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February 1, 2010
The newspaper told the story of the moon appearing 14% larger in the sky, the confluence of two occurrences:
1. full moon
2. moon at perigee
A huge moon - closest to us - makes the moon look bigger than "usual". Take a quarter and put it right in front of your eye. Move the quarter to arm's length. It appears smaller.
Such was the instance with the moon.
But 14% bigger? Where did this number come from?
Let's do some work. To do some work, we need some basic information. Orbit distances. Moon size. Let's get these:
Perigee 363,104 km
Apogee 405,696 km
Orbital period 27.321 582
Average orbital speed 1.022 km/s
Mean radius 1,737.10 km
Since the numbers are so big, let's start with some manageable - intuitive - numbers to build a model.
Let's suppose there is a ball 10' wide, 20 feet from me. This same ball is moved back to 40'. How much bigger does the close ball appear to me?
Drawing the picture, I realize a couple of things. I'm drawing in 2-d, but a sphere is 3-d. When I look at a sphere, is it the same as seeing a 2-d circle?
Let's cross that bridge in a bit.
A first thought in looking at size differences is looking at the change in viewing angles. My viewing angle for a close object is huge, and it diminishes as the object recedes. What happens when I model this?
Let's look at the closer object first:
What happens if I perform this same logic on the ball at a greater distance, and bring all of the data together?
One interpretation: the closer object appears approximately twice as big as the distant object. Here, we doubled the distance, so a natural conclusion is there is a direct relationship between viewing size and distance.
That's just a start. A reasonable guess. Let's plug in our moon data and see what happens:
If our logic holds above, the ratio of the closer viewing angle to the further is 1.26. It's not our 1.14 we're looking for, but it's a start.
For example, the article notes 14% greater than the distance of the moon from the earth - on average. Here, we're using the extremes.
A little research shows the average distance (center of moon to center of earth) to be 384,403 kilometers. Plugging this in leads to a ratio of 5.7%.
I doubt this is the right track, so let's try another. Let's look at the idea of "viewing area" from a different perspective ...
and, during the course of this investigation, look at "viewing area", or "larger", as initially reported in the paper, a bit differently. The area of a circle, for example, is given by:
With the area dependent on the radius - squared - might this account for the "14% larger" report?
More to come!
Thoughts welcomed as well!
February 2, 2010
THE MANDELBROT SET AND THE JULIA SET
In calculating the Mandelbrot Set, we set Z0 = 0, and ran a program with c moving across the grid. At each c, we ran through a series of iterations to see if the distance exceeded 2. If it did not by a prescribed time, we said c was in the Mandelbrot Set. We plotted all of these points.
What would happen if, instead of having c span the grid, we fixed it? What would we do with our Z0? Our starting point could be the points of the grid. That is:
This algorithm gives us Julia Sets, named after Gaston Julia. Let’s play around with this and see what happens, using randomly chosen values of c:
The Julia Set Map
As you can see, for each value of c, a different Julia Set is created. What would happen if we captured the Julia Set for each value of c, and plotted this?
I’ve rotated the graph 90˚ clockwise, and inverted the colors, to see more of the Julia Set Map. Remarkably, it resembles the … Mandelbrot Set!
A VISUAL SUMMARY
The Mandelbrot Set. Remarkable. So much there – from so little formula.
Here are a number of randomly chosen images from the Mandelbrot and Julia Sets.
All of this, and much more, available in ...
The Archimedean Chats
February 3, 2010
THE CASE OF THE EXHAUSTED THEORY
Common Sense is Often Not Common Practice
The scene: lunch, a small deli, in Syracuse, Sicily. Mike Mason (eating lunch) and the great Archimedes (striding across the floor):
Mr. Mason! How good to see you! I see you’re dining alone. May I join you?
Mason: (Shoving some papers to the side)
It would be my pleasure.
It’s been a while since our last encounter. My ideas on buoyancy have been greatly enhanced! What are you working on there, my friend?
I’m doing some work predicting the population of rabbits and other animals, and frankly, having a bit of trouble with it.
What’s the issue?
I’ve created a whole system’s diagram of what impacts rabbit population. Rabbits eat grass, but wolves eat rabbits. More wolves – less rabbits. More grass, more rabbits. And so on. But right now I’m trying to quantify the population over time, merely focusing on birth and death rates. For example, if no baby rabbits live, the population, of course, does not live. However, if all baby rabbits live, eventually there are too many rabbits, the environment cannot sustain them, and likely the population again does not survive!
So there’s some point in-between that surely does the job.
That’s what I thought as well, but there’s some strange things I can’t explain. Let …
Nonsense. It’s somewhere in-between the two ends. In fact, this sounds similar to something I’m working on.
Mason: (seeing it’s no use talking about his own issue anymore)
What are you working on, my friend?
I’ve been working on a problem involving the circumference of a circle.
And this is exciting? Are you going to tell me there’s a “Eureka” moment wrapped up in circles?
After the firm warning from the Judge in my last outburst, I’ve found it’s best just to yell “I HAVE FOUND IT” to myself.
Let’s get to it, then.
The other day, I put a pole in the middle of my yard, attaching one end of a string to the pole, and holding taught the other end. I then walked about the pole, making a circle.
Sounds plenty boring thus far!
I thought so, too, until I asked myself: “How far have I walked?”
I don’t understand.
You see, had I walked in the shape of a square, I could figure out how far I had walked. For example, in this square below, the length of each side is four, and the square, having four equal sides, then has perimeter 16.
And a square as follows – the perimeter 24.
So how far have I walked in this situation?
Off the top of my head I don’t know, but there must be some easy answer!
I thought so, too, but I’ve been unable to find one. This has given me more trouble than the King’s Crown!
Who, by the way, is your client in this case?
I’m my own client – just me, trying to find answers to what should be simple questions.
I thought only a fool employs himself as his attorney … but by the look in your eyes, I trust you’ve found the answer?
Not exactly – I’ve found a way to narrow the answer.
And what did you find?
I couldn’t figure out the relationship between the length of the rope and the distance walked, and it nagged me to no end. I drove my neighbors crazy walking around the yard, with different lengths of rope, tracing paths in the sand.
The shorter the rope, the shorter the distance. The longer the rope, the longer the distance.
That’s obvious, isn’t it?
Of course it’s obvious, but the question is, “How much longer?” What’s the relationship between the length of the rope and the circumference of the circle?
And you’ve found the answer?
As I said, I’ve found a way to narrow the answer.
What do you mean?
Look at what I said above. With the squares, I knew how to calculate perimeter. With the circles, I did not. Something I know – something I don’t. Let’s put them together. (Archimedes drew the following diagram on a napkin):
So if I can calculate the perimeters of the two squares, I know the perimeter (circumference) of the circle is somewhere in-between.
I agree with you, but that’s a mighty big guess. Look at the gaps between the squares. It’s like saying, “The perimeter is somewhere between 10” and 20”.
That’s true – if I stopped at polygons with four sides. But why do I have to stop at 4? What happens to the problem “gap” if I increase the number of sides of the polygon? (Archimedes methodically drew polygons with sides 4, 5, 6, 7, 8, and 9 sides).
The gap erodes. Very interesting. What do you call this method?
I see what you mean, but there’s something missing here …
Nonsense. Now, let’s stop talking about “Sandwiching the Solution” and order our Sandwiches!
Mason (to himself):
(Maybe if I wait until we’re eating – I’ll get a chance to voice my concerns).
to be continued ...
February 4, 2010
Continuing the moon-orbit issue from above, I've come upon an interesting - and not-surprising - issue: rounding area:
Here was the initial chart. The numbers shown indicate how many decimal places they've been rounded to. The above conclusion is added:
If our logic holds above, the ratio of the closer viewing angle to the further is 1.26. It's not our 1.14 we're looking for, but it's a start.
Here's the revised chart, this time with NO rounding:
Our "1.26" factor has been more than cut in half!
Of course, this very issue was the origin of "Chaos Theory", and the Lorenz Attractor ...
Let's move on, recognizing the error and fixing it.
Where we left off, our next plan of attack was to look at viewing area. Remember, we're really working in 3d, so you have to imagine, positioning yourself at "me", you're looking at a sphere. The radius of the closer sphere we know is 1,737 km. What's the radius of the more distant shape (potential viewing area)?
