π day: This year and next

This year has the perfect π day: March 14, at 9:26:53 (and a tad). Also written as 3/14/15;9:26:53, which is incredibly close to the value of π: 3.14159265358979323…..

Purists will object, of course, to the hour: 9. They will insist that time should be written 09:26:53. They will also object to 2015 being written as ’15’.

We will ignore those people.

This date and time happens only once every hundred years. But, like the infamous Y2K “millennium year” of 2000 (the new millennium really began in 2001), we’ll have another chance next year, on March 14, 2016: 3/14/16

There are lesser π days, one every year on March 14, at 1:59 (a.m., of course; the afternoon time would be 13:15).

3.1416 is a really good approximation to π. The difference between that value and π is one part in slightly over 136,000. This is good enough for engineering work, and certainly good enough for government work. Unless, of course, you’re sending space probes to distant planets, in which case you’d better be using all the digits your computer can handle.

For a number innately bound up with circles, π shows up in many mathematical contexts, in physics, cosmology, and higher mathematics. Thre are hundreds of formulas – mathematical series – for computing π, but since it has already been calculated to billions of digits, there’s really not much use for those series (except perhaps to check your arithmetic).

Here’s a website devoted to π.

Here’s theultimate π image

2015’s perfect calendar month

February has 28 days (most of the time).  There are seven days in a week, and 28 days takes up exactly four weeks.  This February’s calendar starts on Sunday, and ends on the Saturday of the fourth week.

This is the only case where a standard calendar takes only four lines (click on the calendar to enlarge):

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This doesn’t happen very often.   The past time was in 2009.   It won’t happen again until 2026.  If you like, you can save this year’s calendar until then.

The Resilient Brain, Part 1

War and other traumas give doctors great opportunities to study brain damage.

The brain seems remarkably plastic – parts that are injured often have their duties taken over by other parts, or the duties of those other parts are enhanced – sometimes remarkably.

The brain seems to be a highly redundant system – parts can drop out and the whole continues, often with little loss in fuction.

Large-scale attacks, like Alzheimer’s, seem to destroy or shut down large areas,
and there, even redundancy doesn’t help.

There are a few remarkable recent cases (and one well-known historical case).

In 1848, a 25-year-old railroad crew foreman named Phineas Gage and his crew were drilling blasting holes in rock. They were using a tamping iron to pack the blasting powder in a hole, when a spark set off the powder, sending the iron (3′ 8″ long, 1.25″ diameter at one end) through Gage’s head, entering under a cheekbone, exiting through the top of his head. (The iron was found some 30 yards away.)

Thanks to the miracle of then-modern medicine, he survived, and about four months later, apparently resumed a normal life.

Unfortunately, his personality changed, from “the most efficient and capable foreman”, to “a complete loss of social inhibitions”. That was attributed to the complete loss of his frontal cortex.

After he recovered, he wasn’t able to hold down a regular job, so he toured the country, with the tamping iron, raising a little money here and there. There is a photo of him with the iron. It’s almost unbelievable that he even survived.

His skull and the iron rod are now at the Harvard University School of Medicine.

The Gage case was one of the forerunners of research into localizing various functions – language, motor control, &c. – to areas of the brain.

Composer of the Day

The Composer of the Day is Ottorino Respighi, born in Bologna, Italy on 9 July 1879.
Like many composers, his early teachers was one of his parents: his father, who also taught piano. He went on to study violin and viola, and, after getting his diploma in violin, went to St. Petersburg, where he was principal viola in the Russian Imperial Theatre orchestra. While there, he studied composition with Rimsky-Korsakov (18 March 1844, who also taught Stravinsky (17 Jun 1882), Glazunov (10 Aug 1865), and Prokofiev (23 April 1891), among others).

He’s best known for three tone poems: “The Fountains of Rome” (1917), “The Pines of Rome” (1924), and “Roman Festivals” (1928). “The Pines of Rome” has the first use of a nightingale at the end of the 3rd movement. Unfortunately, at the time, there were no nightingales in the musicians union, so Otto specified a recording. Immediately following the nightingale comes the “Pines of the Appian Way” section, a stirring and dramatic musical portrait of Roman Legion soldiers marching along the Appian way. The drama is intensified by the organ playing a low B-flat in the pedal.

He also wrote string quartets, five ballets – La Boutique fantasque among them, two Suites of Ancient Airs and Dances. a delightful suite, “The Birds”, and a lot of vocal and choral music.

He died in Rome in 1936.

Composer of the Day

The Composer of the Day is Percy Grainger, born in a suburb of Melbourne, Australia on 8 July 1882. He was a true eccentric, but an extremely talented eccentric. (If you’re weird but not talented, you’re just “odd”.) Like a few other composers (Rachmaninov), his best-known work, “Country Gardens”, was his least favorite. He hated it, but whenever he gave a concert, people insisted he play it as an encore.

