Month

Though pi day passed a few weeks ago, my brother made the interesting observation that this month is "Pi Month." It is March of 2014, or 3/14. Since I spent the month focused on trigonometry, I never had a chance to honor this joyous occasion until today.

Interestingly, pi does play a huge role in trigonometry. From wrapping functions to sine curves to angle measurements, pi is always popping up. Though this is kind of interesting, trigonometry is also one of the areas where tau really shines. Having just finished topics such as trigonometry, polar coordinates, and wrapping functions, I have found that it is a real struggle to use pi. I found myself converting most of my problems to tau before solving them just because pi made it too confusion.

No video describes these sorts of issues better than Vi Hart's "Pi is (still) Wrong" video, which gets into some of the issues involved with using pi, one of these being trigonometry.

I would also like to make you all aware that next month is Math Awareness Month. The theme this year is Mathematics, Magic, and Mystery, in part to honor the centennial of Martin Gardner's birth. At www.mathaware.org, there is a poster with 30 squares on it to represent the 30 days of April. On each day of the month, the next square becomes active. I will try to keep an eye on these webpages, as I will probably use April to comment on the topics posted there. Also, April 1st is a page on magic squares, and I am extremely honored to be hosting that day.

How YOU Can Memorize 2000 Digits of a Number

Since it is two days after Thanksgiving, and many of us are probably eating leftover pie, I thought it would be appropriate to do a post somewhat relevant to the numerical pi. And I could do something very mathematical, but I just finished my first trimester at Phillips Academy Andover last week, and I needed a break after my nearly impossible MATH-380 final exam. As a result, I thought it could be fun to talk about memorizing numbers, pi and tau in particular.

When people hear about my Tau 2000 event, they often ask me if I have what they call a "photographic memory." This is not at all true. I don't even think the types of photographic memories advertised in pop culture really exist (I'm not an expert on neurology, so for more on that, I'd recommend reading this Scientific American article). The way I memorized 2012 digits of tau was all learned and practiced techniques, similar to my mental math presentations. I was not born with some gift or natural talent, it was just learning the methodology and practicing until I could do it quickly. Just like anyone can do mental math, anyone can be a memory expert as well, ranging from being able to remember 57890 digits of pi to being able to remember your car keys as you leave for work. There are techniques for it all.

First, let me introduce you to the Major System. This is a phonetic code that enables you to turn numbers into words. You store them as words, and later retrieve them as numbers. Basically, each digit is associated with a specific consonant sound.

1 is the t or d sound. It can also be either of the th sounds (see note below).

2 is the n sound.

3 is the m sound.

4 is the r sound.

5 is the l sound.

6 is the j, ch, sh, or zh sound.

7 is the k or g sound.

8 is the f or v sound.

9 is the p or b sound.

0 is the z or s sound.

Note: th (both the th in "that" and the th in "thing") is normally paired with 1, but there are other variations on the system that will put it with 8 or not include it.

This looks hard to memorize on its own, but it is actually not that hard. Here are some mnemonics that can help you.

1. A t or d has 1 downstroke.
2. A n has 2 downstrokes.
3. A m has 3 downstrokes.
4. The number 4 ends in the letter r.
5. If you hold up your hand with 4 fingers up and your thumb at a 90° angle, you will see 5 fingers shaped like an L.
6. A J looks somewhat like a backwards 6.
7. A K can be drawn with two 7s back to back.
8. A lowercase f in cursive looks like an 8.
9. The number 9 is a backwards p or an upside-down b.
10. The word zero begins with the letter z.

You will also notice that the consonants that were paired together sound very similar. Your lip movement and tongue placement are the same in any of the consonant sounds chosen for a number (except for the th sounds, hence the inconsistency of its use).

You might be wondering why there are no vowel sounds on the list. There is also no h, w, or y sound. This is because you can insert these wherever you want between consonants and they mean nothing. With all of this in mind, you can begin turning numbers into words. Let's take the number 15. What words can this become?

