Why 2pi is better than pi

1  My opinions


1.1  2π is better than π


I spent an enjoyable few years studying maths. And during that time I kept seeing "2π" in formulas. I came to the conclusion that it is wrong that π has a special name and a symbol, and that 2π should have those privileges instead.


I didn't take it much further than that. Occasionally I would tell people my feelings on this issue in the pub, cos in those days I got to discuss maths in the pub all the time. I certainly didn't go on a massive internet crusade to do anything about it. Then years later, it was pointed out to me that this is not a new idea. And there is indeed a whole lot of people who agree with me. Check it out.


Today is the day when people stop what they are doing and remember "2π". Because in some countries the date looks like "6 . 28", which is the start of the decimal representation of this interesting value. Of course, the day is all rubbish, because some people like to write "28 . 6 . 2011". And there is nothing special about the decimal system either. Whereas the value is some magical intangible object which is completely independent of how we represent it in physical form.

Two pies

1.2 Why is it better?


Well, think about this. The circumference of a circle is πD where D is the diameter. Who uses diameters anymore? (Apart from midwives) Whereas, the circumference is also 2πr where r is the radius. And everybody likes to use radii. The above website will give you lots of information on the topic, so I am not going to bother. But in short, it boils down to the fact whenever you see a a 2π you have gone all the way around a circle. And when you see π you have gone half way, or magicked a half into your equation by some other means. (For example integrating - see below!)


2  Some maths


2.1  Area of a circle


Let's start with the junior school example, the formula for the area of a circle.

How do you derive this? How do you work out any area? You integrate! We want to integrate the constant function 1 over the inside of a circle of radius r, a region henceforth known as B_r for ball. The natural choice is to use polar coordinates, and changing to polar coordinates a sneaky r appears in the integrand.

2.2  Gaussian Distribution

Everybody's favourite probability distribution: the Gaussian distribution. If we take the "standard" one, then it has this formula:

What's that I see? A 2π. How does that get there. Well, being a probability distribution, the Gaussian has the property that

Looking at this another way, you might spot that the area under the curve

(1)

is actually √{2π}. Without going into too much detail, this 2π appears in the same way as the previous example, by integrating around a circle. It turns out that y(x) is very difficult to integrate, but if you square it and change to polar coordinates again, you end up trying to integrate

which, bizarrely, is possible. But not very interesting.


2.3  Stirling's approximation


Stirling's approximation gives you an approximation to the factorial function for large values. Here it is:

So you started by multiplying a few integers together and you end up with a 2π appearing in your maths. What!?! I did a little digging to see how this happens, and it is actually the same as the previous example, you need to evaluate the same integral integral (1). Perhaps it is not so incredible then?


2.4  Bernoulli numbers and the Reimann zeta function


The Reimann zeta function is an interesting thing, and people have written whole books on it. I just want to look at one of its many interesting properties. We only need to worry about the value of the zeta function for positive even integers, and for those values you can express the zeta function using this formula.

When you think about the zeta function at s = 2, you find you are summing the reciprocals of the squares. It's quite fun to work out the value of this sum, but I will just tell you.

Look at that, it's got π in it. Why? No idea. Let's try the same thing with s = 4.

That has a π⁴ in it. So one naturally senses a pattern forming after just two of these equations. And indeed, there is a pattern, it is usually written like this.

(2)

Where B_n are the Bernoulli numbers. Here are the first few...

If this is the first time you have seen the Bernoulli numbers then you might want to take (2) as their definition. Then you could incorporate the factor of 2²ⁿ into B_2n and you are left with a formula involving π and not 2π. However that would be a daft thing to do. The Bernoulli numbers turn up all over mathematics and most of the time, they aren't next to any pis, so you can't use them to sneakily try to change from π to 2π or back again. The appearance of 2π in this formula is pretty strong evidence that we have given a name and symbol to the wrong value.


2.5  The reduced Planck's constant


A little bit of history of physics now. Way back at the turn of the 20th century, before people had "invented" quantum mechanics, Max Planck discovered a relationship between energy and the frequency of some light and invented a new constant h to put in his equation. This came to be known as Planck's constant. Over the years, as quantum mechanics developed, physicists decided that it was better to measure the frequency of light, not in Hertz (cycles per second), but in radians. With the effect that everything gets multiplied by 2π. They found that h / 2π appeared all over their work, so invented a new constant

so that things looked neater. Even physicists want their work to look pretty.

