Sep 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Posting Infrequently This Month As I Continue Work on the Following Books:
Physics in a Couple of Lessons
A Capital Idea
The Mathematical Elephant
The Commerce Connection
Here's a few of the tentative covers:
September 2, 2010
I'll do a few problems first, add a couple for you to do, and then talk about the general method of solving problems like these ...
QUESTION 1 and FOLLOW-UPS
A restaurant occupying the top floor of a skyscraper rotates as diners enjoy the view. Two people notice they began their meal at 7:00 PM, looking due north.
At 7:45 PM, they noticed they were now facing due south.
At this rate, how many degrees will the restaurant rotate in one hour?
KEY: 180 DEGREES = 45 MINUTES
At this same rate of rotation, how long will it take for the restaurant to rotate 225 degrees.
What direction am I now facing?
If they rotate at 1/2 the speed, how long will it take to make one complete rotation?
QUESTION 2 and FOLLOW-UPS
If 12 vases cost $18.00, what is the cost of 1 vase?
KEY: 18 DOLLARS = 12 VASES
I spent $24 on vases, at $1.50 each. How many vases did I buy?
The vases above were discounted after Christmas - 1/2 off. Wanting to stock up, I went with $40. How many could I buy (assuming (ha) no taxes)!
QUESTION 3 and FOLLOW-UPS
A tank holds 5,000 gallons of milk. Each gallon of milk weighs about 8 pounds. About how many pounds does this milk weigh?
KEY: 1 GALLON = 8 POUNDS
Extend this problem (see example below if necessary) to find out "How many tons does this milk weigh?"
After empting the tank into a truck, it's found the milk weighs 38,000 pounds. How many gallons of milk are in the truck?
A STOPPING POINT
First, a note on the General Method, and then on to more difficult problems!
What does 40 meters equal?
5000 gallons equals what?
This goes on the far left.
You figure out how to cancel units, working your way left to right.
What's the stopping point?
When your units in the numerator (or both, if it's a rate or fraction) is what you're looking for!
And this can show up in a lot of places, particularly if you get stuck!
Consider our building problem on Wednesday, where the typical ramp-slope 5 feet-rise for every 100 feet-run.
What's the distance of the ramp from the building, when the building-door is 2-feet off the ground?
I draw a picture, but get stuck.
Let's apply our new method.
I have 2 feet-rise.
Work left to right, using what we know are "equal". We've been told
5 feet-rise = 100 feet-run. Let's use it!
QUESTION 4 and FOLLOW-UPS
How many inches are in 40 meters?
Using the same equality of 2.54 cm = 1 inch, how many feet are in one kilometer?
144 millimeters = how many miles?
QUESTION 5 and FOLLOW-UPS
Traveling at approximately 186,000 miles per second (speed of light), how many miles does a beam of light travel in 2 hours?
How far is one light year?
When Bob Beamon broke the long-jump record in Mexico City in 1968, it was announced at 8.9 meters.
How far is this in feet and inches?
BONUS ... YOU CAN DO BOTH! REMEMBER RULE 1:
FOCUS - AND GET SOMETHING ON PAPER!
QUESTION 7 and FOLLOW-UPS
The measure of each interior angle of a regular polygon with n sides is
What is the measure of each interior angle of a regular polygon with n sides, in radians?
September 15, 2010
The next presentation will be on Waclav Sierpinski. “Mr. Sierpinski” will provide some updates on what we’ve come to know as “Sierpinski Triangles” and “Sierpinski Gaskets”. Please – join me in welcoming Mr. Sierpinski.
HOW SOMETHING CAN BECOME NOTHING
MAKING SENSE OF THE NEW MATH OF SET THEORY
Good morning. I’m very happy to be here to talk about items that interest me – and at the same time, bother me.
My triangle is one of them.
To briefly summarize the general ideas I’ll be talking about this morning, consider a normal, everyday triangle.
We want to take out a middle triangle by bisecting each of the sides. This results in our single triangle broke into four equal triangles, of which we’re going to remove the center triangle.
Of course, once we’ve done this, we’re going to continue doing this, with each of the remaining (darkened) triangles. This is called “iteration”.