Using similar triangles, I can do the following:
Therefore, I have
And incorporating the area of a circle into our table, we have:
Still not there, and this new train of thought seems misplaced when I reread other articles on this issue:
The Scientific American article simply reported "14% larger". I read that as viewing area, and that's what we're trying to capture with this recent thread of logic.
However, other articles are reporting "14% WIDER" ... implying diameter. If this is the case, then area is removed from the equation.
The search continues.
However, note in the process of doing this, we've hit upon tangent, arctangent, similar triangles, radian to degree measure, the origin of chaos theory, ellipses and eccentricity.
Further, just playing with the data raises many questions about the shape of the moon orbit, lunar months, eclipses, etc.
Natural Selection, Variation, Reality, and A New Kind of Science
February 5, 2010
There is an exhibit at Linda Hall Library at UMKC right now, titled: DARWIN @ 200 ... THE GRANDEUR OF LIFE
The layout is a number of books written on nature, from 1500 - 1900. Absolutely beautiful books. Stunning.
Walking through the display, I got to thinking about evolution, natural selection, and variation. As such, let's revisit this section from
THE PROXIMATE EVENT ...
– 10 –
“What’s Going On Out There?”
Watching the movie “Happy Feet”, a question came to me I’m certain I never would have thought of before. The penguin, which once flew, changed into a swimming bird.
Why did it do this – and how did it do this? My initial guess was: there is some rule in the penguin “DNA”, and starting with some definite “code”, proceeds to “grow” into a penguin. Given certain circumstances in the environment, nutritionally, environmentally, the avoidance of predators, etc., it grows into an emperor penguin. That is:
But now, I’m faced with the situation of the penguin “evolving”? What is it that is changing? My first guess is this:
Somehow, the “initial code” changed. This must be the case; otherwise, the same “flying bird” would continue to grow. Why don’t we see that? How did it evolve? It must be the initial code changed.
Is that the only possibility?
I think back to my many simulations of “Rule 110” earlier. They all proceeded by the same rule, but, with different starting conditions, produced different patterns. How might this play into the “pattern of evolution”?
Let’s suppose there are 10 different penguins, each with different “starting conditions” in their DNA:
Let’s assume somewhere in the code is a marker responsible for something to do with how the wings are what they are, just as there’s a genetic market for green eyes and red hair. Let’s say it has to do with “heaviness” of feathers. Lots of light feathers means easy flight, lots of heavy feathers means flight is difficult.
There’s a lot of variation.
“Heavy-feathered” penguins, likely flying less, spend time poking their head in the water for fish. They find it! They live! Flying penguins don’t, and die. The “average” genetic marker changes, not because the initial code is changing, but because of variation in the markers, and environmental factors “selecting” certain attributes.
Is this what nature is doing when we say things are “evolving”? Is this what Darwin said? Was Darwin right?
Or is there another interpretation to all of this?
More work to go on how life evolves!
As I work through a number of things I see in nature, I realize I don’t even have a good definition of what “life” is. What is this thing called “life”? Here’s what the Online Encyclopedia Britannica says about “life”:
A great deal is known about life. Anatomists and taxonomists have studied the forms and relations of more than a million separate species of plants and animals. Physiologists have investigated the gross functioning of organisms. Biochemists have probed the biological interactions of the organic molecules that make up life on our planet. Molecular biologists have uncovered the very molecules responsible for reproduction and for the passage of hereditary information from generation to generation, a subject that geneticists had previously studied without going to the molecular level. Ecologists have inquired into the relations between organisms and their environments, ethologists the behaviour of animals and plants, embryologists the development of complex organisms from a single cell, evolutionary biologists the emergence of organisms from pre-existing forms over geological time. Yet despite the enormous fund of information that each of these biological specialties has provided, it is a remarkable fact that no general agreement exists on what it is that is being studied. There is no generally accepted definition of life.
This isn’t worth much to me, I think. I’ve got a pretty good idea of what “life” is – at least a definition that differentiates between “life” and “non-life”. Thinking back, I realize I’ve already said it!
Life exists when a seed contains both the rule and the information on what to become.
The same is true with the change I see in nature over time. “What’s going on out there?” has become my rallying cry! So many questions – and it’s great!
The whole story is here ...
February 6, 2010
These are some conflicts - dilemmas - I'm working on for a multiple-choice test. Most of these are real, taken from current events, books, movies, history. A few are others I've encountered. The issue of "multiple choice" I'm still dealing with, in addition to a few other topics. All changes will be included here ...
The Super Bowl now over, it may seem we have the benefit of hindsight. That's usually a good thing. But I wonder. The above cloud is from the perspective of the coach: "what should I do?"
Let's look at this tug-of-war from a different perspective: coach vs. fans. Fans, after all, go to see a great game. Their heroes. If some fans are like me, this may be the only game they see live that year - perhaps a lifetime. Sit the best players? Are you kidding me! I want to see the best players!
I'm at odds with the coach.
Which of the following four represents a reasonable place to put the entity "Happy Fans":
C. either "A" or "B";
D. neither "A" nor "B" - "happy fans" is not a good entry.
Which selection seems most appropriate regarding this cloud:
A. It is a good cloud.
B. It could be a better cloud. Item A does not follow.
C. It could be a much better cloud. The real issue here is "If I tell the truth, then I will get in trouble". This is not explicitly in the cloud.
D. This is not a good cloud.
Jamal is angry. He starts to yell at the reluctant Forrester, on why he should help Jamal.
He stops. Is this reasonable? What is the best course of action to follow?
A. Complete the cloud and show it to William.
B. Ask William to complete the cloud.
C. Focus on why William should help Jamal (Entry B).
D. Focus on why William will not help Jamal (Entry C).
What is a reasonable course of action for Mike to take - what is his "next move" ...
A. Construct the cloud regarding core problem / constraint analysis.
B. Insist the proper course of action is to address the system bottleneck.
C. Remain silent and let anger percolate.
D. Leave the room.
What is a reasonable course of action for me to take - what should be my "next move" ...
A. Call a time-out, raise the issue, and ask to construct the cloud regarding "necessary conditions" versus "obstacles".
B. Insist people who don't see this dilemma are wrong.
C. Remain silent and let anger percolate.
D. Leave the room.
There you sit. At the campfire. You're the head of your family. The debate is on the table. The cloud is real - and it's right in front of you. What is a reasonable course of action to take -
A. Insist your family is tired, and wants to get to California quickly. Take the shortcut.
B. Argue for the "tried and true" method: stay on the trail.
C. Challenge the assertion the Hasting Cutoff will lead to a quicker route. Demand proof of some sort.
D. Go to sleep and pray to God things will work out.
Many government agencies wrestle with this dilemma. Is it a legitimate one? Is this cloud reasonable?
What is the best statement regarding the state of this cloud?
A. The dilemma "profile / don't profile" captures the issue well.
B. Item "B" is weak, and needs some work.
C. The ellipse concerning necessary conditions to achieve the goal is well-placed.
D. All of the above are correct.
The cloud above is formulating in your head. You're wrestling with making a choice. As a possible investor, what's your next move?
A. Go with your instinct, and give Edison your money.
B. Go against your instinct, and give Tesla the money.
C. Have Edison explain how direct current can move profitably over long distances.
D. Invest in candles instead.
You're the decision maker, and the clock is ticking! What's a reasonable next move?
A. Shut down the terminal.
B. Keep the terminal open.
C. Act in accordance with the operations manual. If the issue is not covered, kick the decision up to a superior for clarification, explaining the conflict as best you know it.
D. Call in sick and wish everyone a Merry Christmas.
Context: I'm having a conversation with Joe, and Joe, not understanding, says, "I've got a clarity reservation". I hate this language. If you're not sure of something, just say, "I'm not sure", "I don't understand", "This doesn't seem right", etc. ANYTHING EXCEPT "RESERVATION"!
What are possible courses of action for me:
A. Verbalize my disgust for this type of response - follow-up with my response.
B. Take the time to diagram a cloud showing my dilemma in how to follow-up.
C. Answer the question, ignoring this issue, knowing it will continue to arise.
D. A and B only.
What of the following best describes the next course of action ...