Well, one is the t or d sound. Five is the l sound. Insert vowels, and you can get doll. Or tile. Or tail. You can also insert vowels at the beginning or end of the word and make deli, or Adele. You can also insert hs, ws, and ys to get hotel, towel, or yodel. Here are a list of the 66 words that can be made out of  the number 15 (I put the ones that I might use in a mnemonic image in bold print):

Addle, daily, dale, dally, deal, delay, dell, dial, dole, doll, dual, duel, dull, duly, dwell, ethyl, hastily, hostile, hotel, hotly, huddle, ideal, ideally, idle, idly, idol, it'll, italy, oddly, othello, outlaw, outlay, saddle, sadly, seattle, settle, societal, stale, stall, steal, steel, still, stole, stool, style, subtle, subtly, suicidal, sweetly, tail, tale, tall, tally, teal, tel, tell, they'll, tile, till, toil, toll, tool, towel, waddle, widely, yodel

Note that some of the words start with s. Since s is zero, this is referring to the number 015, which is normally still 15. These words do not work if 15 is part of a string of other digits such as in pi or tau.

The ones that I bolded are all nouns that you can create a mental image of in your head. As the Scientific American article that I linked to states, people naturally have a better memory for visuals (the reason why you might remember someone's face, but not be able to place the name). So, you might not be able to remember the number 15, but you can probably picture a doll, or a hotel, or a yodel (for this, I would think of the chocolate pastry, not the verb). If you are trying to remember that it is someone's address or apartment number, picture a relationship between the object and the person. Maybe the person is standing up on a stool shouting to a crowd of confused, awestricken people, or they are on the couch stuffing their face with yodels. The sillier your image, the easier it is to remember.

There are lots of memory experts who will create a list of "peg words," which are essentially 100 words that they will refer to when they are trying to remember a number between 1 and 100. It is certainly not a necessity, but it can often help if you are trying to come up with a word on the fly. Every person has a different list of words that works for them, so this is something that I would encourage you to make on your own. The website www.phoneticmnemonic.com works very well to help create this list.

To memorize shorter strings of digits (something like memorizing 100 digits of pi), the best approach in my opinion is to create sentences out of your words. For instance, take the first five digits of pi: 31415. The only word that can be formed out of this is moderately, which isn't a great start to a sentence. However, it could be turned into "my turtle" or "Madrid law" or "Mother Yodel." The first 24 digits of pi create the sentence:

My turtle Pancho will, my love, pick up my new mover, Ginger.

Say this a few times and you will sadly have it memorized. And since you now know the code, you now have the first 24 digits of pi memorized. If you want to keep going, the next 17 digits are:

My movie monkey plays in a favorite bucket.

The next 19 are:

Ship my puppy Michael to Sullivan's backrubber.

If you want to take it to 100 digits, you can use:

A really open music video cheers Jenny F. Jones.

And my personal favorite:

Have a baby fish knife so Marvin will marinate the goosechick.

This method works great for condensing large quantities of numbers into a small amount of silly, memorable sentences. However, once you get up towards 300, 400, 500 digits, it is really tough to remember the exact prepositions and linking verbs you used, which contribute to the digits. Because of this, the method I used for memorizing 2012 digits of tau is a different variation. Rather than just memorizing plain sentences, I used a technique called the memory palace.

A memory palace is essentially a place that you can mentally visualize that you put the images that you create in. For instance, your drive from your house to work might be a memory palace. Your elementary school campus could be your memory palace. You can even create an imaginary place to be your memory palace. Let's pretend your memory palace is inside of your house. The first ten loci (places to put the images) might be:

1. Your bed (in your bedroom)
2. Your closet
3. Bathroom
4. Hallway
5. Other Bedroom
6. Stairs
7. Living Room
8. Dining Room
9. Kitchen
10. Front Porch

And you might have a grocery list with the following items:

• Grapes
• Carrots
• Corn on the Cob
• Yogurt
• Cheddar Cheese
• Marshmallows
• Cheetos
• Salt
• Pepper
• Ice

All you need to do is mentally "put" each of these items into the corresponding locus in your memory palace. For instance, the first item is grapes. You would put the grapes on your bed. But you wouldn't just put them there, you must do something to make the image stand out. First of all, you must embrace the image. Not only do you see grapes, but you smell the grapes, you taste the grapes. The more of your senses that you alert, the easier the image is to remember. The image also needs to be less dull than just a few grapes sitting on your blanket. Maybe have grapevines growing out of the back of your bed. Maybe visualize the grapes to have legs, and jumping on the bed. As long as it is a silly image that stands out in your mind, you will be able to remember it.

The next item on the list is carrots. The corresponding locus is your closet. Carrots grow out of the ground, so maybe you picture all of the mud that your sneakers have tracked into the closet has carrots growing in it. As long as you pull a carrot out of the mud, you will remember it is carrots. Or maybe there is a snowman inside with a carrot nose, or a carrot shoe-horn. The actual carrot aspect of the image can absolutely be subtle, as long as you can remember the image and this image triggers the thought of carrots in your mind.

