August 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 31
Chapters 1 and 2
August 1, 2008
The Proximate Event
Had it just been a month since his visit to the northeast to visit relatives? How might things be different had he not been able to borrow his uncle's car and travel the 70 miles from Concord, New Hampshire to Jericho, Vermont.
The home of Wilson “Snowflake” Bentley.
It was even better than he hoped it would be.
Michael Johnson was mesmerized. It was even better than he hoped it would be. He, of course, had the beautiful book displaying over 2,300 of the snow crystals photographed by Snowflake Bentley. He, of course, had read biographies and internet postings on Snowflake Bentley.
But here he was!
He walked about the farmhouse once home to the famous “scientist”. Michael marveled at the size of the camera used to capture the crystals.
It was three hours of pure fun – and amazement.
Driving back to Concord, a thought struck Michael. “No two snowflakes are alike”. Snowflake Bentley was the origin of this belief. Who else could make such a claim? He was the only one capable of photographing snow crystals.
But he photographed only 5,500 in his lifetime. It may be the case none of these were identical – though Michael even wondered about that. How could you tell?
But more than that was the nagging thought: in any snowstorm, likely there are billions of snowflakes – if not trillions. Are all of these different? How would you know?
How would anybody know?
A Near Idea – Not So Far Away
“Will you play tic-tac-toe” with me?” It was his 8-year-old nephew, William, paper and pencil in hand, making the polite request. Looking back, Michael was glad he had taken up the challenge.
Game after game of “Tie-game – you go first” had not taken their toll. On the contrary, it had given Michael an idea.
“How would anybody know?” was the question still haunting him regarding all snowflakes being different. Of course, you could never test that, but could you test anything comparable? The “Tic-Tac-Toe” grid had given him an idea. Being fairly proficient in the spreadsheet program Excel, he wondered if he could model snowflakes in a manner similar to the X & O game.
Of course, he knew there was an obvious problem. Snowflakes were hexagonal in nature, and he was talking about creating something in columns and rows.
But it was a start.
He started with the simplest grid that came to mind, one resembling the tic-tac-toe board. A simple 3 x 3 grid:
How to populate it? The one thing he wanted to make sure of, though this grid was not the same as the hexagonal shape of a snowflake, was having some symmetry – like the snowflake. What would this take?
“Which cells do I need to fill in order for all the other cells to be filled – if I require there to be symmetry?” The following was the result. If these three cells were filled in, all the others would become filled in:
With this pattern and symmetry, the rest of the grid is filled:
However, if I do this, it’s hard to keep track of the initial pattern. I’d have to say “black black black”. Instead of this, why not use the analogy of “binary” and encode black cells as “1”, and white cells as “0”. What might this look like?
With just three “open” cells, there are eight possibilities. Further, I can list the rules with 1s and 0s. What happens if, instead of two columns, I use three columns?
Three columns yielded 64 distinct symmetric patterns. Likening the results to ancient hieroglyphics, he had coined the phrase “Binary Symmetric Hieroglyphics”.
Pretty as they were, this was not the answer to his question, “are all snowflakes different?”, though I’m now moving in the right direction.
How many possibilities are there? How many differences, that is? It depends on the number of columns in my grid. If there is just one column, there is just one cell. If there are two columns, then there are three cells open to change. Three columns – six cells.
What is the relationship between the number of columns and the total cells? The question is reduced, then, to:
“Given a certain number n, what is the sum of all positive integers from 1 to n?”
What the great mathematician Gauss did as a child gave rise to the formula widely used today. He quickly reasoned: suppose I want to know the sum of the integers from 1 to 10. I could reverse the series of numbers and add all sums together, and they will all be equal. If I multiply the total number of integers by this sum, I will have my desired figure.
But wait a minute: the sum of the integers from 1 to 10 is 55 – not 110. My problem is I added the whole series to itself – this is why the total is twice what it should be. Therefore, I need to divide the total by 2.
The answer to my question: “Given a certain number n, the sum of all positive integers from 1 to n?” is
All of this work has been in answer to the question:
What is the total number of different “Cartesian snowflakes” I can make with my grid?
Now I know – and I’m halfway done! So, applying this formula to a certain number of columns gives me the following number of “different snowflakes”.
What is that last number? It’s nearly 2.5 OCTRILLION! How can this be? This small grid below contains more different illustrations than any number I’ve ever encountered!
Here are more ..
OK – perhaps it is possible all snowflakes are different! After all, with just 13 columns in a simple spreadsheet, I’ve created a model with 2.5 octillion possibilities. Imagine how quickly this number grows as the spreadsheet grows!
The question was answered! Indeed, it’s possible – no – likely – all snowflakes are different!
It was a Saturday morning in early August when Dad asked me to mow the lawn – again. Why does this stuff have to grow so fast?
My thought moved from a seed of grass growing to a tall blade back my snowflake, and I suddenly was troubled. Why? Snowflakes grow like this, too, don’t they? Not literally, of course, but they grow. They start as something small and somehow grow into those beautiful shapes known as crystals.
My brief spreadsheet had everything appearing at once?
Is the relationship between grass and the snowflake valid? Does my model of the snowflake capture, at all, what I’m trying to do?
What exactly is a snow crystal?
My focus had changed. The original question now answered, a deeper one was now asked: “What was going on, here?”
Rethinking Statistical Education
August 2, 2008
"The Numbers Behind Numb3rs" is a fascinating companion
to the Numb3rs series. In one chapter (Predicting the Future), the
authors present an interesting scenario called "The (Fictitious) Case of
the Hit-and-Run Accident".
In this scenario, a hit-and-run involving a taxi has taken place. The town has 90 taxis: 75 black and 15 blue. A witness claims the taxi was blue.
Blue is pretty close to black, so police perform a test to determine the reliability of the witness. Presented with random colors, the witness can correctly chose 4 out of 5 times.
The question: what color was the taxi involved in the hit-and-run?
Common sense tells us it was blue. After all, there aren't many blue taxis. The witness saw a blue taxi. The witness is right 80% of the time.
