Wednesday, May 25, 2011

Approximating π

The ancient fraction for estimating π, 22/7, is only accurate to two decimal places! Terrible, right? Well, that's about 0.04% too big. The common estimate of π to 5 digits (3.14159) is about 99.9999% of the true value. The fraction 355/113 is 0.0000085% too big. How terrible is that?

If you're trying to estimate the number of square feet in a circle with a radius of 100 feet, the fraction 22/7 adds 12.6 square feet to the true total of about 31,416 square feet. Using π to five digits to approximate the area is less than four square inches short of the true total. Using the ratio 355/113 instead of π to approximate the area overestimates the area by little more than a third of a square inch.

Say you're trying to cut a length of rope to go around a sphere the size of the Earth, and you know that the mean radius of the Earth is 6,371 km. If you use the ratio 355/113 to estimate π, then the 20,015-kilometer rope that you cut will be about a meter and a half too long. Measuring a length of rope that long is probably a bigger problem than the error in 355/113 as an approximation of π.

Wikipedia says:
...the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.

Saturday, May 7, 2011

2011 Derby Disclosure

Midnight Interlude, Dialed In, and Nehro. With three wins out of the last four Derbies, it's hard not to go with Calvin Borel on Twice the Appeal.

It was Animal Kingdom: Nehro placed.

Friday, May 6, 2011

2011 Kentucky Oaks Disclosure

Zazu, along with Lilacs and Lace and Kathmanblu.

Zazu showed.

Monday, May 2, 2011

Road to the Kentucky Derby

The links in last year's Derby post are mostly still useful this year. But in the months leading up to Derby, there are a number of important Triple Crown prep races that test and prepare the 2011 Kentucky Derby contenders This year, Uncle Mo was an early favorite and the first to get his own Wikipedia page.

Before the colts are three, they're two years old. The Breeder's Cup Juvenile is held in November the year before. Uncle Mo won in 2010 (video); Street Sense had won in 2006 before winning the Kentucky Derby the following year. Also, the American Champion Two-Year-Old Colt is awarded as part of the Eclipse Awards (which also has an award for fillies); Uncle Mo won in 2010; Street Sense had won in 2006.

The major prep races leading up to the Kentucky Derby begin at the end of March and run through April; all are 9 furlongs on dirt, except the Blue Grass Stakes, which is now run on synthetic polytrack.
  • The Wood Memorial Stakes (top three) is held in mid-April at Aqueduct Racetrack in New York. In 2011, Uncle Mo was heavily favored to win, but in a big upset showed while Toby's Corner won and Arthur's Tale placed (video). In the past, 11 winners have gone on to win the Kentucky Derby, four of whom captured the Triple Crown.
  • The Arkansas Derby (top three) is held in April at Oaklawn Park in Hot Springs, Arkansas. In 2011, Archarcharch won (video).
  • The Blue Grass Stakes (top three) is held at Keeneland in Lexington, Kentucky. Brilliant Speed won in 2011 (video).
  • The Florida Derby (top three) is held at the end of March or the beginning of April. In 2011, Dialed In won (video).
  • The Louisiana Derby (top three) is held in early March at Fair Grounds in New Orleans, Louisiana. In 2011, Pants on Fire won (video).
  • The Santa Anita Derby (top three) is held in April at Santa Anita Park in California north of Los Angeles. The winner in 2011 was Midnight Interlude (video).
  • The Illinois Derby (top three) is held in April at Hawthorne outside Chicago, Illinois. Joe Vann won in 2011 (video).
Looking at the videos, Nehro had good performances at the Louisiana Derby and Arkansas Derby, which would seem to bode well for the long, 10-furlong Kentucky Derby.

Other notable prep races include the Tropical Park Derby (January, 2011), Fountain of Youth Stakes(March), Tampa Bay Derby (March), and Holy Bull Stakes (April) in Florida; the San Felipe Stakes (March) in California; and the Southwest Stakes (February) and Rebel Stakes (March) in Arkansas.

2010 Kentucky Derby

In 2010, the top three finishers were Super Saver, Ice Box, and Paddy O'Prado.