THE ITERATIVE PROCESS
Continually Removing The Middle Triangle of Every Triangle That Had Previously Been Left
Let’s take a look at what the iterative process looks like, after running through seven iterations:
September 16, 2010
PATTERNS TAKING PLACE IN THIS ITERATIVE PROCESS (with the area of the triangle =
THE APPLICATION OF THE FORMULAS TO ADDITIONAL ITERATIONS
This looked, to me, like the “total area removed” was approaching 100%, while the “total area left” was therefore approaching … none? How could that be? After all, at each iteration, I’m only removing ¼ of the space?
So I thought: OK – let’s be rigorous in our math here, and find the algebraic solution to the problem.
To be clear, instead of looking at “n” iterations, I want to consider ∞ iterations to determine if there is a limiting factor to how much area I’m removing.
That is, I want to see whether this is the case – or not: Does:
Which equals, when I expand the series by a few terms:
What do I do with this? The latter part of this formula is really a geometric series – a series with a common ratio between each term – and I know how to find the sum of a geometric series:
This, integrated into my earlier formula, gives:
Zero? No area left?
"The Mathematical Elephant in the Living Room"
September 17, 2010
Our first presentation will be my interpretation of the work of Michael Barnsley, and the title of his presentation is “2,500 Years Too Late: Cleaning Up the Mess of Zeno”.
2,500 YEARS TOO LATE
CLEANING UP THE MESS OF ZENO
Good morning. I’ve very happy to be here today to talk about a topic dear to my heart – fractals. But rather than start with “fractals”, I want to tell you a story about a name famous in the annals of mathematical paradoxes: Zeno. You all know the name.
“THE PARADOX” PARADOX
Zeno of Elea is well known from ancient times for formulating interesting paradoxes regarding motion. Perhaps his most famous paradox is the “Tortoise and the Hare”, where he purportedly demonstrates a slow-moving tortoise, if given a head start, can never be overcome by a speedy hare.
How can this be?
Well, we’re told, surely the hare, in pursuing the tortoise, must move half the distance to the tortoise.
But in the time it takes the hare to move this distance, the tortoise itself has moved. Hence, when the hare again attempts to overtake the tortoise, it must again move halfway to the tortoise. Clearly, every time the hare moves halfway, the tortoise has moved, albeit slightly.
Hence, we’re told, the always-moving tortoise will never be overtaken by the rapidly-approaching hare, which must infinitely make up “half-distances”.
Of course, we know in reality this is ridiculous. We know in reality the hare does overtake the tortoise, just as a fast-moving runner overtakes the plodding jogger. Why did Zeno himself not recognize his logic did not conform with reality, and wonder himself where he went wrong?
Richard Feynman, the great physicist, verbalized this wonderfully in “Surely You’re Joking, Mr. Feynman!”. While at Princeton pursing his graduate degree, Feynman was talking with the mathematicians, who claimed you could cut up an orange into a finite number of pieces, and, putting it back together, arrive at something as big as the sun.
“Impossible”, claimed Feynman.
When given the mathematical explanation about cutting the orange, Feynman interjected: “But you said an orange! You can’t cut an orange peel any thinner than the atoms.”
When given further mathematical justification about being able to cut continuously, Feynman concluded, “No, you said an orange, so I assumed that you meant a real orange.”
Indeed – dealing with reality.
A GEOMETICAL PARADOXICAL PERSPECTIVE
Rather than deal with this specific paradox, let’s modify the behavior of the tortoise, and say he doesn’t move at all. What of the course of action of the hare? How can we visualize it? With the ending point stable, we need only graph the halfway point between the ever-changing starting point and the stable ending point. Let’s see:
This gives me a visual idea of what’s going on, but now I’d like to change the rules a bit. Rather than continuing in the same direction, always halving my distance to the goal, what would happen if I go halfway, and then wherever I am, I choose randomly: to continue on in the same direction, or turn around, going in my new direction half the distance to the starting point in that direction. What would this look like? Let’s graph a few points to get an idea:
This new rule seems to have me going back and forth to many, many different points. What happens if I continue the pattern for a 1,000 movements? Let’s see:
As expected! I eventually hit every spot between the starting point and the ending point.