The authors continue:
"Bayes' method shows that the facts are quite different. Based on the data supplied, the probability that the accident was caused by a blue taxi is only 4 out of 9, or 44 percent ... What human intuition often ignores, but what Bayes' rule takes proper account of, is the 5 to 1 odds that any particular taxi in this town is black. Bayes' calculation proceeds as follows:
1. The "prior odds" of a taxi being black are 5 to 1 (75 black taxis versus 15 blue).
The likelihood of X = "the witness identifies the taxi as blue" is:
1 out of 5 (20%) if it is black
4 out of 5 (80%) if it is blue.
2. The recalculation of the odds of black versus blue goes like this:
P(taxi was black given witness ID) / P(taxi was blue given witness ID) =
(5 / 1) x (20% / 80%) = (5 x 20%) / (1 x 80%) = 1 / .8 = 5/4
The Bayes' calculation indicates that the odds are 5 to 4 after the witness' testimony that the taxi was black.
I honestly have no idea what has just been said.
Let's see if I can create a "foothold", by which I can understand what has just been said. I'm told there are 90 taxis - 75 black and 15 blue. Fine. Let's create a table of 75 black and 15 blue taxis.
These are my 90 taxis. But we know something more, as the police did a "credibility" test on me, with the results coming back 4 out of 5 times I'm write. That means (4/5) x 75 = 60 times, when the taxi is black, I guess black (and the other 15 taxis I guess blue - wrongly). Further, for the 15 blue taxis, (4/5) x 15 = 12 times, I guess correctly on the color of the taxis, and 3 times I guess black when it is blue.
Let's add our "guess" to our "taxi-table":
Now, I'm only concerned with the last column - taxis numbered 61 to 90. These are the taxis I've numbered where either the taxi is blue - or I've guessed it's blue - or both. What conclusions can be drawn from this column?
So the author's were right! OK - I had no doubt they were right. I also know I had no doubt I could not follow "math talk". Not like this, at least. But even this is not a valid statement. I can follow the logic once I have an idea of what's going on!
So what proceeds what? Does a model or table of data come before the formulas? Vice versa? Is one always needed? It all depends!
August 3, 2008
A Good Analogy?
I had started with the simple task of seeing if it were possible all snowflakes were different. Now, not only am I convinced it’s possible, it’s the most likely scenario.
But in the course of modeling that result, a nagging thought crept over me. Was I modeling what actually takes place in a snowflake?
I didn’t think so.
My “models” were of square sections of a spreadsheet, and they were populated at the same time. Surely, snowflakes don’t grow like this.
But how do snowflakes grow?
Snow, I know, is just frozen water. But what is water? I know the chemical designation is H2O, but what does this mean? Two hydrogen atoms have bonded with one oxygen atom.
But does this help me? I have no idea why “lots of water molecules” accumulate to “some starting point” to create the snowflake I see when it hits the ground.
How does the hexagonal shape appear? Is it always hexagonal? Thinking back to my original picture, these all were, so let’s assume “yes – they are all hexagonal”.
As before with my spreadsheet, is there a way I can “model” snowflake growth? This is made harder than my model before because I don’t know how the thing “grows”.
Let’s make a guess. Let’s start with a single populated cell, and grow outward – hexagonally – from there. If I do this for one “iteration”, here’s what would happen:
What happens if I continue in this manner? Let’s see:
This looks interesting, but hardly like a snowflake – plus, where is the change for all the variation I saw above? None!
There must be something I’m missing.
What could it be?
I’ve assumed outward growth just goes on and on without any conditions. What if I add a condition, and see if I can change the result?
What’s open to change?
What if I grow outward, not to every cell, but to only certain cells? For example, two cells touching a cell cannot “grow” into that cell. That is:
What happens if I continue in this manner? Let’s see:
Something at least resembling a snowflake! But this seems to be the only way I can create with my rule above, the rule being:
If one cell is colored, so is the next one;
If two cells are colored, the next one is not.
Are there other rules? There must be, else this would be the only snowflake (with the one earlier) I could create. Let’s see.
Let’s take a portion of our hexagonal grid, and see just how many “alternatives” there are. The question: what scenarios exist for the hexagon with the “?” inserted below?
Actually mapping out the possibilities, I see there are several:
In fact, merely doing this step has led me to realize there are actually a lot more possibilities. For example, the following four situations are quite different – and this is just considering one neighbor:
Rather than continue trying to list all the possibilities, let’s just
take one at random and see what happens:
When I try to automate this in Excel, however, I run into a problem. Above, I was exporting hexagons into “Paint” to shade the figures. I don’t want to do that for all the possibilities I’m running into – I just want to set up a program and see what’s possible.
How can I do this in Excel?
Can I use the column / row format, counting the number of neighbors, and all the different possibilities? Here’s one possibility:
Rather than continue trying to list all the possibilities, let’s just take one at random and see what happens.
and another ...
and a third ...
OK. I’ve got the idea. Now, rather than capturing each step, let’s just run a series of rules and see what the final image looks like:
The generation of a snowflake – growth, if you will – can produce an unbelievable variety of snowflakes. I still don’t know how the water molecules do this, nor what role weather, temperature, and humidity play. I’ll save that for another day.
Growth from a simple starting point and simple rules, I see, can produce a great deal of variety.
For a while.
The thought of the seed of grass comes back to me. From the seed of grass comes the blade of grass – all of the information in “becoming” a blade is embedded in the seed’s “DNA”. This is the case with us as well – with flowers, trees, lots of things. But this is not the case with my snowflake above. There is no such thing as a “snowflake DNA” embedded in the initial water molecule, telling it how to grow! That’s ridiculous!
Are these two situations different?
But does it matter?
August 4, 2008
Every (educated) individual older than 14 knows Cy Young's career win total of 511 - it's a magical number. Walter Johnson? For the longest time, it was 416, until updated records showed the total to be 417.
The great Grover Cleveland (Pete) Alexander of Elba, Nebraska is tied for third with Christy Matthewson at 373 (incidentally, Matthewson's total was 367 when I was growing up)..
But these were all players long ago.
How do modern players compare?
How would data best be presented to get "the lay of the land"?
One method ...