A. Seek out a company in a different industry to see how they deal with this conflict.
B. Applaud the direction of the arrows, noticing the dilemma is inherent in many business models.
C. Do an analysis of membership versus rates to find the rate where profit is maximized.
D. More than one of the above is a correct answer.
OTHER CLOUDS TO SUBMIT / IN THE WORKS
"YOU'RE NOT LISTENING TO ME"
Objection on Capturing My Objection
You're a consultant working on an "after-event analysis" on the Die-Hard 2 dilemma above:
Shut down the terminal / don't shut down the terminal.
You're questioning the Director on his actions. The conversation follows:
Consultant: "I can understand you're caught pretty quickly in a dilemma."
Director: "You're telling me - a no one situation."
Consultant: So I'm clear: the officer wanted the terminal closed to conduct a thorough investigation. You wanted to keep the terminal open because of Holiday travel."
Director: "You bet - he (pointing at his boss) would have had my butt if I would have cleared the terminal."
Consultant: (Writing "keep holiday traffic flowing" under "need").
Director: "Wait a minute - that's not what I said."
Consultant: "That's what you just told me."
Director: "No - that's what you just told me! I told you I knew I'd get in trouble if I closed the terminal. Of course holiday traffic would be a big problem, but my problem was I knew I'd get in trouble."
The question here will revolve around capturing the "real" need of the involved parties ...
OTHER CLOUDS COMING
"My Store" Cloud
Worker on Assembly Line
The Archimedean Chats
February 7, 2010
Mason (with mouth half-full, while both eat):
Let me tell you about this modeling problem I’m working on. I was working on the idea of modeling the life – and death – of animals – and came upon this thing called “The Logistic Map”.
I’ve not heard of it – what is it – and how does it work?
How it works I’m still working on. It looks like this, however:
What is going on, here?
Slow down – I know the terminology is a bit odd, but it’s not even that I’m concerned about. However, for you, I’ll go into just a bit of detail.
I don’t know if I feel complimented or offended!
Here’s a start – I think. If I wanted to know how many of something – anything – I have right now, one thing it depends on is how many I already have. If I had 10 apples yesterday, there’s more chance I have 10 today than 0, right?
And I don’t have any right now, likely I may not have any tomorrow, right?
This is called “iteration”, where the result of one iteration is used to help calculate the value in the next iteration.
But this is no different than what traditional scientists and mathematicians do. You’ve just put a fancy word to it. For example, consider this sequence of numbers 2, 4, 8, 16, … Each one depends on the prior, and increases by a certain geometric or arithmetic amount.
I wasn’t trying to slip anything by you, my friend. Both are correct! But is it realistic to say, regarding a population, it grows by a factor of 2 all the time? Take fish in a pond. If they keep growing, there, at some point, has to be a scarcity of food. In most circumstances, a simple progression is unrealistic.
So there has to be a way to temper growth. On the other hand, if there’s too small a population, there’s an opportunity for growth!
Indeed – this is why the formula is laid out in the form:
The is tempered – or kept in check – or balanced out – with the factor .
So let’s put some numbers on the table. What are we doing here?
Let’s tart with a simple one: Let’s set the initial population value equal to 0.5, and let the parameter r = 1.00.
Not a very promising future for this species. But you said “for a value of r = 1.00”. I guess that means the behavior changes as r changes.
Exactly, my friend! You catch on quickly! Let’s see what happens when r = 2.00:
At least the populations over the years do not die out! That shows the model is a bit realistic, right?
Maybe / maybe not. What species has the same population year after year? Surely, there’s some fluctuation, right?
Let’s change the parameter, then. What happens when r = 3.00?
Very well –
Now we’re talking! The population does not die out, and there’s some variation in the population from year to year! r = 3.00 must be it!
Aren’t you a bit curious what happens when r increases even more?
You’ve got me there. What happens when we increase the value of r?
Here’s the behavior when r = 3.90. Look at all the variation!
I discern the pattern immediately.
That’s what I thought, too, until ...
Until nothing! The pattern is complete!
IS IT? STAY TUNED FOR THE CONCLUSION ...
February 8, 2010
Returning to the "nautical mile" for a moment ...
Where did I leave off?
In watching a documentary on meteorology, I was told the conversion factor in going from nautical miles to regular miles was 1.151.
Where did this come from, I thought? The investigation followed.
The "nautical mile", I was told, was the length of one minute of latitude arc length. Fair enough, I thought. How much is this?
Here's where things got tricky ...
Notice the nice, round number for the circumference of the earth: 40,000,000 meters. Where did this come? By definition!
In standardizing units, it was decided to let 90˚ equal 10,000,000 meters.
But the earth is not exactly round, so the distance of one minute of arc length varies, greater at the poles, less at the equator. The average was used as the definition of "nautical mile": 1,852 meters.
But there is a circumference of the earth in English measurements -
But we just said the earth is not a sphere - exactly. But what measurement is this mileage valid? The meridian going through Paris.
So we've got two metrics: 40,000,000 meters and 24,859.82 miles. What does one mile equal in terms of meters?
One mile equals 40,000,000 meters / 24,859.82 = 1,609.022 meters.
At last, the 1.151 figure!
And in the process, I've discovered the origin of the term "meter", the idea of navigation as the key to "nautical mile", of the sphericalness of the earth, latitudes, and a lot more.
For example, this measurement makes sense when you consider people in the air or at sea --- "mile" is meaningless. "Nautical miles" - distances between latitudes is everything!
But how does one calculate the distance between two points on a surface?
Stay tuned for Part 3!
A Moral and Economic Elephant in the Living Room
February 9, 2010
There's an elephant in the living room - and nobody is talking about it.
Let me explain.
The Olathe School System is facing a funding crisis of epic proportions. Early retirements, reassignments, and district cutbacks have only slightly addressed the loss in funding from the state. Kansas, like many states, has its own budget crisis. Cuts must be made. And the state, according to the school district, provides 74% of the revenue for education. Cuts were inevitable.
The district has gone on an information campaign to inform the public what the district is up against.
So far so good. As I've always said: "More information is good".
But it must be information. It must be credible. And it must provide the whole context.
Don't read too much into those last two paragraphs. Read my earlier journal entries and you'll see I simply like to verify what I'm told.
For example, watching the streaming video over the internet, I did a screen capture of this graphic titled: MORE STUDENTS - LESS MONEY:
Superintendent All accompanied this graphic with the explanation: "Funding has paralleled enrollment, until now".
This seems to make sense. The two lines are relatively parallel.
But I'm suspicious. My suspicion is always elevated when I see two y-axis' in play. To see why, let's do our best to extract the data from these lines, and put this data into a table:
As I suspected. Looking at the percentage change, finance has left enrollment far behind! Money has increased 40.9% over these 5-years, while enrollment has only increased 13.5% over this same period. But how can this be? What about the graph?
The graph employs a characteristic error when two y-axis' are in play: the scales are different: the Finance-Scale goes from $160 to $250, a 56.25% difference, while the Enrollment-scale goes from 20,500 to 26,000, only a 26.8% difference. Further, why is the minimum value on the enrollment axis so low?
Let's reset both axis so they each have a scaling factor of 1.50:
Just as the table told us. Revenue has not paralleled enrollment - it's greatly exceeded enrollment.
It may seem there's something very wrong here.
Five years ago, there was a certain relationship between operating revenue and enrollment. Presumably, five years ago we were fairly well off as a district. Since that time, finances has soared past enrollment, yet now we find ourselves in the midst of significant cut-backs? Had finances merely kept pace with enrollment over this period (13.5%), current operating revenue would now be $186 million. Yet it's $231 million! A $45 million difference!
How to account for this?
Let's remember operating revenue takes into account not only more students, but also teacher salaries. These go up annually. How much? Let's take a stab at calculating it:
Annual salary increases of 4.4%? Sounds reasonable. And this amount probably includes other factors outside of salary, so the annual increases are likely less than this.
Which makes sense. Finances should outpace enrollment. A simple system of one teacher and 20 students from year to year shows this: enrollment change is zero, but the teacher gets a salary increase, right? Finances outpacing enrollment. That's OK.