SHIFTING TO TWO DIMENSIONS
I’ve focused on one dimension. What happens if instead I can go in two dimensions? What happens if I have a square? My intuition tells me if, in one dimension I eventually landed on every point on the line, in two dimensions I should cover every point in the square.
Carrying out the procedure, I get exactly what I expected – a completely filled square:
This seems natural and intuitive: if I bounce around randomly within a certain area, eventually I will hit every point. As this was confirmed by both a straight line and a box, I suspect every shape follows suit. To be safe in confirming my theory, I decide to try the method with a triangle ...
Knowing what you know about the process, and the results in one-dimension and two-dimensions (square), what would you expect to see with the triangle?
"The Mathematical Elephant in the Living Room"
September 18, 2010
This seems natural and intuitive: if I bounce around randomly within a certain area, eventually I will hit every point. As this was confirmed by both a straight line and a box, I suspect every shape follows suit. To be safe in confirming my theory, I decide to try the method with a triangle, and am astounded by the result:
How can this be?
This makes no sense, particularly given the solid straight line and the filled square earlier. But this was the result of moving 50,000 times. Let’s “slow it down”, and capture the results to see how this took place:
RANDOM STEPPING IN TWO DIMENSIONS
From 10 to 50,000 Steps
A Series of Movies Brought Together ...
September 19, 2010
A Major Re-Write Part I
September 22, 2010
A QUICK START
THE SCATTER PLOT
I like spirals, like the one above. I’ve seen many like it on the internet. Do a google image search on “spiral” and you’ll see many more. Lots of different ones.
How is this made? And how might I do this? Let’s try to figure out what’s going, first.
The points being plotted are merely numbers, so they must represent a distance. But what kind of distance? Let’s assume this is the distance from the origin (0,0). Maybe it’s this:
Assuming I’m right, is this enough information for me to graph the data? As I move the distance vectors … wait a minute! That’s what’s missing! How much is this moving? What is the angle that changes for each line drawn?
Since I don’t know, let’s assume it’s something simple: 10˚. Is this all I need? Let’s plot the data and see what I get. But how?
Let’s take the degree shift of 40˚ and a distance of 5. What can I do with these two facts? In my triangle below, I'm looking for distances ‘x’ and ‘y’ to plot. If I had these, then I would have the coordinates to plot any of the vectors. That is:
An Important Note
Let’s briefly stop and look at that last diagram. In this booklet are several such logic diagrams. You must pause and read them – out loud if possible, silently if you must – but read them – and slowly. An example:
This is read as follows:
IF I made three baskets and two free throws;
AND IF: Each basket was worth two points, and each free throw worth one point;
THEN: I scored eight points.
It must make sense to you before you move on. GO BACK AND READ THAT LOGIC DIAGRAM ONE MORE TIME – SLOWLY!
Finding ‘x’ and ‘y’
Back to business. How do I find ‘x’ and ‘y’? Do I know anything about my triangle? Trigonometry plays a role here. I know the cosine and sine functions have a relationship between the adjacent, opposite, and hypotenuse sides of the triangle. Since I know two of the three facts, I can solve for the third. That is:
Applying these formulas to my data earlier, I have the following coordinates I can plot:
After looking at these figures, however, I see funny things happening. These aren’t reasonable. The formulas, I’m certain, are correct, yet the results are not. For example, I know cos(90) = 0, but above it’s not showing anything close to that, so something’s wrong somewhere. What’s going on here?
A bit of research reveals Microsoft Excel does not perform trigonometric calculations using angles, but rather by radians! Therefore, to properly use my formulas, I must convert all degree measurements into radians.
What are radians – and how do I convert degrees to radians? Let’s find out - next time ...
A Quick Look at the Seasons
September 23, 2010
I once thought I understood the reason for the seasons: the orbit of the earth around the sun.
After all, this is the image ordinarily displayed as the orbit of the earth - moving elliptically - about the sun:
Then I realized this is a gross exaggeration of the orbit of the earth about the sun. Sure, the actual path is elliptical - barely. Here's the relationship between a circle and the earth's orbit.
It's not the "elliptical" orbit, I found out, but instead the axial tilt of the earth responsible for the seasons.
All is well - or so I thought.