Speaking of the great Grover Cleveland Alexander, the American Poet Ogden Nash penned the following poem:
Lineup for Yesterday (Ogden Nash)
A is for Alex
The great Alexander;
More Goose eggs he pitched
Than a popular gander.
August 5, 2008
Watching the Royals last night, I again watched as players round first - heading for second, second for third, and third - racing home. What strikes me as about this common phenomenon is the amount of space wasted.
But is it?
Rounding a base clearly is not a direct line to the base - but is directness a goal here, or speed to the next bag? Clearly, getting to the bag as fast as possible is the goal, and, in going directly at a bag and turning 90 degrees, one loses speed. Shorter distance? Yes. Speed? No.
To the former - definitely. To the latter? Let's not dismiss it just yet.
How much additional distance are we talking about?
A typical field, with both the diamond marked, as well as a complete circular path, as if the runner were rounding each base in a perfectly circular manner:
Likely, runners do not run perfect circles when rounding the base. For now, let's assume they do, and see what the difference is - between a straight path and a circular one. A straight path has total distance 360'.
What of a perfectly circular distance? I know the formula for the circumference of a circle is C = 2πr. But what is 'r'? Above, I've called this distance 'x', so let's maintain that nomenclature.
It's the radius of the circle. How do I find it?
A runner runs about 40 feet more rounding the bases than they do running directly to the base. How much speed is lost by stopping? That's the purpose of an upcoming study.
Another thought comes to mind. When the runner rounding third heads home - with a ball hit to the left fielder, there is a "play at the plate". The throw usually is on a direct line to the catcher, with the runner comfortably out of the way - as they are rounding the base. Supposing the study above confirms it's better to round the base than move directly from one base to another, that's assuming speed is the goal.
In the situation above, is speed the goal, or is getting home safely the goal? The latter is obviously true. Therefore, might it be reasonable, in this case, to run directly home upon reaching third, as this obstructs the throwing angle from the left fielder to the catcher?
Where else might this be the case - where running strategy is dictated by the position of the ball in the outfield?
The two possibilities seem to be as follows:
The Geometry of The Game ... More to come on both studies!
August 6, 2008
Today's issue of the Kansas City Star included a
wonderful article titled "Down the Road,
Brick is Back",
by Nathan Gill, discussing the re-emergence of paved roads, nearly a
century after their glory days.
The article discussed the aesthetics of paved roads, the safety quality of paved roads artificially slowing down vehicles because of the "bumpiness".
The article provides the cost of the roads being built in Lawrence, Kansas ($670,000 for two blocks), though 75% of the salvaged bricks will be reused, and it's unsure if the 6-figure cost is artificially low because of the salvaged bricks, or if the cost would be higher if a community were to start from scratch. The article also provides no baseline for the cost of simply paving a road, so we have no idea whether the above cost is reasonable or not.
Most importantly, the article makes no reference to Olathe, Kansas at one time being home to the brick-laying champion of the world:
James "Garfield" Brown - an Oneida Indian known as "Indian" Jim!
But first, a bit of history:
The beginning of the 20th century saw the growth of the automobile as the dominant means of transportation. Good transportation meant reliable roads, and they were hard to come by.
At this time, commerce between a growing Olathe and Kansas City increased, but the time and troubles traveling via automobile were a deterrent to vibrant commerce.
The Westport route of the famed Santa Fe Trail was this route, but cars would get stuck on this road in the mud.
What was needed was a paved road. A paved road meant a brick road. But not just any brick road, but one built via a contest between the two greatest brick layers in the world!
Indian Jim versus Frank Hoffman. The date: September 12, 1925.
The winner? Indian Jim, laying 46,664 bricks (each weighing approximately 8 pounds) in 7 hours 48 minutes. You do the math: about 100 bricks a minute for a full shift. Try just moving your hand nearly twice a second as though you were laying bricks. It's nearly impossible. But to do this 46 thousand times?
Nothing is left of the paved road, of course. It's been "paved" over. No bricks remain. Nothing remains, except this memorial at the corner of Kansas City Road and Poplar Street (501 E. Kansas City Road), identified as the site of the 1925 bricklaying contest.
THE KC STAR --- 1925
The achievement of Indian Jim Brown in the bricklaying contest on the Olathe Brick road Saturday was of heroic proportions. It deserves to be celebrated in song and story. In the days of the ancient Greeks it would have been.
If Indian Jim had lived back in the twilight of history he would have gone down to posterity in an epic written by Hesiod as a dweller on Olympus.
KC Star Editorial
August 7, 2008
We're all different (thank goodness) with our own
strengths (and weaknesses). With a co-worker one day, we were
planning a joint presentation to a client on the nature of rising health
care costs. Bill (not his real name) and I discussed where the
presentation would go, how it would be presented, who would be doing the
Bill was a good speaker, and the client had heard Bill speak on many occasions, so I had no problem letting Bill take the lead.
During the presentation, questions came up. Here, Bill's skills were not as sharp. Bill was great at dispensing an answer right away, though it was clear "speed" was no virtue towards validity.
But what could I do?
Have you found yourself in a situation where "clarifying" an answer by a co-worker diminished their credibility? One clarification? Maybe. But following up repeatedly will diminish their credibility, make them hostile at you, and diminish both in the eyes of the client.
But what is the alternative? Say nothing?
What a dilemma!
Thinking further about this, one of the issues causing the dilemma is the co-worker is a friend. Suppose we were not friends. I wouldn't hesitate to provide clarifying answers to issues.
I think England was in this same boat regarding the Iraq War. Immediately after 9/11, and the documentation Iraq was a threat, England enthusiastically joined the coalition.
However, when a great deal of the "evidence" turned out not to be reliable evidence, I think England found itself in the position of me above as a co-presenter: What to do?
Might it be the case, in solving one problem we solve both? Or solving one helps solve the other?
But why stop here?
In an earlier post, I talked of a tri-partite approach to such a process, linking personal, historical, and literary examples with a "conceptual common denominator".
What could be a literary example?
"The Lifeboat" was the first thing to come to mind. Suppose you're adrift in a lifeboat with 7 others, rations running low. There are 7 person-days of rations remaining. Food and water for all seven today exhausts our rations. Food and rations to just me, and I can live a week.