He didn’t like the standard Italian annotations (“molto allegretto” &c), and wrote them in English (“somewhat pertly”). He lived recently enough that you can find recordings of him playing – there’s a YouTube cut of “Irish Tune” (better known today as “Danny Boy”) . The sheet music goes by as he plays. The tempo is marked “Slowish, but not dragged, and in wayward time”. (Either he or the editors put in the Italian notations.) The dynamics run from ff to pppp. It’s lushly harmonized, with the melody in the upper left hand. And made many piano rolls – so you could have Percy playing in your parlour.

A true eccentric, he rated himself the 9th-best composer ever – between Mozart (27 Jan 1756, not quite so good,) and Delius (29 Jan 1862, a little better), with Bach (21 March 1685) – to no-one’s surprise, Nr 1.

He met Grieg (15 June 1843), and performed his piano concerto in concert. During the 1920s, he earned the equivalent of $60,000/week on the concert stage. A bit of an overachiever, he spoke 11 languages fluently (I wonder if there’s a connection between language and music). When he married, it was at the Hollywood Bowl.

He was one of the first to go seriously into electronic music – he wrote a piece called “Free Music No. 1 (For Four Theremins)”, which has recently been adapted for 4 iPhones.

He died in 1961 in White Plains, New York. There is, of course, a Percy Grainger Society.

Circles and Spheres

Standard formulas for the area of a circle, and the surface area and volume of a sphere, involve the constant π. Most of us remember that π is about 3.1416. That’s pretty close, but not exact. Even closer, π is about 3.1415926536. Now that’s really close, but still not exact. π has no exact numerical value – it’s been computed out to more than a million decimal places, but no matter how many we compute, there are always millions more. So there’s really no point in asking for the exact value of π. We can come as close as we want – or that we can afford (in terms of computer time).

Another way of looking at it is to realize that there is no pair of integers, a and b, such that a/b = π. There are a few that come close. The first is 22/7 = 3.142857…. (the decimal part repeats). Then comes 355/113 = 3.141592920353982….  (that one goes on for at least 30 decimal places with no repeat in sight)

This gives us a hint that there may be a way of calculating approximate areas and volumes that are “close enough”.

Start with the standard formula:

Circle area: A = πr2

It’s not always convenient to measure the radius directly. We can measure the diameter and divide by 2, but suppose we could work with the diameter directly:

A = πr2
r = d/2
A = π(d/2)2
A = πd2/4

which we rewrite as

A = (π/4)d2

We can look up the value for π/4; it’s almost exactly 0.7854.

So our area formula becomes

A ~= 0.7854d2

We use ~= to mean “approximately equal.

A very good approximation is

A ~= 0.8d2

this is within slightly less than 2%

It’s easy to multiply something by .8: double it three times, then divide by 10.

If d=12 inches, then A = 0.8*12 = 9.6 in^2.

Let’s see how that works out:

Suppose you have a garden with a 10-foot diameter circular area you want to plant with flowers.  You’ll need the area.   The actual area is 78.53982 square feet.  Using the “very good approximation”, we get 78.54 square feet.

The difference is 0.00018 square feet, or about 1/4 square inch.  And, it’s always a little lager than the exact value, so you never have to worry about running out.

We can also get the area if we know the circumference. The circumference of a circle is not so easy to measure – unless you’re working with a cylinder (like a tin can) and you need to get the volume. But the formula will be useful later on, so let’s develop it here.

A = πr2
c = πd
c = 2πr
r = c/2π
A = π(c2/4π2)
A = c2/4π

Moving right along to spheres

First, the formulas:

As = 4πr2

You may recognize the “πr2” part as the area of a circle with radius r. The surface area of a sphere is exactly 4 times the area of the circle whose center is  the center of the sphere (sometimes known as a great circle).  An old Greek guy figured that out, before anybody knew what π was.

Rewriting that to use the diameter:

r = d/2
As = 4π(d/2)2
As = 4πd2/4

Which simplifies to

As = πd2

That’s nice, but no help from the world of approximations.  But note that the area of a sphere is 4 times the area of a great circle.  We start over:

As = 4Ac

Now, we just go back to our circle area approximate formula:

Ac ~= 0.8d2

Plug that into the sphere area formula:

As ~= 4*0.8d2
As ~= 3.2d2

Finally, let’s find the area from the circumference. That’s a lot easier to measure, on a sphere. (Think of a basketball.)

As = 4Ac
As = 4(c2/4π)
As = c2

Calculating, 1/π = 0.3183, which we can round off to 0.3, or

As ~= .3c2

This gives a result that is just 6% too big.

Finally, the volume of a sphere:

Vs = (4/3) * πr3

From the diameter:

r = d/2
Vs = (4/3) * π(d/2)3
Vs = (4/3) * π(d3/8
Vs = (π/6) * d3

Calculate π/6 = 0.5326 Using 0.5 is too far off, so we’ll need to use 0.53:

Vs ~= 0.53d3

Now let’s step back and ask if it’s possible to calculate the area of a circle without using π.

Start with the basic formula

A = πr2

Rewrite that as

A = πr*r

From

r = d/2

replace one of the r’s:

A = π(d/2)*r
A = πd*r/2

But πd = c, so

A = c*r/2

The trick is that π is contained in c – because c = πr.

Ikea’s Magic Elevator

Seen in an Ikea store in Southern California:

 

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