But the graphic is very deceptive. And the explanation accompanying the graphic is simply wrong. But we now see there's nothing improper with the proper explanation!
In search of truth ...
Let's move on.
One question from the audience echoed my thoughts: "We're being told to write our legislators. Write them what? What should we say?"
You see, the issue people are wrestling with is: "What can one do?" The school financing formula is set. It was set in 1992. Each district gets a certain amount of money per child. Even this is not entirely correct. Johnson County pays into the state far more than it gets back, while rural schools receive a windfall.
Can't local areas raise their own money to make up for shortages?
To a point, and, by law, no more.
Again, this goes back to the 1992 school financing law.
What about building up reserves over a period of time in case of economic crises like the one we're facing now.
The contingency reserve percentage is, you guessed it, set at a maximum: 6% of the General fund budget. The same law restricts this. You cannot hold more.
You can't set much aside. You can't raise much by yourself. You must depend on the state for the bulk of your money.
All in the name of ---- equality.
Democratic Governor Mark Parkinson, taking over for exiting Governor Sebelius, has mandated these massive education cuts.
Then State House Representative Parkinson, newly elected in 1990, was the sole Johnson County Republican who voted for the school financing bill - in 1992, reports Fred Logan in the Kansas City Business Journal, in his wonderful column last June ...
People watching the presentations rightly wonder, "What can one do while the Governor is making these cuts? Why can't there be more local control?"
Because he's against that, too.
Other Johnson County districts apparently recognize the obvious flaws in the school financing formula. For those of us who've lived here prior to the law, the flaws were immediately obvious - morally and numerically. Welcome to the party, Blue Valley and Shawnee Mission.
Our district, however, reportedly is silent on the issue. From Kansasliberty.com:
"Another Johnson County-located district, the Olathe School District, is appearing to be quieter in its criticism of the school finance formula. In its 2010 legislative agenda, Olathe does not urge for an overhaul of the formula but does advocate for having strong local control."
Why are we quiet on this issue? There's an elephant in the living room, yet we're pretending it's not there. This is the issue. Morally and financially.
Former "Olathe Schools First" Steering Committee member Mark Parkinson: this would be funny if it didn't impact my kids. But it does. And I want an apology.
The Archimedean Chats
February 10, 2010
The story concludes -
That’s what I thought, too. However, I was playing around with some numbers as things became more random. Look what happened with r = 3.78 and 3.88:
Exactly as I predicted. As r increases, the randomness increases. But what are these points on the far right of this graphic?
The behavior, after a certain period of time, seemed to oscillate between numbers, so I started to plot these as well. For lower values of r, there was no oscillation – the behavior settled down to a single number.
However, when r was large, the behavior resulted in a lot of points.
I see. But the theory still holds, and this new graphic just confirms it, visually, right?
Indeed! It is exactly as you theorized. But I want to make clear your theory. If we’ve got some randomness at 3.78 and more randomness at 3.88, and if your “Sandwich Theorem” from the Method of Exhaustion is correct, then we’d expect to see “in-between” randomness at 3.83, right? That is:
Here’s what we got:
Wait a minute – that doesn’t make any sense …
Mason: (not stopping):
And if I plot the results for 3.78, 3.83, and 3.88 side by side, I get this:
How can this be? There must be …
Mason (on a roll):
And If I plot not just these values, but run the program for hundreds of values of r between [3.78, 4.00], I get this!
Would you stop! Let me breathe! What is the meaning of this?
I have no idea, but you didn’t want to listen at first!
I suddenly know how Pythagoras must have felt when he was asked the length of the diagonal of a square with sides of length 1.
Now you’ve got me curious: what happens when you plot this for all values of r, and not just these between [3.80, 4.00]?
I’m glad you asked! It looks like this:
This is an unusual map. And your counter-example seems to have disproved the validity of my Theorem. So what are we saying here? Are you saying my “Sandwich Theorem” is wrong?
Maybe. Maybe not. Maybe something’s missing. It seemed to work for your circumference problem. However, it didn’t work at all in the population growth example. Something’s missing somewhere.
Let me take a stab at verbalizing it: I had a pretty good grasp on how polygons behave. There’s no surprises – that I know of. In circumstances like this, the Sandwich Theorem works.
But how can we be sure what “pretty good grasp” means? After all, we thought we had a “pretty good grasp” of how our simple equation worked, and we were wrong!
Maybe it’s enough just to have the knowledge we may not know what we think we know!
Maybe it’s the case “Common Sense is Sometimes Not Common Practice”.
Maybe. And I’ll get the tab!
February 11, 2010
The investigation of the "knot" has led to "nautical
mile" and "navigation", among many things.
There's a question on the table: you're at Point X on earth, I at Point Y. How far apart are we? How can we describe our relative positions? Latitude and longitude is one example ...
But I'm working in three dimensions to find the distance. None of the angles are intuitively obvious as to how to find them.
What can I do to get started?
Let's reduce it to two dimensions.
Let's choose two points on the circumference of the circle.
What I'm looking for is this arc length. How can I find it? Lots of ideas come to mind, now that I've got something on the table. For example, if I merely connect the points, I think, "surely there's a relationship between this chord and the related arc length."
In search of the relationship, I try a couple of examples, starting with the simplest ones: If my chord length is equal to the diameter - that is, it goes straight through the center of the circle, the corresponding arc length is half the circumference.
This, of course, requires I know the length of the chord in my example. I know this. This is merely the simple Euclidean formula for the distance between two points in 2-d:
Knowing the chord length, and knowing the formula for the circumference of the circles as C = 2pr = pd, I can substitute into the formula I found above to find the related arc length:
Fine. Given any two points, I now know how to find the length of the arc connecting the two on a sphere. The arc length of two points A, B on a circle with diameter d is given by:
Let's remember my convenient choice of points. I did this so I could get up-and-running quickly - to get the "lay of the land".
Most of the time, this is good. It keeps the intellectual ball rolling.
But there are drawbacks as well.
For example, above I needed the diameter, and I conveniently knew it. Suppose I didn't know it. Suppose my points weren't conveniently chosen? What then?
Let's look at this a different way, and set the stage for Part II of this brief analysis regarding the issue of finding the length of the arc connecting any two points on a circle, leading to the ultimate problem of finding the length of a geodesic between any two points on a sphere ...
A "QuickStart" Book
February 12, 2010
A "QuickStart" Book
February 13, 2010
A "QuickStart" Book
February 14, 2010
Applying Baseball to 2-Dimensional Cellular Automata
February 15, 2010
A NEW KIND OF BASEBALL
Applying Baseball to 2-Dimensional Cellular Automata
Stephen Wolfram, in his book A New Kind of Science, discusses the computational nature of the universe, the idea that complex behavior can evolve from simple rules and simple processes.
As I’ve modeled crystal growth in other applications, while writing these articles, the thought came to substitute a baseball for the hexagonal crystals used in my models.
What would this look like? Could one even tell if a baseball were being used, as the image is reduced by a factor of about one hundred?
So, when you see the images to follow, keep in mind all of the “cellular” elements – the really small things – are really copies of a baseball!
For example, here’s a baseball continually reduced. You can see the essence of the image pretty well even when it’s spilt from one baseball into eight.
Keep this in mind!
How does this work – in practice? Starting with a hexagonal grid of baseballs in the middle of the grid, we work our way outward according to a rule.
I’ll let you research these rules in another of my books. For now, recognize there are many different ones. Here’s how one rule might progress, step by step:
This is merely steps one through nine of a single rule. What happens if I let the rule run for a number of steps, and then capture just the last image? Here’s one:
Here are a couple more different rules, capturing just the last iteration. Following these two are many more, showing the complexity of a system of simple rules:
February 16, 2010
For those who don’t enjoy math, why wait for the profession to come up with another way to teach math. Time is too short! For those who do enjoy math – what else is there for us?
When I think of fractals, modeling, simulation, probabilities, chaos theory, etc., I wonder why this is nowhere to found in the curriculum.
What can be done – for both groups of people? What can we do – by ourselves?
A FIRST STEP
The Graphing Calculator, as it is used now, is the enemy. No, this is not an attack on technology. It’s an attack on a tool that has taken the joy out of learning, of programming, of understanding.