All Together, Now!
Which one should I start with to solve? Does it matter? If all three share a common thread, likely the solution is similar in nature!
And the solution, as Sly and the Family Stone said, Takes Us Higher! The solutions above collapse / fold-up to a tetrahedral pyramid of success / understanding!
August 8, 2008
The Hungarian biologist Aristid Lindenmeyer noted nature proceeds according to rules. How can one put a "grammar" to these "rules of nature" to distinguish one system from another? Lindenmeyer created such a formal-language, called the L-System, capturing relevant variables of the system.
In this example, the number of branches, the angle, and the growth reduction of each branch are captured.
For fun, what happens if we vary the internal angle of the tree, but maintain the growth factor:
What happens if we maintain the internal angle of the tree, but vary the growth factor:
August 9, 2008
While in Hutchinson, Kansas, last week, we visited the Cosmosphere. The Cosmosphere has the actual command module for the Apollo 13 mission, beautifully restored.
The following dialogue from the "Apollo 13" movie might be the best example of one of the uses of the Cloud, not in reaching a "compromise", and not in reaching an injection, but rather, for simply getting the ideas logically on the table.
APOLLO 13 TRANSCRIPT
Okay, people, listen up. Gentlemen. I want you all to forget the flight plan. From this moment on, we are improvising a new mission.
How do we get our people home? They are here. We turn 'em around? Straight back? Direct abort?
No! - I can't guarantee the burn yet.
No, sir, no, sir, no, sir. We get them on a free-return trajectory. It's the option with the fewest question marks for safety.
I agree with Jerry. We use the moon's gravity to slingshot them around.
No! The LEM will not support three guys for that amount of time. It barely holds two. I mean we've got to do a direct abort. We do an about face. We bring the guys right home right now.
Get 'em back soon.
No. We-We don't even know if the Odyssey's engines even work, and if there's been serious damage to this spacecraft, they blow up, and they die.
That is not the argument. We're talkin' about time, not whether or not these guys...
I'm not gonna sugarcoat this for you.
Okay, hold it. Let's hold it down. Let's hold it down, people. The only engine we've got with enough power for direct abort is the S.P.S. on the service module. From what Lovell has told us, it could have been damaged in an explosion. So let's consider that engine dead. We light that thing up, it could blow the whole works. Just too risky. We're not gonna take that chance. About the only thing the command module's good for is reentry, so that leaves us with the LEM. Which means free-return trajectory.
I've slipped in "double arrows" here. Why? The logic - and how one arrives at it - works both ways.
The logic of Gene Kranz seems sound. We need to get the ship home, and to do this, we need power. There's been an explosion, so we have to assume the service propulsion system aboard the service module has been damaged. If we do an "about face", we'll have to use the SPS. That's too risky.
Is this the end of the story?
Let's not forget the other argument. Is the "about face" only said to "get them home now"? Is that the only reasoning these smart guys are using? We learn more later in the discussion:
If we use a free return trajectory but the LEM can't support three astronauts for an extended period of time, then the astronauts will die.
Here is, to me, one of the genius decisions of the mission. Mission Control, caught in this dilemma, weighed the probabilities above and thought as follows:
the SPS will likely cause an explosion. The air problem? WE CAN SOLVE IT SOMEHOW.
This, of course, led to this famous dialogue ...
Oh, Gene? We have a situation brewing with the carbon dioxide. We got a CO2 filter problem on the lunar module. Five filters on the LEM. Which are meant for two guys for a day and a half. So I told the doc, and he ...
They're already up to eight on the gauges. Anything over 15 and you get impaired judgment, blackouts, the beginnings of brain asphyxia.
What about the scrubbers on the command module?
They take square cartridges. And the ones on the LEM are round.
Tell me this isn't a government operation.
This just isn't a contingency we've remotely looked at. Those CO2 levels are gonna be getting toxic.
Well, I suggest you gentlemen invent a way to put a square peg in a round hole. Rapidly.
August 10, 2008
"I'd like to share a revelation during my time here. It came to me when I tried to classify your species. I realized that you're not actually mammals. Every mammal on this planet instinctively develops a natural equilibrium with the surrounding environment but you humans do not.
You move to an area and you multiply and multiply until every natural resource is consumed. The only way you can survive is to spread to another area. There is another organism on this planet that follows the same pattern. Do you know what it is? A virus."
Agent Smith, from The Matrix
I'm reminded of this scene from The Matrix after watching Olympic basketball this morning, the USA vs. China, and also hearing star basketball players may decide to play overseas in the coming years, lured by huge contracts
What has The Matrix have to do with International Basketball?
THE HIGH PICK AND ROLL
It's hard for me to watch NBA basketball anymore. The inconsistency of the referees. The non-calls at the end of games, for fear of "deciding the game" (a non-call where there should be a call DOES decide the game). The flagrant traveling, palming the ball, and block/charge variation are three further reasons. The horrible "flopping" is out of control.
But these are incidental parts of the game relative to a philosophy coming to dominate NBA play: the "high pick and roll".
It's not, usually, a pick and roll. Properly called, it's an illegal screen. OK - 1 time out of 100, it's not. 99 times, the offensive player is committing a foul.
Fine. The ball-handler dribbles off the illegal pick. What does he do? Of course, he's not close enough to the screen for it to rub his man off properly. I guess that doesn't matter much, since his defensive player goes beneath the screen anyways.
Who picks up the ball handler? The screening defensive player, of course, who jumps out and 75 times out of 100 himself commits a foul not called. This takes place, of course, as one (or both) of the defensive players is clawing and holding the offensive players. It's ugly.
Fine. We have a mismatch now, right? What's our illegal screening player doing? First off, 99.5 times out of 100, he's turned his back to the ball in moving to the basket, so he can't see what's happening, allowing the small guard to play both players momentarily, while his teammate, now done fouling the ballhandler, can "recover".
OK - lots of things going on.
You might be wondering: these are the actions of 4 players. What about the other 6? Well, the three offensive players are typically spread about the floor, standing. Their defensive players? A bit more active, because they need to be in a position to help, should something happen with our pick-and-roll.