Continue through the list, and you will have ten images in your head that will in fact be stuck there until you use other techniques to remove them (yes, there are techniques people use to forget things). Try this out a few times, and I'm sure you will find it very useful. If you have a list of things to do at work, you need to remember when to pick up your kids and bring them to their activities (you may even use the major system for translating times into words - if you need to bring your son to baseball practice at 4:15, you may just picture your son swinging his bat at a "hurdle" (r=4, d=1, l=5) in the appropriate locus), or anything else, the memory palace is a great way to go.

How does this help one memorize the digits of a number, like tau? Well, what the major system does is turns numbers into words, which can then be turned into images. The memory palace then acts as a place holder for those images. For instance, take the digits of tau:

6.28318530717958647692528676655900598...

The first two digits are 62. What words can this form? You can say chain, gin, maybe you know someone named Jane or John. I ended up choosing the word ocean.

The next three digits are 831. This forms the word vomit. Yes, it is disgusting, but it is a word that will create a memorable image.

The next two digits are 85. From this, we can create the word waffle. So the first image will be "an ocean vomiting a waffle." It sounds very silly, but it will be memorable. The smell of the saltwater, the taste of the waffles, the sound of the ocean waves crashing. This all will go into your first locus. My memory palace for tau was my middle school campus, so I remembered this image in the back parking lot of the school.

The next image is comprised of the digits 30717958. This can be turned into "a mask tugging on a bailiff." This was put inside of a staff room that the back parking lot has a door to. It is a very weird image, but still memorable. Picture the bailiff really struggling to get away from this mask, while still fearfully reciting his lines: do you solemnly swear to tell the truth, the whole truth, and nothing but the truth. Make yourself feel scared of this moving mask, and sympathize with the bailiff. The more you relate to and embrace the image, the more memorable it will be. Especially when you are memorizing 2012 digits of tau (which took me 272 images), you need each image to be extremely vivid.

To retrieve the numbers from this memory palace, all you do is go back to the image, find the subject, root verb, and object of it, and translate the consonants back to numbers with the major system. With practice, this becomes easier and easier to do. I strongly recommend practicing at least memorizing grocery lists and to-do lists with the memory palace, and if you want to take it further, learn to convert numbers to words with the major system for more advanced lists and situations. Maybe even memorize your family and friends' phone numbers with the major system and memory palace. These are all great exercises for your mind, and will definitely give you a better memory.

What Sounds the Coolest? Pi or Tau

You may have noticed that today is Pi Day (or Half-Tau Day). Because of the special occasion, I thought I should post something. Since π ≈ 3.14159, I decided to set the post to go up at 1:59.

I have posted numerous times about the Pi vs Tau debate. All of the reasonings involved mathematics. However, it is fun to analyze completely non-mathematical representations of these numbers.

There have been videos posted on YouTube of a musical representation of both pi and tau. If you want to celebrate pi day without a mathematical burden, listen to these and decide which number sounds cooler.

What Pi Sounds Like

What Tau Sounds Like

Comment which number you think sounds the best!

Why Does πr^2 Work?

Recently, I have been alluding to the pi vs tau argument, which is debating the proposal of replacing pi with tau (the equivalent of 2π). Since both sides have pretty convincing points, it is a fun thing to talk about.

Pi fans argue that by changing pi to tau, we would ruin the formula for the area of the circle. It would go from:

A = πr^2  –>   A = 1/2τr^2

As you can see, the formula looks a lot sloppier.

Tau-ists rebut this by saying that the proof of this formula requires one to multiply 1/2 by 2πr^2, thus proving the significance of 2π. I mention this a lot, but I have never actually proven it.

Rather than writing out the proof, I think it would make more sense to watch this YouTube video. It made it a lot more interesting for me.

I find this proof fascinating on its own. Also, you may have noticed when the 2π was present and was cancelled out by the 1/2. For the pi vs tau argument, this specific instance can fall to either side.

Pi vs Tau: Pi's Rebuttal

As you may know, I have been strongly intrigued by the tau movement. This movement is promoting the idea that we should not be using pi as the circle constant, but rather 2π, which has been renamed with the greek letter tau.