In my mind, however, I still have the solstices taking place when the earth is at the greatest distance from the sun, the equinox' when the earth is at the closest distance from the sun.
And I don't think that's right.
And I've been sitting in my office spinning a globe around me trying to find an appropriate definition of "equinox" and "solstice".
I think I've found them.
Suppose I take that figurative "stick through the earth" that's 23.4˚ off center and drop a plumb line from the top of it. I connect the two lines to form a triangle. How much of the triangle do I see at the equinox? The whole triangle!
What about the other three seasons? What do I see? Take a globe and do the experiment yourself!
And one thing you'll notice: you must hold the tilt of the earth constant as you simulate the orbit. That is, the "stick" is pointing in the same direction the entire orbit.
More to come ...
An Article to the Mathematical Philatelic Society
September 24, 2010
The Beauty of Line Designs
"Line Designs" have an interesting story. To many, they are merely neat designs done by young kids.
That's true. Here are several:
To others, they can serve a much greater purpose.
Mary Everest Boole, wife of famed mathematician George Boole and known by many as the origin of this type of activity, said the following regarding this process:
“The beauty of some of the designs is unquestionable; and there can be no second opinion about the value of the method, as training, from the point of view of geometry as well as from that of art. What is not quite so obvious at first sight is its bearing on the training of the unconscious mind for science. Without the slightest intellectual strain it puts the children through that normal sequence of orderly attention to classification and detail, interspersed with nodal points of synthesis, which may be called the very breathing-rhythm of the scientific discoverer.
But to make this exercise of any use there must be no copying from diagrams; the value of it depends on the child evoking a curve, watching it growing, under his fingers, from mere obedience to a law … and beauty has resulted, not from understanding but from obedience … the act of evoking a curve ‘out of the everywhere into here’, by simple obedience to a rhythmic law, lodges an impression on the unconscious mind which will be ready to surge up in ten years’ time."
Clearly, what we as adults consider “good” qualities are necessary conditions for a good activity, but are they sufficient? Ms. Boole addresses ‘orderly attention’ and ‘classification of detail’, leading to ‘beauty’ as the result, with a particular benefit the training of the mind for future excellence. How is this process captured in a rubric concentrating on product? To emphasize this point, Edith Somervell said the following about the “process versus product” dilemma:
“Beautiful curves are produced by a process so simple and automatic that the most inartistic child can succeed in generating beauty by mere conscientious accuracy; and the habit of doing this tends to produce a keen feeling for line. It has also been noticed in some cases, where clean, pure, and strong colour has been used, that a remarkable sensitiveness to colour relation has grown.” “The results obtained by a child, of exquisite curved and flower forms on the ‘back’ of his card, by faithful obedience to a dull little rule in making straight stitches on the ‘front’, is of the nature a miracle. It should, therefore, be hardly necessary to insist that the less said the better, when the little worker produces anything especially beautiful or unexpected.”
Here are several from Ms. Somervell's book, "A Rhythmic Approach to Mathematics":
Is pedagogy, however, where the usefulness of line designs begin and end? An interesting article on Pierre Bezier, inventor of the Bezier curve, claims otherwise:
"In today's computer aided world, the applications (of the Bezier Curve) are numerous. Not just in obvious applications like computer graphics and animation (animation often uses Bezier curves applied to the fourth dimension to describe smooth motion), but also in robot controlled manufacturing. The Bezier Curve changed the world."
And the Bezier Curve is really just a line design ... here's a 3-point Bezier Curve ...
What has all this talk of line designs to do with stamps? I recently came upon this stamp on the work of Emma Kunz, who lived in Switzerland from 1892 - 1963:
Ms. Kunz' used design, as I understand, as a healing tool. To anyone who has actually drawn these designs by hand, there is a rhythmic feeling - a soothing feeling - to the process. I don't pretend to understand how she used them, but I do marvel at their beauty! Other designs by Ms. Kunz include:
A Tribute to a Remarkable Man
September 27, 2010
I'd like to pay tribute to a remarkable individual, one who blended intellect with action, with significant consequences not only immediate but long-standing.
He was born on December 14, 1896, in California, lived for a time in Alaska before returning to California, living in Los Angeles.
The new century brought in the Airplane. The Wright Brothers. Flight.