So the game has become a dribble-dribble-dribble-dribble - high-screen - game, with lots of standing around.
I hope international teams don't copy this nonsense passing itself off as basketball.
I hope our players, going overseas, don't carry the disease.
August 11, 2008
It's a big one, I assure you! Actually, while
playing catch with my son in the front yard the other day in late
afternoon, I was marveling at the shadow I was casting. It seemed to
stretch 50 feet! The sun was low in the west, just above our
Of course, when the sun is directly overhead, I cast a little shadow - if any.
How does this change during the day?
So Isaac and I went out every two hours, marking my shadow. It was amazing how little it changed midday, but grew rapidly as the sun disappeared over the horizon.
I'm now inside, trying to model via spreadsheet what took place in the yard.
A "quick-start" looks as follows:
What I need is to model the sun moving from east to west overhead, with "me" standing in the middle - certainly a "Mike-Centric" view of the universe! I need to model the sunlight projection from the sun to me - and past me to the earth. The "shadow" cast, then, is the length from where I stand to the projection of the sunlight vector to the ground.
The key feature, now, is to find a way to simulate the sunlight moving across the sky, as in:
I've found the best way for me to model and present this information in a spreadsheet is to develop points, and graph the points as a line-graph. But these points are changing. How do I proceed?
A Starting Point
Rather than start with a general formula, let's start with a specific example. Let's assume a 45˚, when, plotted on a circle with radius 100, gives me a point P located about (71,71). This is the "incoming sunlight".
Let's plot "me" as "Ground Zero", so my "x-coordinate" is 0. Let's assume we're not to scale here, so I plot my height as 25. Therefore, the sunlight ray passes through points P and Q, onward to Point R, along the ground. Being on the ground, it has y-coordinate 0, and some unknown distance 'x'. That's what I need to find.
(more to come)
August 12, 2008
Architects of Their Own Future
An Educational Action Novel - A Viable Vision - The Goal for Education
I watched an old movie the other day, the relevant issue being an industrial "incident" prompting a regulatory commission investigation. A plant supervisor tells a foreman for the plant (who is working with the commission), "This shutdown is costing us money. Give them everything they want. Be thorough." He pauses, and then adds, "But be fast. Everyday we're down costs us money."
What a dilemma the foreman suddenly found himself in! Sure, the solution may present itself rapidly, but what if it didn't? What if "thorough" meant "very slow"? What's he to do?
Dilemmas, though not with the consequences of a faulty power plant, tug at us daily. Should I, as a programmer, document my work when I'm getting paid by the amount of code I write? I'll leave the debugging to the next poor fellow!
My bonus is tied to the number of subscribers I enroll? Don't blame me when many cancel their plans 6 months down the road when they can't pay. I've already cashed my check!
In other words, it's not my fault when my behavior is a product of how you will measure me.
I believe the K-12 educational system is similarly between "a rock and a hard place".
The principal goal of "Architects" is not to "solve" the problem of education, but rather to define the reason education does not improve like we think it should.
The Principal Conflict
Imagine you're teaching math to a group of 25 kids with differing abilities, having to get through material dictated not by you but by the district curriculum head, with state tests around the corner. In comes "consultant x" or "person y" carrying "great materials z" to revolutionize math. What would you say? "It looks great, but not right now". What else could you say? You can't possible do anything with the material, given your situation, right - even if you wanted to!
What's the conflict? You see the great things possible out there. New things. Exciting things. Great things. You want to integrate them into your class. You know things could be a lot better. You'd prefer things be taught in an interdisciplinary manner. The realities of your situation don't allow you to, however. You must focus on the existing curriculum, your specialty, and the important tests right around the corner.
In focusing on the existing curriculum, however, we should be able to improve so much we can integrate the "great stuff" out there - eventually - right?
If we could, why haven't we?
What if we could? What if we did?
What is the relationship between constraint management, the core problem, and simple and complex systems?
The Format of the Book
A word on the layout of the book: there are three "themes" intermixed. One regards high-stakes tests, beginning not with "how do we do better", but rather, "Why isn't anybody doing better?", particularly when all of the material is right in front of the kids (in the reading and science sections). The progression is through the reading, math, and science sections.
The second thread is provided as a "rest" from the first, and it deals with a basketball team struggling with the loss of key players and the eminent shutdown of the school. How can they compete? Though seemingly different than education, the issues involved are identical. Can we (and how) improve rapidly - now?
The third thread is my philosophical thread. These are Principal Ragnar's nightly walks / self-discussions, trying to make sense of systems, of improvement, of stagnation, etc. The six "peripatetic adventures", called in the book "Chautauquas", are:
Other Issues Considered
A non-exhaustive list of things considered in the book are as follows:
The relationship between the core problem and the constraint;
The "have a seat" injection, where, instead of trying to solve a conflict, you simply show your conflict to the other party;
The brutal consequences of injecting your "solution" on someone else's problem (The words "do you know what you should do" should be banished from one's vocabulary);
The "foothold", allowing one the ability to "get grounded" in a body of material;
"Conduction", integrating "induction" and "deduction" in solving math problems with perfect documentation;
"Iterative Effect-Cause-Effect", seeking causal explanation for natural phenomena, and integrating new experiences into one's understanding;
"System A" versus "System B" thinking;
Debate and Logic, not taught as separate and confrontational disciplines, but integrated into everyday materials.
The Plot Line
A retired principal. A struggling basketball team. Stagnant ACT scores. Washington High was running out of time. To make matters worse, the community was clamoring for more! Foreign language programs. Hands-on science. More technology.
What to do?
This is their story.
A story of survival. Of optimism. Of the possibility of improvement, now AND in the future. Of an achievable goal for education, with students as ...
ARCHITECTS OF THEIR OWN FUTURE.
August 13, 2008
We Watch The Game - But What Do We See?
Do We See All This?