This movement was started by Bob Palais of University of Utah when he wrote the article Pi is Wrong which was published in the Mathematics Intelligencer. Physicist Michael Hartl proceeded to write The Tau Manifesto, and founded Tau Day on June 28th (instead of Pi Day on March 14th) as well as the website www.tauday.com. This gave dozens of reasons in geometry, trigonometry, physics, and statistics why tau is more practical and natural than pi. On a side note, I currently have the world record for tau memorization at 2012 digits, but that's nowhere near the world record for pi memorization at 67890 digits.

If you have some more time, The Tau Manifesto is a fascinating read. I'd also recommend watching this video done by Vi Hart for an overview of the tau movement.

You can also click here to see all of the blog posts I have done giving details on why tau is better than pi.

Just two days before writing this, I found that someone was inspired by Hartl's Tau Manifesto to write The Pi Manifesto, which presents an interesting rebuttal for tau. This wasn't recent news in mathematics, but it was definitely news for me!

I found that all of the reasons for either side can actually be debated. I think this is the first post where there is actually a debate over what is the right answer.

An argument for tau is that the radius is what a circle is measured by. Dictionary.com defines circle as a closed plane curve consisting of all points at a given distance from a point within it called the center. Clearly, the radius is the main measurement here.

Pi supporters would then argue that the radius can only be found by taking the diameter and dividing it by two. When looking at a circle on paper, it is impossible to pinpoint the center and find the measurement out to the end.

Yet, constructing the circle requires knowing the radius. If you were to use a compass, you must put the point where you want the center to be and trace a line with a constant distance around it. This is the true way to form a circle. For practical uses, you would need circles to see what restaurants were within ten miles or something. In this case, ten miles is the radius, and you are constructing a circle with this measurement.

This example requires the choice between constructing the circle easier or deconstructing the circle easier. The construction requires the radius, while a circle already given uses the diameter.

An argument for pi is that the area formula, which is one of pi's most common uses, is messed up by this change.

A = πr^2  –>  A = 1/2τr^2

Yet, tau-ists argue that this is more natural. First, there are many other shapes that have 1/2 at the beginning of their area formula. Triangles, trapezoids, and n-sided polygons all use formulas with a 1/2 in it. Second, the area formula can be proven (I will in a future blog post), and this proof's last step is to multiply 2πr^2 and 1/2. It turns out to be more natural just to tack on the 1/2 rather than hiding the 2π with it.

This is again an argument where we need to choose between having a natural number or an efficient number. It is clear that tau is the constant that belongs, but we have an opportunity to simplify the equation.

Both manifestos list other reasons that take longer to explain, but are very interesting (I think a lot more interesting than the two above). These involve finding measurements of the unit circle, graphing trigonometric functions, and rewriting Euler's identity.

Because math is such a definitive discipline, it is rare for Cool Math Stuff posts to have comments. I strongly encourage you to comment on this post. After reading both manifestos, watching several YouTube videos, and seeing lots of press coverage, it is difficult to take a side. Comment what you think about the different arguments (I think the three I listed earlier are the closest arguments to call), and if you are a Tau-ist or not. This is one of the few chances where you can get into a debate about mathematics!

Pi, Lie, Same Thing Part 3: Tau 2000

This week, I wanted to take a break from the hardcore “Cool Math Stuff” and talk to you about something a little different. Since we have just been talking about the Pi vs Tau argument, I wanted to bring up an event that I am hosting that involves Tau.

On May 6, 2012, I will be establishing a new world record by reciting from memory 2000 digits of Tau. This event will be a fundraiser for the Bethel Public Library. The event will also include magic, raffles, contests, and more.

If you would like to find out more about this event and see how you can help out, please go to www.Tau2000.com.

Bonus: When I was teaching at Westport Minds in Motion this past Saturday, I heard a puzzle that isn’t super mathematical, but does involve some mathematical logic and reasoning. It was a really cool one, so I thought I would share it.

You are in your basement and there are three light switches. Each switch controls a bulb upstairs, but you do not know which switch controls which bulb. You also cannot see any of the bulbs, or have any mean of finding out if the bulb is on or off (helpers, really well trained dogs, face time, etc.). You are also feeling kind of lazy, and will only go up the stairs once. With all of this in mind, how do you figure out which switch controls which bulb.

I will post the answer to this puzzle in about a month for anybody who wants to try to figure it out, which I definitely suggest trying.

Pi, Lie, Same Thing... (Part 2)

Last week, I did some explaining on why tau is better than pi. However, that post was relating it to past posts more and using the popular reasons less. Let me bring up some of the more popular reasons that are a little easier to follow.