This young man, aged 13, attended the 1910 Los Angeles International Air Meet, what the LA Times then called "one of the greatest public events in the history of the West":
It's difficult for us living now to understand the significance of such an event. Arthur C. Clarke, in writing the book "3001", talked of humans being transformed a thousand years into the future, and wondered if the 20th-century man would be astonished - intellectually - by what they found. He doubted it. Modern man is used to the radio, TV, the DVD, internet, cell phones, flight, transcontinental travel, space travel. But the 19th century man transformed into the 20th? Yes. Everything - technologically - was new. But the 20th century man into any part of the future? He doubted it. There wouldn't be that great a paradigm shift, intellectually, whatever changed.
When this individual was born, of course, none of this was around, and most would not be around for more than half-a-century. The country was mesmerized by flight.
An 11-day air-show! A quarter-of-a-million people in attendance! And look at the variety of planes in the promotional flyer above! Indeed, in his autobiography, this individual wrote:
"What was interesting to me was to see the radical differences in the construction and design of the machines; yet, amazingly, they all could fly."
After graduating from high school, he studied engineering in college, before taking a leave to enlist in the Signal Corps Reserve as a flying cadet.
It was 1917. World War I was raging across Europe.
He became a flight instructor. A gunnery instructor.
The "Great War" ended, he returned to complete his degree.
And he continued to fly.
In a article I wrote titled "Johnson County, Kansas - The Aviation Capital of the World", I laid out - chronologically - "breakthroughs" in the aviation community, three centered right here in Johnson County, Kansas:
To this list, I'd like to add three more breakthroughs. Yes, Calbraith Rodgers was the first to make a transcontinental flight, but his flight took 49 days! In fact, the quest was in response to a challenge by published William Randolph Hearst to make it across the country in less than 30 days!
This individual, in September 1922, became the first to fly across the country in less than one day - 2,163 miles in a total elapsed time of 22 hours, 30 minutes.
DeLlyod Thompson was the first to perform 8 rolls. Stunts were inevitable. Loops were exciting. "Inside" loops, that is, where the pilot is on the inside of the circle, like a car performing a loop on a roller-coaster ride. But an "outside" loop, where the pilot is facing out? Unheard of. Undone.
He did it - in 1927.
He set a speed record of 232 MPH in 1925.
He would break this record in 1932, reaching 296 MPH.
And after having won the three big air-racing trophies of the time, he officially retired from air-racing.
I started this tribute with a note about the intellect as well.
Remember, not only was he a flight instructor and a gunnery instructor, he was before this going to school as an engineer. After World War I, the army gave him two years to complete his Master's Degree in Aeronautics.
He enrolled in MIT, and completed the program in one year, becoming the first person in the United States to receive this degree. Having done this in one year, he spent his second year earning his Doctorate!
Our individual is one remarkable individual.
But I'm just getting warmed up!
As a test pilot setting speed records and performing stunts, one of the things he realized was the inability of the pilot - perceptually - to keep up. It was easy to become disoriented, particularly as speeds increased - and they would, he knew. Further, weather conditions hampered one's ability to fly, as did the mere setting of the sun.
Again, we're use to 24 / 7 / 365 flying, a sky perpetually filled with blips. Pilots able to set the instruments to automatic and let the computer take over.
There were no computers, of course, in the first half of the 20th century.
In fact, there were few instruments! Flight was done by sight. "Contact" flying.
He knew this was a problem. He also was in a unique position to do something about it, with his background of test pilot and Doctor of Science!
He designed many of the instruments that still exist today in the cockpit of an airplane.
Of course, not only did he design these, he was the first to fly - takeoff, fly, and land - by instruments alone!
He was in his mid-30s now, and had already accomplished a lifetime's worth of achievements! Retired from the racing-season, inactive militarily, he was now a vice-president at Shell Oil.
Retired, but still paying attention.
And something caught his eye. One racer now was winning more than others. Why was this? Observing his engine, some noticed higher-compression pistons, and had them installed in their planes. The pistons in their planes, however, burned out.
Why was this?
This man found out. It wasn't the pistons by themselves. It was the pistons and a new type of fuel - a high-octane fuel.