This science of defensive work which enables four men to cover 180 feet of ground is the most fascinating part of modern baseball. It has become so intricate and involved that the spectator at a game of baseball between two highly developed teams really does not see the game at all. He sees the plays, the stops, the throws, the catches. He sees men shift and swing, change position, move forward, move back, move to the right or left, and then move back again, but all the beauty of the inside game is lost to him, nor does he imagine that behind each move is the mastermind of a field general. The spectator yells himself purple in the face because Johnson fumbles a grounder and wonders why the manager "doesn't release that big stiff?" for fumbling. Then he sits indignantly striving to imagine why the manager is plastering language upon Smith for failing to stop a ball he "couldn't have got anyhow."
"Inside baseball" is merely the art of getting the hits that "he couldn't have got anyhow."
Now watch this play closely. See whether or not you can discover what is going on. "Pat" Moran stoops behind the batter and hides his right hand back of his mitt. Ed Reulbach, pitcher, shakes his head affirmatively. Johnny Evers stoops, pats his hand in the dust, touches it to his knee and rests it upon his hip. Jimmy Sheckard trots twenty feet across left field angling in toward the diamond. Steinfeldt creeps slowly to his left; Tinker moves toward second base and Evers takes four or five steps back and edges toward Chance, who has back up five feet. Reulback pitches a fast ball high and on the out corner of the plate. Mike Mitchell hits it. The crowd yells in sudden apprehension. The ball seems a sure hit - going fast toward right field. Evers runs easily over, stops the ball, tosses it to Chance and Mitchell is out.
You saw all that. The ball was hit in "the groove" directly at the 7 1/2-foot gap the geometrician will say is vacant, yet Evers fielded it. Now this is what happened: When Moran knelt down he put the index finger of his right hand straight down, then held it horizontally on the top of his mitt. Evers saw that Moran had signaled Reulbach to pitch a fast ball high and outside the plate. He rubbed his hand in the dirt, signaling Tinker, who patted his right hand upon his glove, replying he understood. Then Evers rested his hand upon his hip, signally Sheckard, the outfield captain, what ball was to be pitched. Sheckard crept toward the spot where Mitchell would hit that kind of ball 95 out of 100 times. While Reulbach was "winding up," swinging his arm to throw the ball, Evers called sharply to Chance (whose good ear is toward him), and Tinker called to Steinfeldt. While Reulbach's arm was swinging every man in the team was moving automatically toward right field, in full motion before Mitchell hit the ball. The gaps at first base, between first base and second, over second base and between third and short, were closed hermetically, while the gap between Steinfeldt and the third base line was opened up 22 feet. The ball, if hit on the ground, had no place to except into some infielder's hands, unless Reulbach blundered and Mitchell "pulled" the ball down the third base gap. Every man on the team knew if Reulbach pitched high, fast and outside, Mitchell would hit toward right field. The only chance Mitchell had to hit safe was to drive the ball over the head of the outfielders, or hit it on a line over 7 feet and less than 15 feet above the ground. If Reulbach had been ordered to pitch low and over the plate, or low and inside, or a slow ball, the team would have shifted exactly in the opposite way.
Every club worthy the name uses the same system, but it is in the major leagues (the American and National) that it reaches its highest perfection.
The Science of Baseball
John Evers & Hugh Fullerton
Yes, this is the same "Evers" in "Tinkers-to-Evers-to-Chance", here writing on "The Science of Baseball" in the phenomenal book "Touching Second". Incidentally, it was also Evers who, while playing second, pointed out the base-running error of Fred Merkle in the 1908 pennant race between the Giants and the Cubs. A ton can be written about that "heads-up" play, and the actions taking place earlier in the year that made that "heads-up" play possible!
But I divert.
Reading the above excerpt from Touching Second, I'm struck by the amount of "intellectual work" going on during every play. Unbelievable.
But a thought creeps into my head - everybody is moving during the play. Everybody knows what's going on. How does this affect the hitter?
A simple example to illustrate.
Mike is batting and Jim is pitching. Jim's coach (team name Cougers) has statistics showing I'm an inside hitter, and I can't go with the pitch. The coach signals to Jim: low and away. The signal is given to the fielders: the "shift is on". They're all moving, knowing the pitch will be low and away, and if I do hit, it's likely to be hit to the right side.
What am I doing at the plate? I see all this, don't I? Seeing shifts and knowing what they mean, won't I scoot up to the plate, so I can pull the ball, and I don't have to reach as much?
Have I negated all the strategy?
Is strategy like this necessary, if we're each guessing at the potential actions of the other party?
The Metaphor of The Strange Attractor
This thought generates the image of a "strange attractor" from dynamical systems ... here are a few, randomly generated ...
August 14, 2008
Michael Phelps' phenomenal success, workout regimen,
diet, and physical anomalies have been well documented during these
His career total of 11 gold medals will possible rise to 13 this weekend.
This shear total has earned him "The Greatest Olympian Ever".
The 1912 Summer Olympics (the Games of the V Olympiad) were host to what may be the greatest athletic performance, certainly in the history of the Olympics, but also world athletic competition!
The legendary Jim Thorpe won not only the decathlon, but also the 5-event pentathlon!
For the 1912 Summer Olympics in Stockholm, Sweden, two new multi-event disciplines were on the program, the pentathlon and the decathlon. A pentathlon based on the ancient Greek event had been organized at the 1906 Summer Olympics, but the 1912 edition would consist of the long jump, the javelin throw, 200-meter dash, the discus throw and the 1500-meter run.
The decathlon was an entirely new event in athletics, although it had been competed in American track meets since the 1880s and a version had been featured on the program of the 1904 St. Louis Olympics. However, the events of the new decathlon were slightly different from the U.S. version. Both events seemed a fit for Thorpe, who was so versatile that he alone had formed Carlisle's team in several track meets. He could run the 100-yard dash in 10 seconds flat, the 220 in 21.8 seconds, the 440 in 51.8 seconds, the 880 in 1:57, the mile in 4:35, the 120-yard high hurdles in 15 seconds, and the 220-yard low hurdles in 24 seconds. He could long jump 23 ft 6 in and high-jump 6 ft 5 in. He could pole vault 11 feet, put the shot 47 ft 9 in, throw the javelin 163 feet, and throw the discus 136 feet. Thorpe entered the U.S. Olympic trials for both the pentathlon and the decathlon.