First off, we looked at trigonometric reasons last week, but the “homeland” so to speak of both of these numbers is the circle. This week, we will only look at circles.

People measure angles in degrees usually. There are 360° in a circle. Mathematicians, however, measure angles in radians. There are 2π radians in a circle. No wonder. In a third of a circle, there are 2π/3 radians. A third of a circle corresponds to this mess!

What about a quarter circle. This is equivalent to 1/2π. Still, a quarter corresponds to a half. That’s also crazy.

What if we used tau? A third of a circle is a third of tau. A quarter circle is a quarter of tau. 28/53 of a circle is 28/53 of tau. Does it get any easier?

You might be thinking that I am only showing you one side of the story. Obviously, this is partly true, but let’s look an argument that a pi person might bring up. What about the area formula? This is just a plain old pi.

Yes, but one of the most common proofs for the area formula (I will definitely post this at some point) ends up with the following equation, which you have to simplify:

1/2 x 2πr^2

So basically, there was a 2 there, but it got cancelled out. The new area formula is:

1/2τr^2

So, the 1/2 does us a favor. It helps us remember the proof.

This might not be good enough for you, so let me bring up how the 1/2 makes it more natural as well. What is the area formula for a triangle:

1/2bh

What about a trapezoid?

1/2h(b₁ + b₂)

Even the universal polygon area formula starts with a 1/2. So, the new formula just makes the circles fit in more with the other geometric shapes. I think this is a good thing, not a bad thing.

Next week, I won’t be talking as much about tau, but about what I have done with tau over the past few months. I’ll bet some of you might already know what it will be about. For a pretty good hint, check out www.Tau2000.com.

Pi, Lie, Same Thing...

Last week, you might remember that I mentioned something about Euler’s Identity getting better. In fact, it makes it more natural too, as I will show you. But before we go all the way there, let’s begin with the basics.

First off, let’s start with pi. Pi is approximately equal to 3.14 and is the ratio of a circle’s circumference to its diameter. It is probably considered to be one of the most important numbers in geometry and trigonometry. You might hear pi referred to as the “circle constant.”

The true circle constant should be a number that comes out of the definition of a circle, correct? And what is the definition of a circle? According to dictionary.com, this would be:

A closed plane curve consisting of all points at a given distance from a point within it called the center.

This definition just defined two lengths in the circle:

The circumference: A closed plane curve

The radius: A given distance from a point within it called the center

In other words, the circle constant should be the ratio between the circumference and the radius, right? This number comes out to the number tau. Tau is in fact 2π, but it is called tau since it is quicker and easier to say and write it that way. There is no doubt that tau should be the circle constant.

What about all of pi’s trigonometric significance, and other properties involving circles. Pretty much all of them are made better with tau. Next week, I will show you that even things that are made more complicated with tau are still more natural or helpful.

However, I didn’t want to keep you waiting too long about Euler’s Identity. Here it is again:

e^iπ = -1

Its rewritten form that we call God’s Equation is below:

e^iπ + 1 = 0

What if we use tau? It looks like this:

e^iτ = 1

Pretty good! Tau just took the little ugly thing and fixed it. Some people are actually upset that we lost a little zero, since zero is significant too. Well, no worries!

e^iτ = 1 + 0

I actually found out something about the new version of Euler’s Identity that was really surprising, and mixes in another past cool math stuff post which can never be bad. Remember the very first post on imaginary numbers back in October when we learned about the cube roots of one? If you haven’t definitely check it out. It was a really awesome post. Just search “cube root” and it will come up.

This post taught that there are two square roots of one, three cube roots of one, four fourth roots of one, and so on. Well, turns out that these roots can be found with the new Euler’s Identity.

e^iτ/1 = 1

e^iτ/2 = -1

e^iτ/3 = -1/2 + i√3/2

e^iτ/4 = i

e^iτ/6 = 1/2 + i√3/2 

e^iτ/8 = √2/2 + i√2/2

e^iτ/12 = √3/2 + i/2

When I saw that, all I was shocked that that actually worked, but it also makes tau such a beautiful number. Don’t get me wrong, e and i definitely get credit, but they don’t have such promising numbers to take their place.

Since tau has so many cool things about it, I am turning it into one of my little three week series, like I did with graphing calculators and at CTY. Even though it’s kind of late, have a happy half-tau day!