If the graphing calculator is the enemy, who can we call “Friend”? The blank spreadsheet in a program most everyone has access to: Excel.
But isn’t Microsoft Excel for keeping your checking accounts and business applications? What good is it for the student? What can possibly be done in Excel?
This series provides an answer to that question.
The goal of “The Geometric Mind”, Part I, is to whet the appetite of the student, to let them know good things are possible in this environment. It’s a learning environment where the novice can be up-and-running in a short time, not relying on the crude algorithms of the graphing calculator.
Part II of the “The Geometric Mind” series will explore how specifically to do all of this – plus more.
The presentation will be quick, explicit, with plenty of examples. That’s the nature of the game. Get up-and-running quick, and let the joy in learning commence!
Part III: The briefest of examples. By this time, the veil has been lifted, and the reader should be able to grasp possible ways the graphics were done. At this point, a "recipe" or "instruction booklet" is not needed. In fact, it's a hindrance to understanding what's going on. With an idea of "what to do" and "how to do", you're "off and running"!
THE GEOMETRIC MIND
In Search of …
So what exactly am I calling “The Geometric Mind”?
Is it the use of visual logic tools to understand the flow of the argument? Yes.
Is it the use of math in – and out – of these visual logic tools to understand the mathematical flow of the argument? Yes.
Is it the use of ambitious targets to diagram and understand what needs to be done to solve a problem? Yes.
Is it programming, animation, simulation, graphics? Yes.
Is it the use of all of these tools – plus many more – to better understand and solve problems?
What’s possible when any one of these intellectual “pistons” is firing? An example is to the right …
For more information ...
February 17, 2010
My life's had ups and downs ...
Quite a voyage it has been.
"Reincarnation"? Maybe -
Already I am rambling,
In doing so I cheat.
The fabulous stories of my life,
Of them I am replete.
You see I'm not a person
But just the same I love.
My goal in life's to warm five fingers.
Yes, I am a glove.
A leather glove, to be precise.
My color? Brown and glowing.
And when I do my job correctly ..
Fingers' blood keeps flowing!
But let me start right from the start
There's much more than cold weather.
Silk lining with a leather coat,
Skillfully stitched together.
A sewing machine is quite a marvel.
A needle and some thread.
The bobbin is the crucial link ...
Below - and shown in red.
Once complete, I'm wrapped and packed ...
A "Made in China" tag.
I'd like to take a moment here, as
Economic knowledge seems to lag.
A trade deficit cannot occur.
Despite the TV shows.
The Exchange Rate checks the rise and fall ...
And balances the flows.
Across the Peaceful Pacific -
I'm ready for my trip.
Destination? Long Beach Harbor
Aboard a container ship.
A gantry crane drops me ...
A B.N.S.F. train.
Across the heartland to Chicago -
Through sleet, snow, fog, and rain.
Unloaded and delivered
A truck straight to a store.
The second shelf from the top.
My price tag? $24!
And yet I do not sell.
A discount is applied.
25 percent? I wait ...
I'm both eager and petrified!
A woman picks me up.
She's clumsy, rude, obnoxious.
I'd rather sit upon this shelf.
Than deal with all that fuss.
A nice old man, and with a cane!
Pulls me from the shelf.
Tenderly he tries me on.
He feels just like myself!
A home! My home!
I hear his name is Max.
He takes me to the checkout stand
"$18 bucks - plus tax!"
He wears me around town,
I keep his fingers warm.
Through thick and thin, warm and cold,
A blizzard and snowstorm.
His granddaughter came to town.
A tournament: Volleyball!
To the auditorium ...
Games - wall to wall!
He walked in through the door.
Dropped me in his pocket.
But something didn't feel quite right
I felt me start to slip.
Down and down I sensed me fall
I looked in horror as he went ...
He didn't even miss me!
People walked on by.
"Please pick me up" I pled.
But as the moments came and went.
My destiny? I dread.
People gazed and gawked
Tall, short, big, small, and fat.
I started to get an attitude!
"What are you looking at?"
A man was talking on his phone.
And walking like a clod.
Dirty boots! I braced myself!
Upon me! Yes, he trod.
Finally a workman,
Picked me from the ground.
"Rescued!", I thought, until I saw
A box titled: Lost and Found!
He tossed me to the box
With others of my ilk.
Surely I'll be rescued ...
Remember ... I'm made of silk!
I waited and I waited ...
For a searching hand above.
But not many have a need
For a single, dirty, glove.
And reality struck me.
Resignation replaced hoping.
I would not be chosen.
And that could mean one thing.
To the land fill for years and years
I would not decompose.
I started reflecting on my life ...
With all it's highs and lows.
And then I heard a voice.
Break through the arena air.
No, it wasn't Max I heard,
But I could hear it clear.
"Why take these to the dump.
Though an odd lot, I agree.
Perhaps some kids who don't have much.
Would wear these gloves with glee!"
"We care they don't match ...
But who are we to say?
A child with two warm hands.
Can go outside and play."
And yes, that is my story
A child wears me now.
And when we go outside to play ...
I smile, when time allows.
February 18, 2010
Recalling our earlier discussion regarding how to find the distance of two points
on a circle, moving about the circle, we found the distance to be:
As I write this, I know something's not right - or, at a minimum, something is missing: I can go clockwise around the circle - or counterclockwise. There should be two answers, yet I only have one.
Something to remember ...
I wanted to move on and consider this problem from a different perspective. Suppose I have these two points on the circle, with a common point in the middle. Down the road, I want to think of this middle point as the center of a sphere.
What can I find out about - well - anything? Let's see:
What can I say about the sides - or the angles - or the relationship between the two? A starting point, I've found, helpful for problems like this, is to drop a perpindicular. Why? It provides a common link between two different angles. Let's see where it leads us:
Drawing a perpindicular from B to b, and A to a allows me to generalize the "Law of Sines" ...
Let's suppose I knew one of the angles --- how would I find the other two? Easy enough:
So what? I don't feel like I'm progressing much, and part of the problem may lie in 2-dimensional representation versus 3-d. All of the problems I'm encountering above I seem to be able to handle with the math I know, yet I know something's missing regarding latitude and longitude and 3-d.
A simple image of the earth and a simple triangle may provide some help. You see, this simple triangle shows in spherical geometry the sum of the angles in a triangle may not be 180˚!
What are the implications?
We'll see in a bit.
For the meantime, I'm going to find how others have gone about finding the distance between two points on earth, and go from there.
Integrity, Thoroughness, Progress, and Economics
February 19, 2010
John Lienhard is a professor of mechanical engineering and history at the University of Houston. He also hosts a remarkable program titled "Engines of Our Ingenuity". He has traced the word "engineer" to the Latin word ingeniare - "to devise".
A related word is "ingenious".
Let's try to be ingenious, and devise a solution to a real problem.
There is a huge river - hundreds of feet across.
Train traffic on both sides of the river is great, but must go hundreds of miles down-river to cross another bridge, and then come back to the city.
I want a bridge across the river.
Simple enough: Build a bridge:
"Wait," I'm told. "I forgot to tell you. This is a major commercial river. Barges. Big boats. Lots of traffic. You build this and nothing will get under!"
OK - a minor setback, if it is a setback at all. We'll merely elevate the bridge a bit:
"I'm sorry - you didn't understand the implications of the reality regarding trains. They literally are on the banks of the rivers. This same elevation that allows ships to get under prevents trains from going over!"
Fine - back to the drawing board.
I draw a pivot bridge. Centered on a mid-river pier, the whole bridge pivots parallel to the flow of the water, allowing ships to pass when there's ships, and trains to cross when there's trains. Perfect.
"There's a lot of car and pedestrian traffic we want to keep moving, and a pivot bridge brings everything to a halt!"
Great. It would have been nice to be told these conditions up front. Maybe this is my issue as well. They want a bridge that's not too high but not too low. It can't pivot, lift, or draw.
I put something on paper. Whenever I'm stuck, I put something on paper. Rail underneath. Car and pedestrian traffic on top.
I need to get the lower part of the bridge out of the way for oncoming river traffic, but in the process of getting it out of the way, I need to keep the top section undisturbed.
If I could just lift part of the bridge up, but keep a section of it stable ...