He easily won the awards, winning three events, and was named to the pentathlon team, which also included future International Olympic Committee (IOC) president Avery Brundage. There were only a few candidates for the decathlon team, and the trials were cancelled. Thorpe would contest his first—-and, as it turned out, only-—decathlon in the Olympics. Thorpe's Olympic record 8,413 points would stand for nearly two decades.
King Gustaf V of Sweden
Legend has it that, when awarding Thorpe his prize, King Gustav said, "You, sir, are the greatest athlete in the world."
Chapter 4 (part 1)
August 15, 2008
In Search of Meaning
Grass seeds and snowflakes. What was going on here? They both “grow”, but at the same time, they grow differently, don’t they?
Driving into town, I see a massive oak tree in the distance. I decide to drive over. Scattered about the ground are myriad acorns. They are everywhere. Many will be eaten. Others will rot. But a few will find their way to becoming oak trees like that in front of me.
But what does that mean? How does it happen from this simple acorn comes the massive oak tree appears?
Back to the snowflake. It’s remarkable in the beautiful symmetry and complexity of its patterns, but this is the result not of information embedded in the initial water molecule, but … what? Where does it structure come from?
If there’s nothing “from the outside” telling the snowflake how to grow, and if there’s nothing “from the inside” telling the snowflake how to grow, how does it grow?
“Information and rules” seems to be the common theme here, but is it? Oak trees, snowflakes, and grass clippings … is that it? I’ve never really thought of the world like this, but maybe it’s time. Are these the two main classifications of things?
I decide to take a survey of the environment, first in search of anything that comes into being because it has some form of “DNA”, or information.
I’m surprised at the variety …
Amazing! But what about the other side, the side governed by "rules". Is there anything here? I'm off to a slow start
(more to come)
August 16, 2008
THE ROAD NOT TAKEN
Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;
Then took the other, as just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,
And both that morning equally lay
In leaves no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.
I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I—
I took the one less traveled by,
And that has made all the difference.
So spoke Robert Frost in "The Road Not Taken", telling us why "The Road Less Traveled" made all the difference. Frost came to mind recently when I was out for a nighttime stroll ...
A Random Road Normally Traveled
A casual nighttime walk.
The road a narrow line.
I came upon a boulder,
That changed my state of mind.
This rock caused me to stop
And to my choice give sight.
Which direction should I choose?
To the left or to the right?
My good friend, Robert Frost,
Suggests the road not taken.
Since I was off the beaten path,
BOTH routes were forsaken!
I didn't see it mattered.
The logic of my choice.
"Just choose it randomly!"
I said in a quiet voice.
To my left I jutted,
And to my fright I saw.
Seemed to be the present law!
I continued as I had,
each move an equal chance.
in math parlance!
I finally arrived
A jagged, ragged, route.
The logic of my process ...
Was sound, I had no doubt.
But then I got to thinking,
Suppose I start again.
Where would I end up?
Anywhere from One to Ten!
Just as I suspected
The flight of a different bird.
Repeat the process many times.
"Iteration" is the word!
To the left and right I end,
Though the center seems frequent.
What I need is tabulation.
No counting accident.
The data is complete
And to my wondering eyes.
The random route I took.
Is normal in disguise!
August 17, 2008
THE PROXIMATE EVENT
A Science Fact / Fiction Novel Introducing the High-School Aged Student to the Plausibility of the Computational Universe
The Center for autoSocratic Excellence
The notions of dynamic systems, cellular automata, systems dynamics, fractals, variability, adaptation, etc., have become commonplace in the past few decades. Academics and professionals alike use the concepts in the creation of models and simulations to improve their work and understanding of reality.
However, this intellectual structure has made little inroads into K-12 education, and what are often included are simple and independently taught techniques. Is there a role for a “computational curriculum” for youth? This presentation introduces a “novel” novel, seeking not as a primary goal the introduction of these concepts to schools, but to high-school age students outside the school.
It’s tempting, when seeking to communicate a new idea, to immediately “get it to these people”. A new math idea? Let’s get it to math teachers! A new approach to teach dialogue? Let’s get it to the English teachers. How have such methods worked in the history of education? The history of math over the past ½ century gives us a good idea. It’s not a good idea.
Why not? Recognizing the teacher in the classroom, with 20 kids of varying ability working with a curriculum outside their control, it’s no wonder they won’t incorporate new ideas. They can’t.
But does this doom the new idea?
The goal of this novel is to introduce these ideas to the high-school-age student in a different arena, by way of a scientific novel.
Will a student pick up a book like “A New Kind of Science” and read it? Likely not. It’s huge, and it has a unique starting point: 1d-ECA. This was the crucial experiment, so why not start there? Complex behavior from incredibly simply rules.
This grabs the adult, but would it grab the student?
Likely not. What would?
This novel starts with “A Proximate Event”, an innocent visit to the home of Snowflake Bentley. Attempting to determine how likely it is “no two snowflakes are alike” leads the protagonist, Michael Johnson, to the idea of modeling and the nomenclature of cellular automata. He travels through a series of natural steps – of differentiating between organisms that grow because of the “information” within them, and as the result of processes external to them.
His focus becomes one of intensely looking at reality, for how and why it works.
Variability, adaptability, computational equivalence and irreducibility, are all naturally introduced, leading Michael to a profound definition of “life” and a belief the “computational universe” is plausible.
This 70-page novel is intended to be read with ease within an hour, with any of the 7 chapters providing the reader the opportunity to create the models and simulations themselves – on their own time.
August 18, 2008
No, Not "Phelps IS Leonidas", but rather "FROM LEONIDAS TO PHELPS" ...
Stay with me ...
Everybody knows Darius I of Persia wanted to conquer Greece and bring it under the Persian Empire. In 490 BC, the Persian Fleet landed at the Bay of Marathon, and despite superior forces, were defeated.
To announce the victory to his fellow Athenians back in Athens, Pheidippides ran the distance, proclaimed "We Have Won!" and died on the spot.
King Darius left the task of defeating the Athenians to his son --- Xerxes I.