The Life and Times of Paul Daniels

I have just made a shocking discovery. Paul Daniels' blog only has 12 followers - and I am one of them. I have seven followers, that's over half of his.

I am deeply shocked by this discovery - perhaps Google have just got confused and the numbers are rubbish. Or perhaps followers and readers are different things. Like perhaps people read his blog but don't publicly follow him, cos they are ashamed and worried their friends might find out. Surely one of the most popular magicians of recent times will have literally tens of people interested in what he has to say. I know I am.

Like you can learn about how he is preparing for his tour. You get to hear about his boat "The Lovely Debbie McGee" and his wife of the same name. I like it, but not a lot.

How long is this post?

There are three enormous books that I have struggled through in the last decade. By far and away the most enjoyable was Godel, Escher, Bach by Douglas Hofstadter. The other two were The Lord of the Rings (Tolkein) and The Brothers Karamazov (Dostoevsky). GEB covers many and varied topics, but there is one in particular that I want to consider here. Hofstadter was a little bit annoyed that when you read a book you are constantly aware of how near you are to the end. This removes the possibility of a sudden surprise ending, or a surprise extension to the story. I guess he must be in a minority because Amazon decided this idea of knowing where you are in a book was important enough that their Kindle constantly gives you your progress as a percentage. However, Hofstadter's idea has lots of appeal to me. There is a similar phenomenon with films or TV shows. TV shows are always a fixed length, and you can easily tell when the story isn't going to reach a natural conclusion in the remaining time, and will likely end in a cliff-hanger. And one way or another, I always seem to find out how long a film is before I settle down to watch it. So again I can tell that we aren't going to have a sudden explosion that kills all the characters, or that the director has decided to drag the ending out for some uncomfortable length of time.

Godel, Escher, Bach and Eternal Golden Braid.
Hofstadter's magnus opus is not a victory of style of substance, although it has an incredible amount of both.

Hofstadter has a possible solution to this. He suggests trying to disguise the actual end of a story. Your first iteration in this direction might be to add a hundred blank pages at the end of a book, so you might fool some people into thinking they have more to read than they really do. But this obviously wouldn't work as people tend to flick through the remaining pages and would spot this. So maybe you add extra pages with works on them, set into paragraphs and chapters like the real story. But this probably wouldn't work very well either. You would have to use the same vocabulary, and the same names as in the real story. Even then, there is a good chance somebody would spot that you had written "The End" somewhere in the middle of the text. He suggests that you can communicate the actual end of the story to an attentive reader by (firstly bringing the story to a conclusion) and then continuing, but changing the behaviour of the characters to something completely unprecedented. You would need something sudden so that it is clear the story had ended and the rest is nonsense, but need to keep writing in the same style so that even if people look at later pages and read words and sentences, they don't realise they aren't part of the story until they have actually read the real story.

Looking at this another way, I enjoyed playing Portal. You start out progressing through numbered test chambers, each one clearly labeled and telling you there are 19 in total. Not only this, but the narrator tells you there will be a party when you successfully complete all the tests. However, if you are really paying attention you should pick up on the hints that the same narrator is actually insane and intends to murder you. And then during the 19th chamber (which is surprisingly short!) you get to realise a true jump of levels (in the style of GEB) as you discover you were in a game within a game. And although you thought you were near the end, there is actually an unspecified amount left as you explore the testing facility.

This trick only works once, and although Portal 2 had twists and turns in the story, I knew before I started that it was approximately 6 times longer than the original.

In other news, I got some "Blue Shropshire" cheese for 75p in an impulse buy this evening. It made a nice supper. I am wondering if Blue Shropshire is the same as Shropshire Blue, or some Morrison's brand trick that has nothing to do with the famous regional cheese. Just like Brie has to be made in France, unless it is Cornish.

The thing that I really like about Portal 2 is the way Hofstadter uses the portal gun to shoot cheese at things. And, after watching TV in the 19th chamber, he read Dostoevsky on his Kindle.

Fahrenheit is rubbish

There are many reasons why the Fahrenheit scale is rubbish, here are just a few of them. Obviously it isn't used so much these days, except by old people and Americans who just can't let go. The bilingual weather forecasts we have to endure are very irritating, but not actually a property of the silly temperature scale. I learnt the some following facts in GCSE science class, therefore they are probably not true.