That's it! Lifting the bottom section INTO the top! The car-floor remains in tact, while the supports - OK - how do I interweave the supports of the lower section into the upper?
How do I lift the sections?
A counter-weight pulley system. Easy enough. But where do the lower supports go? Why not pull them THROUGH the upper supports? Make the upper supports hollow steel, with a cable going through them, and connected to the corresponding support on the lower level?
Car and passenger service undisturbed.
Clearance for ship traffic.
A bridge for rail traffic.
What do you think of my "ingenious" solution? Reasonable? Anything extraordinary about it?
Here it is in practice:
The ASB Bridge in Kansas City, spanning the Missouri River. Completed in 1911.
It actually is extraordinary! In the history of bridge building, as I understand it, this and the Steel Bridge in Portland are the only two built like this - the latter more extraordinary as the upper lift can be lifted higher in the case of high tide or an unusually large ship.
The common link?
The King of the Movable Bridge: John Alexander Low Waddell. Central to his thinking:
INTEGRITY, THOROUGHNESS, PROGRESS AND ECONOMICS.
Other Waddell bridges:
The Vertical Lift Halstead Bridge in Chicago .
The Steel Bridge in Portland:
Closing note: the car / pedestrian section of the ASB bridge was removed, once the Heart of America Bridge was complete.
I have a meeting with an engineering professor at a local university Monday morning who was a working scaled model of the bridge. I'll report back!
From the front
The back, where the cables attach to the counter weights ...
From the top, showing how the lower-level telescopes into the upper ...
Starting this week, I, with the kids, will try to re-create this, but with a couple things:
1. the upper road level where the lower level telescopes into;
February 20, 2010
THE ACT IS RIGHT AROUND THE CORNER
The August 19th Kansas City Star lead headline told it all:
“DESPITE BIG PUSH, ACT SCORES REMAIN FLAT”.
How can this be, particularly when the company writing the test provides the principal study materials on how to pass the test?
The next two test dates are September 12th and October 24th. Is your child ready? What are the stakes? Scholarships? Financial assistance? Mere acceptance? The stakes are enormous.
And how is your child feeling? Confident? Nervous?
Unlike a normal test, where “cramming” usually helps, it’s hard to cram for the ACT. In the Science section, for example, your child may be asked about the efficiency of illumination, blackbody radiation, and polypeptide molecules. What’s the use on studying?
You judge for yourself about Reading and Math. Everybody hopes they’ll do well, but decade-long stagnant results suggest something’s not working. The headline above only confirms this.
WHAT TO DO – IF ANYTHING?
Can something be done – in a short period of time – leading to the dramatic improvements we’re hoping for? In Reading? In Math? In Science? In English? Can your child raise their score from this stagnant 21.1 to a 28.6 – or higher?
Though my background is math (BS, MS), I’ve used the same simple thinking processes to write many of the books / booklets below, ranging from advanced math to Shakespeare:
But more important than this are two governing thoughts making the improvement I claim possible a reality:
INSANITY: Continuing to do the same thing while expecting different results.
THE BUTTERFLY EFFECT: A small change in the right spot can have dramatic effects.
AN APPROPRIATE CONCERN
You may still be worried. Why purchase other materials when the current ones don't work? Why waste my money with no guarantee of results?
Let's guarantee results - or your money back!
THE NEXT STEP
What is “the right spot”? What "small change" can make this possible? Find out!
But before you call / e-mail me, download this brief explanation of the program. It has a simple exercise for your child.
Center for autoSOCRATIC EXCELLENCE
February 21, 2010
Consider the two diagrams below:
Which statement below best describes the logic above:
a. A is correct: if I have a dog, then I buy dog food.
b. B is probably correct. If I buy dog food, then I probably have a dog, though I might be buying the food for someone else's dog.
c. Both A and B are valid.
d. Neither are reasonable statements.
Consider the following statement:
A reasonable employee might say, "If I stand idle, and my boss sees me standing idle, I might be fired". Such a belief is an example of::
a. a negative branch reservation.
c. unwarranted workplace griping.
d. none of the above.
James claims "People hate math". Mike counters: "Wait a minute - that can't be right, because I like math". What "reservation" am I making?
d. entity existence
My company's finance director refuses to attend TOC training, as does the head of marketing. Assembly line operation and project management fall ourside the perview of their departments. My company is, in the long term, destined to:
a. succeed - they are right.
b. fail. All departments are linked to the goal.
c. improve marginally.
d. no impact.
The Federal Reserve, to spur economic growth, drops the lending rate 2 points. What is a possible unintended consequence of this action ... a "negative branch"?
c. there are no unintended consequences.
Company X is trying to meet a deadline. 20 pieces in 5 hours. The boss believes this can be done, as both work stations can do their job at 15 minutes per piece.
Station A is on break for 30 minutes, but returns to produce all 20 parts at 10 minutes / piece - a new record.
Did Company X ship?
a. Yes, with Worker A getting a bonus for extraordinary performance.
b. Yes, with both workers getting bonuses.
c. No. Worker B sat idle for 30 minutes, and therefore, the system only produced 18 pieces.
What statement best summarizes this issue?
A. Toyota should have designed a better product initially.
B. Auto makers should recall cars every time something is found wrong.
C. A chronic conflict implies a recurring issue likely to be addressed best by seeking new solutions.
D. Toyota doesn't care about customer safety.
The Hatfield / McCoy feud is like many feuds - you hurt me? I'll hurt you? ... and on and on. What statement best describes this feedback loop:
a. self-fulfilling prophecy
b. a feedback loop cannot go on forever.
c. this is not a feed back loop.
d. situations like this are often resolved by the two parties sitting down together.
February 22, 2010
Consider a circle, I was dared (hurrah - hurrah);
Area is given by pr2 (hurrah - hurrah)
If I differentiate A by r
I arrive at circumference 2pr
What I want to know is - what does it all mean?
What does it all mean?
Lars Erickson of Pi Symphony Fame (and Lars' Creative Warehouse) pointed out this - and several other - anomalies to me. I call them "anomalies" because they seem strange yet they're right in front of my face!
And no, he didn't "dare" me to investigate! After a few revisions to the above poem, I settled on "pi-r-squared" as the close of the second line. Struggling for a rhyming word to close line one, "dared" seemed to work. And I took it as a challenge!
What might it mean? Is there a relationship, or is it merely coincidental?
As always, let's get something on the table.
A circle. Since we're talking about derivatives, we're talking about limits as something approaches zero. What's changing? Let's make our radius change:
Let's pursue this from an intuitive approach. If I have a circle, and I increase this circle by an infinitely small amount, then the amount I have added is really only the ring. Therefore, the derivative of the area with respect to the radius equals the circumference!
Let's be more rigorous, and see if rigor confirms intuition. We want to see - mathematically - how area changes as radius changes. Rates of changes. The change in A relative to the change in r. This is calculus. Infinitesimal changes. Let's see.
Considering the two circles above, the question is:
How does the area of the outer circle compare with the area of the inner circle as we make the outer circle approach the inner circle?
OK - this relationship works with a circle. Let's see if it works elsewhere? What about a sphere?
Indeed! Can we generalize? In 2-d, we go from area to circumference. In 3-d, we go from volume to surface area. What are we generalizing? For the moment, how about we blandly say "from one metric to another"? That is:
Sounds reasonable. But let's check. What if we apply the simple square to the test. How does our test hold up?
Riddle Solved - Not!
Consider a circle, I was dared (hurrah - hurrah);
Area is given by pr2 (hurrah - hurrah)
If I differentiate A by r
I arrive at circumference 2pr
What I want to know is - what does it all mean?
Looking at circles and spheres, I think (hurrah hurrah)
Differentiation is quite succinct! (hurrah hurrah)
But when I extend my theory hence;
The simple square - it throws a wrench ...
Lesson learned: Beware of Generalization!
The Remarkable Bob Beamon
February 23, 2010
The Winter Olympics and a coincidental article on non-record-setting performances gives rise to this thought:
The remarkable Bob Beamon.
29 feet 2 1/2 inches.
Here it is ...
It's mostly remarkable both for how it shattered the previous mark, and how long it took to be broken itself. A graphical depiction of the record over time ...