He knew more powerful fuel would be necessary as flight became more popular in the United States. And he knew a more powerful fuel would help military craft.
But the military was against it. Their thought was to have a single fuel during wartime to simplify supply problems in wartime.
And Shell Oil wasn't on board, either. It would cost millions and millions of dollars for Shell Oil to retool their machinery. Could he really make the argument this was the way to go?
Tests were run. 15%-30% more power. 15% better fuel efficiency. He made the argument - well. He said later:
"I think there are two great benefits of an advanced degree: one is the increased knowledge and greater capability that you have, and the other is the prestige it gives you with your associates, particularly those who also have advanced degrees, so I took a calculated risk in pushing vigorously for what I believed was not only good for Shell but good for the military as well. I believed in 100-octane fuel, but I would not have been able to sell the idea had I not had the educational background that the Army had given me."
The Army subsequently ordered all aircraft engines built after January 1, 1938, but designed for 100-octane fuel.
Shell Oil had gambled as well. They went from supplying 20 million gallons of airplane fuel annually to 20 million gallons of high-octane fuel daily with the onset of World War II.
What would our planes have done, with 30% less power? 15% less fuel efficiency? We can only guess.
Which brings us to World War II. December 7, 1941. Pearl Harbor. The Japanese have attacked our Pacific fleet, and sought a campaign of conquest in Southeast Asia along the Pacific coast.
Months of losses - US losses - following the attack at Pearl Harbor sank US morale. What could be done to counter these losses? These defeats? To boost morale? To put the Japanese on notice they were vulnerable?
Not likely. Though through luck the US carriers had not been at Pearl Harbor that sad December day, they could only launch fighter planes with a limited distance. With this limited distance, the carriers would have to get closer to Japan. Proximity to Japan, of course, would put them at risk.
And then the idea came to load the carriers with bombers. It had never been done. Bombers needed long run-ways to get their massive weight up. And they needed even longer run-ways to land.
It couldn't be done.
But what if it must be done?
They found a way to have B-25s take off from a distance equal to the length of the new carrier being built - the Hornet.
They decided to have the bombers drop their payloads and fly on to China, then at war with Japan.
Massive training. All volunteers. It was a dangerous mission.
And the military had brought back our individual to lead the team.
Now 44, he was old enough to be the father of most of the other flight teams and crew. But he was the man for the job.
After months of training, they steamed towards Japan. Secrecy was the crucial element They needed to get close enough to Japan before being spotted. Should their position be radioed to the Japanese coastline, not only would that put the planes at risk, but the carrier and accompanying ships as well.
And they were spotted.
800 miles out.
The order was given: launch away!
The seas were rough. 35' foot waves. Heavy winds. And all eyes were on - yes, our man! He was initially brought on board to lead the team only, but he would have none of that. He was lead pilot on the first plane off! Yes, all eyes were on him, because every pilot knew if he didn't make it, nobody would.
And he did!
They all did! And after they dropped their payloads, they flew on to China. Because of the extra distance they needed to fly, they were unable to get to their pre-determined landing points. They had to bail out over China, letting their planes crash.
After bailing out, he and his crew found the wreckage of the B-25. He sat sadly by his plane, believing the mission a complete loss, believing he would be court-martialed when he got back to the US.
He wasn't, of course.
He was awarded the Medal of Honor for the Tokyo raid, and ultimately would receive the Presidential Medal of Freedom as well, the only person to be awarded both the Medal of Honor and the Medal of Freedom, the nation's two highest honors.
And the mission did exactly what it was intended to do. Morale rose immediately, in the military and in the country. The Japanese, once believing they were invulnerable, now knew otherwise.
And they must act as well ...
They had their rendezvous - at Midway.
The course of the Pacific Campaign was now altered - and the course of the War changed.
This is the famous "Doolittle Raid", of course, made more famous by the film "THIRTY SECONDS OVER TOKYO":
This is how most people know of the name "Doolittle".
Yes, Jimmy Doolittle, record-breaking pilot, MIT-graduate-in-Aeronautics, Instrumentation designer, fuel-proponent, and finally, the Doolittle Raid.
Jimmy Doolittle died this day, September 27, 1993.
Well done, sir.
September 30, 2010