The Fahrenheit scale was defined ages and obviously made more sense at the time, and was (not unsurprisingly) defined so that 0 and 100 were important temperatures. They decided that 0F should be "the coldest temperature possible" which sounds like a good idea. But they didn't know so much about physics as we do now, and now we think of 0F as the coldest temperature people could make in a science lab 200 years ago. Which is even more stupid when you consider that you don't have to explore the arctic very hard before you find temperatures below 0F. On the other hand, have you ever noticed that a healthy persons temperature is nearly 100F? Mr Heppenstall (chemistry teacher) explained that 100F is the temperature of a healthy cows mouth.

Now, obviously, the scale is defined in terms of the modern temperature scales and seems like a bit of an historical oddity. Chemical properties of water are fine, but with Celcius you end up with absolute zero being some silly irrational number. Or you could go with Kelvin, but then the freezing point of water is some silly irrational number. So why not come up with a new scale where the freezing point of water, the boiling point of water and absolute zero are all sensible values? Here's how you do it...

First you need a name for your new temperature unit, let's call it an "Onkion". You say that 0C is 0Onk, then you start taking logarithms so that absolute zero is negative infinity. You still have the freedom to scale your new scale so that 100C = 100Onk, then you are done. Here is a simple formula for converting from Celcius to Onkions. I have used T_C to represent the temperature in Celcius, and T_C0 to represent absolute zero measured in Celsius.

 Here is a simpler, but imprecise version of the formula with numbers and things in it.

Here is a table of some interesting temperatures with both Celcius and Onkions in it. It seems that this mental new temperature scale that I just made up it quite good. Instead of worrying about lots of decimal places of temperatures that are very close to absolute zero, they are all nicely spread out. And for very hot temperatures you don't really need to worry about using scientific notation for very large numbers, as the numbers don't really get that large.

Description	Temperature in Celcius	Temperature in Onkions
Absolute zero	-273.15	Negative Infinity
Superfliud liquid helium	-271	-1560
Boiling point of oxygen	-223	-541
Boiling point of nitrogen	-196	-405
Warm day on Saturn	-139	-228
Freezing point of water	0	0
Warm day in England	20	23
Healthy person	37	41
Boiling point of water	100	100
Medium oven	180	162
Warn day on Venus	460	316
Melting point of iron	1538	606
Surface of the sun	5500	978
Centreof the sun	15,700,000	3500

In other news, please enjoy this picture from my garden.

My red hot poker is poking very well this year. It's actually a metre tall.

Arise Sir Bruce

I heard some exciting news on the radio this morning: perennial dirty-old-man Bruce Forsyth is to be knighted. Yay.

Didn't he do well


When asked about it, Bruce said, "...there's been too much talk about it, so I'm thrilled at last it has happened." Then in his next sentence described the award as "unexpected". He's so modest.

Bruce Forsyth is the one in the middle

I heard recently that The Queen doesn't actually say, "Arise Sir Bruce", but something much more mundane, perhaps "I liked that thing wot you did." Perhaps he will retire now that he has (possibly) achieved all his life-goals. Or perhaps he will subject us all to another few years of catchphrase based humour.

That is all.

More Cookery Show Based Obvservations

It's the end of an era. Or maybe the BBC are just having a short break in their evening cookery show slot. Great British Menu has reached it's climax with a street party in London (where else?)  Through the course of the series they have really gone to town on their food based puns. Some were so contrived as to be unrepeatable. Some were so good they went completely unnoticed by mere mortals. Here are a selection of middle-of-the-road ones that I managed to jot down.

• Turning up the heat
• Upping the steaks
• More seasoned opponents
• Who has reeled in top marks? [In the fish course no less.]
• Made a hash of the corned beef
• Chickened out [with her chicken dish]
• Bitten off more than he can chew

The food puns were the most entertaining, but once I had started looking, I noticed them overusing these metaphors as well.

• Step up a gear
• Pull out all the stops
• Keep him in check
• Keep his head above water

In other news, I have a new favourite person. Or at least I did for the first 20 minutes of Portal 2. Now his character is really grating on my nerves. Here is a picture...

Stephen Merchant