A couple things stand out: the record lasts a long time, then is broken by a significant amount, and then there's decreasing - but continual improvement, until the next "great leap".
This is true in the first part of the 20th century, and the mid 20th century. It's not true after Beamon, however.
Why is this?
The altitude in Mexico City, we're told, is one possibility.
Mexico City is about 7,500 feet above sea level. The air is thinner. Jumps should be better.
But if this explanation is reasonable, we should expect to see other tremendous leaps from the 1968 Olympics. Did we?
Here are the other jumps:
The claim the cause was "altitude" doesn't seem to hold much water. Likely it helped Beamon, but he already must have been head-and shoulders above the competition; otherwise, we would have seen other great jumps. We didn't.
One final note regarding the chronological graphic above. It's remarkable Beamon's mark stood for 23 years. However, there was a longer stretch earlier in the century: 25 years.
Who was this?
The great Jesse Owens!
The World Record Over Time
A side note about Jesse Owens and this world record above, from Wikipedia:
"Owens's greatest achievement came in a span of 45 minutes on May 25, 1935 at the Big Ten meet in Ann Arbor, Michigan, where he set three world records and tied a fourth. He equaled the world record for the 100-yard sprint (9.4 seconds) and set world records in the long jump (26 feet 8¼ inches, a world record that would last 25 years), 220-yard sprint (20.7 seconds), and 220-yard low hurdles (22.6 seconds to become the first person to break 23 seconds). In 2005, both NBC sports announcer Bob Costas and University of Central Florida professor of sports history Richard C. Crepeau chose this as the most impressive athletic achievement since 1850."
A CLOSING THOUGHT
I'd always thought the world record was 29'2.5". However, all records show it as 8.9 meters. How do I convert metric to British?
I'm not alone, here. When his distance was announced as 8.9 meters, Beamon was unaffected. He didn't know, either! It wasn't until teammate and coach Ralph Boston told him the distance in feet Beamon collapsed to his knees, hands over his face!
How do I convert metric to British?
Here's my favorite way, which requires only the knowledge of 2.54cm = 1 in. Every other conceivable combination I can derive from this, using unit fractions ...
Choice & Context
The Legacy of Aesop
February 24, 2010
Once when a Lion was asleep, a little Mouse began running up and down upon him. This soon wakened the Lion, who placed his huge paw upon him, and opened his big jaws to swallow him.
"Pardon, O King," cried the little Mouse. "Forgive me this time, I shall never forget it. Who knows. I may be able to do you a turn some of these days!"
The Lion was so tickled at the idea of the Mouse being able to help him, that he lifted up his paw and let him go. Some time after, the Lion was caught in a trap, and the hunters who desired to carry him alive to the King tied him to an Ash tree while they went in search of a wagon to carry him on. Just then the little Mouse happened to pass by, and seeing the sad plight in which the Lion was, went up to him and soon gnawed away the ropes that bound the King of the Beasts.
"Was I not right?" said the little Mouse. "Little friends can prove great friends!" After all, "One good turn deserves another!"
You may think that's the end of the story. It's not. What became of the eves-dropping Ash tree that overheard the miniscule mouse conversing with the mighty lion?
Years later, a Peasant was looking around in the woods, and came upon the Ash tree, now surrounded by many other trees, in the forest. The old and wise Ash tree asked him what he was looking for. "A straight piece of wood," said the Peasant, "to mend my axe handle." All the other trees warned the grand Ash about the pending danger, but the Ash, remembering the story of the Mouse and the Lion, decided "One good turn deserves another", and "Little friends can prove great friends".
Owning the straightest and mightiest branches, the magnificent Ash dropped one at the feet of the eager Peasant. The Peasant nailed the Ash branch to his axe blade and began to cut down all the Trees.
The Oak Tree, waiting for the axe to strike, yelled at the Ash: "YOU FOOL! Do not help your enemy to hurt you!"
As the Ash Tree watched friend after friend fall helplessly to the ground, he asked himself, "How did it come to this?" Hadn't he learned, from the story of the Lion and the Mouse, "one good turns deserves another", and "Little friends can prove great friends"? In doing this, the mouse had HELPED his enemy! But here he was, about to fall, with the lesson learned, "Do NOT help your enemy to hurt you"!
The Ash - and his friends - were not heard from again.
What is going on here?
There does seem to be contradictory edicts in the "lessons to be learned" of Aesop. Here, the conflict might be as follows:
This is but one of many conflicts arising in "general lessons". For example, the one who "looks before he leaps" is beat to the punch by the person who adheres to the edict "he who hesitates is lost".
What is going on here?
There's merit, usually, in both claims to "general lessons" or advice, so it must be the case there are times when each are relevant. What are these times? What is the context?
How should I act? How should I choose?
What are my needs? In the forest, for example, I should not lend a branch to one in search of an axe handle. But let's suppose that same Peasant was in search of a branch to build a fire to warm his family? It always depends on context. On prediction. On the future reality.
Let's suppose I, as the mouse, came upon a cat tied up. Would I chew through the ropes binding the cat, knowing the cat will likely pounce on me? I think not!
Morals to the story? You bet! But choice and context! That (should be) the advice of Aesop!
February 25, 2010
Miscellaneous Thought 1
On The Calculus
An astute gentleman and frequent reader of my columns pointed out a problem in my "area & circumference" post of February 22.
A summary of the post might be: take a circle with radius r, and increase it just a bit - an infinitesimal bit. How much have you increased it? By the circle itself, because the addition was really adding the circumference of the existing circle to the area.
"Wait a minute," he said. "You said you increased the radius, but then set it, essentially, to zero. That doesn't make sense."
It is a contradiction, I admit.
An old one.
When the calculus was invented approximately three centuries ago, these three talked a lot about it.
The first two dismissed the arguments because the results "worked", or "made sense". The latter refused to let the issue drop, and published, in 1734:
A DISCOURSE Addressed to an Infidel Mathematician
The "infidel mathematician" was believed to be either Edmond Halley or Newton himself. Berkeley was attacking the seemingly contradictory ideas of "fluxion" and "infinitesimal change", which Newton and Leibniz had used to develop the calculus, and used the phrase Ghosts of Departed Quantities to make fun of the apparent contradictions.
Miscellaneous Thought 2
"Highway Trust Fund"
There's really no "trust funds". No "Medicare" trust fund. No "Social Security" trust fund. No "highway" trust fund. It all goes into the general fund, and is spent accordingly.
It's a big lie.
But that's not my miscelaneous thought here.
In the February 15, 2010 Transport Topics, it's noted Washington state's highway trust fund is not bringing in as much money as it once did.
Fuel-efficient cars. People actually driving less.
Predictable? Entirely. Easily. Unintended? Sure. But as has been noted earlier, it's the folly of ignoring the laws of reality to enact legislation with both predictable - and unintended - consequences.
Miscellaneous Thought 3
Thoughts on YRC - and GM - and Social Security
YRC is a "less-than-truckload" (LTL) local trucking company. They've been having financial problems for some time.
Their demise has been postponed.
Benefits, I understand, are largely to blame.
Sound familiar? GM has high retiree costs that are unsustainable. They drive up the cost of cars currently made, and make the company untenable in the future.
Social Security and Medicare fall into this same category, which might be summarized as "Promises Made".
What can be done?
This image comes to mind: a wagon being pulled by six oxen on level terrain. As it approaches a hill, two decide to quit pulling and hop in the wagon. The burden on the other four? Greater. As the slope of the hill increases, the number of oxen pulling decreases, as two more decide to hop in the wagon.
What happens if the two pulling oxen, straining "to make it work", finally decide: enough is enough. This system is not working. You're causing me to fail!
Suppose the pulling oxen decide "to shrug"? An ultimatum: either help pull or lose weight, or I'm leaving?
Johnson County, KS?
February 26, 2010
December 17, 1903. Kittyhawk, North Carolina. The Wright Brothers. Flight.
Twenty-four years later, Lindbergh made the first trans-Atlantic flight, flying the The Spirit of St. Louis from New York to Paris.
The idea of flight likely is old as man himself. Leonardo Da Vinci made many attempts at triumphing over gravity.
Of course, people had "defeated" gravity for centuries. Balloons. But the Wright Brothers? We were now "heavier than air" and in the air!
