Monday, April 4, 2016

The World's First Polar Skyway

While doing some research on science journalist of the '50s, Michael Gladych, I come across other interesting science journalists as well.  One of them is Ansel Talbert of the New York Herald Tribune.  Looking into Mr. Talbert's travel in the '50s, I found that he'd taken a flight from Denmark to Alaska.  At first, the only remarkable thing about the flight was that HRH Prince Axel of Denmark and Iceland was on board, (if you're from the Sates like me, I bet you didn't know there was a prince of Denmark and Iceland!)

A little investigation made even this fact seem trite when Prince Axel was revealed as the CEO of SAS airlines, the carrier for the flight in question.  Not all was lost though... Far from it!

It turns out the flight was the first commercial flight to take advantage of the U.S. Air Force's newly released polar maps!  For the first time ever, a group of passengers flew from Denmark to Tokyo, stopping only in Alaska, in just 8,000 miles, removing 2300 miles from the old route!

Thursday, March 3, 2016

Harold Daw Inventor of the Air Table!

D.r Harold Daw
In the late 1950’s, my uncle, then a teenager, found himself at odds with the Law in Las Cruces, NM.  Simply put, the police had decided he was, in fact, his brother, wanted for fleeing a drag race in the same car my uncle had just driven home.  Despite his repeated pleas that they were arresting the wrong Carter boy, the police persevered, first handcuffing my uncle, and then shuffling him into the police car.  Unbeknownst to them, the woman who lived across the street from my dad’s family had observed everything.  Turning to her husband, she said Harold, they go the wrong boy!  You go down to the station and help them straighten this out!”  That’s how my uncle found himself riding back to his house in the car of Dr. Harold Daw, head of the New Mexico State University physics department. 

The Hot Rod in Question
The reason I bring this story up now, isn’t because of its somewhat topical nature in relation to the state of police arrests in this day and age.  (Although I do suspect prisoners can no longer be liberated on the say-so of a physics department head.)  Nope, it’s because I found an air table, at the Excelsior Science Workshop[3] this week, and well… Dr. Daw in addition to being the ‘great liberator’ of wrongly accused teens in Las Cruces, New Mexico, is also the documented inventor[1] of the air table!  Ever played air hockey?  Dr. Daw’s invention for studying two-dimensional mechanics in physics lab ultimately became best known as the air hockey table!

From AJP article [1]

Dr. Daw first publicly documented his invention in a 1963 issue of The American Journal of Physics, known colloquially to physics students the world over as the AJP.  In this article, he pointed out the advantages of the device for studying two dimensional collisions, and gave explicit instructions for how to build one. In the original implementation, they floated glass pucks on the air cushion.  In our local incarnation, the glass pucks have become CDs!

Like many-a-rock star, in 1987, Dr. Daw topped his original offering by adding... wait for it... fire!  The images in the AJP article does’t due the experience justice, but the video by +Jared Ficklin who built a fire-table of his own does!

Wednesday, March 2, 2016

Making Change and... Partitions?

Today’s math fun involves portions of yesterday's, but with a few more steps.  It may also—dare I say it?—involve partitioning!  I might be using the word partitioning in an incorrect way, and if so, then pardons please, (also, please let me know).  The question is, what are the fewest number of coins you need to make change for up to a dollar.

Here's how I worked, using an iterative algorithm, (fancy words for: "I'm going to use the same trick over and over").  It was all about granularity of coins, and getting quickly from one amount to the next.  The quickest way, (where quick is defined by using the smallest number of coins), to get to a large amount of change is with large coins.  So, as we move around within 99 cents, the biggest step we can make is with a half dollar. Using yesterday’s method, we can fit one half dollar into 99 cents.  That leaves us with 49 cents left move around in.  The quickest way to make progress within that interval is with a quarter.  We can only fit one quarter into 49 cents.  That gives us our next interval which is moving around within 24 cents.  We use our next biggest coin, a dime, and figure out we can get two of those into 24 cents.  Within the remaining 9 cents, (the space a dime can't make up), the quickest way to move is with a nickel.  One of those will fit.  That leaves us a four cent interval to move in.  We can fit four pennies into that, and we’re done!  So, the answer winds up being

1 50 cent piece
1 quarter
2 dimes
1 nickel
4 pennies.

The next step, would be to prove that I'm correct, but how?

Tuesday, March 1, 2016

Combinatorics and LCMs

Working through the problems in Niven's book on combinatorics, I came across the following one that cleverly introduces least-common-multiples without saying any of those words.  The book asks the following question:

How many numbers that are evenly divisible by 11 exist between 1 and 2000?  How many that aren't also evenly divisible by 3?  How many numbers that are evenly divisible by 6, but not by 4 exist between 1 and 2000?

The hastily scribbled answer can be seen below, with each of the answers boxed in succession down the screen.

By simply dividing 2000 by 11, we find out how many integers between 1 and 2000 are evenly divisible by 11.  In other words, we ask how many multiples of 11 can fit between 1 and 2000.  When we want to eliminate the multiples of 3, that's when the least common multiple comes in.  We already have the answer for all numbers divisible by 11, but how to eliminate those also divisible by 3?  By first asking what number divisible by 11 are also divisible by three, we arrive at an answer.  The first number divisible by both is 33, or 3 X 11.  Then next is 66, and then 99, and so on, (if you're working it out by hand, it's easier to count by 11's than 3's).  As it turns out, only multiples of the least common multiple (LCM) of 3 and 11, (33), are divisible by both 3 and 11.  Now, we just need to figure out how many copies of numbers divisible by 33 are between 1 and 2000, and then subtract that number from the total number of integers divisible by 11 we calculated earlier to get the answer to the 'divisible by 11, but not 3' question.

The divisible by 6, but not 4 questions is similar.  In this case, the lowest common multiple of 6 and 4, in other words, the smallest number they'll both divide evenly, is 12.  By removing all the numbers divisible by 12 from 1 to 2000 from the count of integers between 1 and 2000 that are divisible by 6, we wind up with the answer to "How many integers between 1 and 2000 are divisible by 6, but not by 4?"

Friday, February 19, 2016

Seen on the way to work

Yesterday was gorgeous here in San Francisco!