From "off the ground" to "across the ocean" in 24 years:
Of course, in any paradigm shift like this, there are incremental steps. Look at the Wright Brothers picture, for example. Yes, they were off the ground - barely. When did real altitude first take place? 500 feet above ground? 1909. Charles K. Hamilton.
Lindbergh took us across the Atlantic. Who took us across the country? Calbraith Perry Rodgers: 1912.
And in-between, don't forget, is likely a whole lot of experimenting. Loops. Rolls. Playing around with this new technology. Who was the first to do eight rolls? DeLloyd Thompson. 1911.
Hamilton's flight took place at Aviation Park.
Rodgers' headquarters for his trans-continental flight was - also - Aviation Park.
And Thompson's rolls? You guessed it. Aviation Park.
Where is Aviation Park?
Overland Park, Kansas! Of course, if you planned a visit to see this remarkable structure and site of history in the making, you'd be disappointed. Nothing remains. It's been gone for decades. All that remains is this marker ...
You can read elsewhere about William Strang and the Strang-Line Interurban Railroad, but a word is relevant to continue a different story here. You may be thinking "this is the end of the story". It's not. There's another line of thought I'd like to pursue.
The great Kansas City flood of 1903, you may recall, gave Strang the idea of building an interurban railroad to a bluff southwest of Kansas City.
He was a man who was tired of the continual flooding of the town. Of residents being uprooted, of homes being destroyed, because of the low-lying downtown area.
Why were there residents in Kansas City in the first place? You see, though downtown Kansas City has approximately 1/2 million people now, in 1850 there was essentially no one here. However, by 1900, the census count shows the population grew to almost a third of what it is right now!
A population explosion.
To ask this question may be to ask the wrong question: Instead, let's ask why did people previously not come here?
And to answer that question requires us to know a bit about the geography of Kansas City.
Downtown Kansas City lies south of the mighty Missouri River. Uncrossable except by ferry. Until 1869. When the great Hannibal Bridge united north and south.
It was the first bridge across the Missouri River.
Kansas City exploded.
And what is that truss under the bridge, you might wonder? Let's remember: this is the Missouri River. Steamboat travel dominates. This is a connecting structure for the two piers supporting the swing span. Yes, this was a swing bridge!
More remarkable than the bridge is the man who designed it: Octave Chanute.
You may, at this time, be thinking a couple things:
1. Kansas City has a unique history regarding rail bridges across the Missouri. The Hannibal. The ASB.
2. I thought this story was about Aviation.
You see, Chanute was born in 1832. By 1875, he became interested in flight.
As told in Wikipedia, "Following his systematic engineering background, Chanute first collected all the data that he could find from flight experimenters around the world. He published this as a series of articles first published in The Railroad and Engineering Journal from 1891 to 1893, and collected together in Progress in Flying Machine in 1894. This was the first organised, written collection of aviation research."
He not only collected research, he applied it:
If the biplane hang-glider image looks eerily familiar to the Wright Brothers image earlier, it should. They copied him. He consulted with them. And at his funeral in 1910, Wilbur Wright gave the eulogy.
Who first platted the town of Lenexa, Kansas - in Johnson County, Kansas.
And so I dedicate this article to two local men who changed the world. Intellectual and Technological Pioneers:
An Injection to Address an Urgent (to me) Problem!
February 27, 2010
My story is not unique. It's a common complaint. I pay for an airline ticket. I expect a reasonable seat. There are none for me. You see, I'm 6'7". There's no leg room. And here comes the seat in front of me, crashing down. I respectfully ask if they would mind keeping their seat up. "Everybody's cramped," I'm scolded, "and everybody reclines their seats."
I'm left to count the minutes to touchdown.
Instead of bitching, let's look at this problem a bit.
I don't fit - comfortably - in an airline seat. What would I like to see? Good seating. Clearly. What would it take to get good seating? The answer is obvious - when your knees are crammed into the seat in front of you: an increase in the distance between seats.
But if it were that obvious, it would be done - and it's not. Why do airlines maintain the status quo: to not increase the distance between seats.
Why do they do this? The answer is just as obvious: they've looked at the number of seats on a plane, the price per ticket, the average height of a person, etc. They've run the numbers. The current arrangement exists to increase profitability.
This is quite a narrative to explain a problem. Let's integrate it visually ...
In order to be a good airline, we must have good seating for our riding customers. To do this, we should increase the distance between seats.
On the other hand ...
In order to be a good airline, we must have profitability. To do this, we should not increase the distance between seats.
This seems an unbreakable dilemma.
What have airlines down to address this? Charge more for emergency row seats with extra leg room? Add first class seating?
There's one connection I've been thinking about above: in order to have good seating, we must increase the distance between seats.
Clearly - or maybe not so clearly.
Let's take a look at the issue. Here's me on a simulated living room airplane seat, 18d, one row behind a non-gentleman in 17d.
Why is he a non-gentleman? Because here comes the recline, right on my already crunched knees!
What to do? We already said airlines have run the numbers to arrive at the current spacing. Increase it and that reduces the number of seats, which impacts profitability.
Can nothing be done?
It doesn't seem like it.
But look again at the first image. What strikes me as odd about the spacing as the amount of space under the seat in front of me! It's inaccessible to me, though, because my knees are already hitting against the seat in front of me!
What I'm really looking for is knee clearance, rather than seat distance.
Is seat distance the only way to achieve knee clearance?
What if instead of moving my seat back, I move myself up? Increase the angle of my knees?
There you have it ... the "air-cliner", achieving good seating while not jeopardizing airline profitability!
The key: get oneself UP rather than BACK!
Verbalizing the dilemma, and the assumptions underlying the logical connections. The injection here? The obtuse angle!
The Earth and the Sun
February 28, 2010
The earth orbits the sun "once every year". That's our definition of "year" - one complete orbit. If we left our calendrical units here, there is no issue. However, the further use of "days" gives rise to a slight discrepancy. You see, one year doesn't equal 365 days, but instead 365.242199 days.
But let's round this to 365.25 days to get a better idea of what's going on here. After one "of our years", our calendar has moved 365 days. However, for the earth to get to it's initial starting position, it needs an extra 1/4 of a day. It's just short of the starting point. After the second year, it's short 1/2 day. A third year moves the distance another 1/4 day, and finally, a fourth year leaves the earth one complete day from it's orbital "starting point". Leap Year intends to correct for this "drift".
What happens if we do this? What does this look like - graphically?
But let's return to reality, now. One year does not equal 365.25 days. More precisely, it equals 365.242199 days. What happens if we add a day every four years? We over-compensate slightly! That is:
So, we've got to somehow correct for this "over-compensation". But how? Right now, the rule is "every 4 years, add a leap year unless the year is divisible by 100". Let's see what this looks like:
Correcting for this 100-year glitch seems to put us on track - or does it? It certainly resets the bar, but does it do so accordingly? By the graph below, we seem to be losing ground now! Let's take a guess at how the rule was modified: at 400 years, it's clear we're at a point where, if we ignore the "100 rule", we're back on track.
The New Rule
every four years is a leap year - we add a day to get us back in line. But this day over-shoots the target. Therefore, every 100 years, skip the rule. Doing this, however, erodes the "over-shooting", so every 400 years, we need to add back the leap day!
Well - let's implement our algorithm here and see what happens. Graphing the results for the next 10,000 years yields some amazing results. The process does not stabilize! We get out of whack around the 4,000 year! Now, likely, calendar-adjusters know this, but see no reason to make this an issue, but it is interesting to see what happens if we ourselves make it part of the rule.
This latter thought was not considered until the 19th century (John Hershel), and has not been adopted to this day.
So the rule: if divisible by 4, then a leap year,
except if it's divisible by 100, which is not a leap year,
unless the year is divisible by 400, which is a leap year,
unless the year is divisible by 4000, which is not a leap year!
What makes this most interesting to me is the official rule, regarding every 4/100/400, was made by Pope Gregory XIII in 1582 (hence the name 'Gregorian Calendar') long before there were calculators, satellites, and GPS systems.
The Logical Haiku of the Week
This weeks "logical haiku" is in recognition of this cosmic / calendrical calibration process!