March they did towards Athens and Sparta. What to do? Here comes the mighty Persian empire to smash them! What to do? To head them off, one must get to the bottleneck. Thermopylae! And not just anyone, but the might Spartans - led by Leonidas!
What might have been ...
The original Olympics Games, of course, are Greek in origin. From Olympia, Greece, and originating in 776 BC and running through 393 AD, they were resurrected as the "Modern Olympic Games" in 1896, the first games held, of course, in Athens.
A Visual Summary
August 19, 2008
Eccentric behaviour is typically "not normal" behavior.
What got me thinking about this word was the discussion earlier regarding
the length of my shadow as the sun passes overhead throughout the day.
Of course, the sun does not pass overhead. We're rotating about our axis, while revolving about the sun - the former giving rise to night/day, the latter the seasons.
But what is the relationship between this and "eccentricity", you may wonder? Our orbit, we've always been told, is elliptical, and the level of "flatness" of the ellipse is its eccentricity.
The Path of the Orbit
The path of orbit is dictated principally by foci. They are the "focus" of the path. The combined distance between a point on the ellipse to each focus is constant. That's why it's called an ellipse.
So a question: what happens if the two foci are not apart (like above), but are instead together?
My friend, the circle! That's one end of the spectrum - the focus are as close together as possible. What happens why I reach the other end of the spectrum, where the focus are as far away as possible?
New definitions for "line" and "circle" to add to my growing collection.
A thought creeps into my mind: a while back, I was talking about fuel trucks and volume. How much gas to these beasts carry? In that case, the truck driver told me the capacity of the truck, and I assumed the truck was spherical. It was a reasonable estimate, and I was just playing around, then.
What would it take to figure out the area, volume, and perimeter of this truck, now that we are on the subject?
Where did the formula for the area of the ellipse come from above? I didn't know it, so I looked it up. Why is it that way? I'll figure that out later. I need to get something on the table, so there it is. The circle provides a good check, however, as setting the two distances equal to one another, I see I get my familiar formula for the area of a circle. Good. It's not proof the formula for the area of the ellipse is right, of course, but it's OK for me right now!
To summarize these two areas, plus what I know about the perimeter (circumference) of a circle (where the radius = A), I have the following table:
But what is the perimeter (circumference) of an ellipse? I cannot find this formula anywhere! Surely, given the relationship between the other three formulas above, it must be something relatively easy.
It isn't. In fact, there is no exact formula for what I'm looking for!
How can this be?
How eccentric is that!
August 20, 2008
I know how to draw a circle. Use a compass. Trace the bottom of a drinking class. Easy enough.
How could I create a circle randomly?
Suppose I take a grid and, within that grid, randomly pick points. Is the point on the circle? What would I need to check for? I need to check the distance from the point to the center of my "imaginary circle".
Fine. Here, I've simulated 500 points and checked their distance to the origin. Here's the ones that "made it":
What's happened here? To be "on" the circle, the radius has to be pretty precise. Miss by "a little", and the point is not included. In fact, to even "be selected" for the circle above, there was some "tolerance".
Let's make it variable:
Now the randomly selected point does not have to be a particular distance, but only in a given range. OK - fine in theory. Let's strap the theory on our backs, run down the runway, and see if it will fly:
A NEW DEFINITION OF CIRCLE
the accumulation of points randomly selected falling within two tolerance limits, as that tolerance difference approaches zero.
I'm often asked, with something like the above, "what good is it"? Likely I would have asked myself the same question years ago. No longer. In many math things, "relevance" does matter. In others, it's clearly just "playing around". But to what end? Joyful investigation? Why not?
But in moving forward "joyfully", many skills are used. Math. Thinking. Programming. Error correction. Graphing. A ton of good things.
So, don't be offended if, next time you ask me "the relevance of it all", I respond with "I'm flying high - don't try to clip my wings". My wings can't be clipped without my permission.
August 21, 2008
August 22, 2008
Some work I did on The Montessori Method, about a decade ago.
August 23, 2008
August 24, 2008
The riddle is often told: from my campsite, I walk 1 mile south, 1 mile east, 1 mile north, and, arriving at my campsite, find a bear. What color is the bear?
The answer, of course, is a polar bear, because the coordinates above take place at the north pole (they actually take place at an infinite number of places on earth [find general, please], and the "bear" clue narrows the possibilities from "infinity' to "one").
But the joke brings to mind a thought about random walks. What would happen if I randomly took 1 step north, south, east, or west? Where would I end up?
A Starting Point
What happens if, from this point, I start walking, randomly, one step at a time, north, south, east, or west? I can retrace my steps if I want, or I can go off "to the milky way", as Dr. Deming would say.
Here's one scenario:
That was 24 steps. What happens if I let my walk continue for 100 steps? Instead of capturing each step, let's let the algorithm run, and simply capture the entire set of 100 steps. Each image below, then, equals 100 steps, and, after completing these 100 steps, I return to the origin (0,0), and start anew:
Sometimes my finishing point is right back where I started. Others, I wander aimlessly across one of the four quadrants. I sometimes crisscross across the grid, all over the board.
What would happen if I plot only the ending points of these simulations? Let's see.
Another Definition of Circle
Can it be, after all of this "meandering", I've arrived at a new definition of circle? It seems the more simulations I perform and plot the ending points, the closer I get to a circle.
Might a circle then be defined as the set of ending data points of a random-walk process, as the number of scenarios approaches infinity.
Chapter 4 (part 2)
August 25, 2008
What at first had been a clumsy search became an unshakable habit.
Everything presented itself in terms of rules and information.
But was the classification right?
The idea of “informational DNA” seemed right. After all, a person is essentially this kind of “data” at birth. That classification seemed OK.
What about the other category? Sand dunes and spider webs. Rainbows and canyons. Are these simply the result of “rules” outside the objects themselves? We don’t see rainbows with green outside of red. We don’t see sand dunes where the peaks and valleys are perpendicular to one another. Why not?
Because it’s contrary to the nature of the objects – to the rain and sun in the first example, to wind and sand in the latter.
Things are created in accordance to their nature.
The pattern of creation had changed.
I walked about the back yard, happy with my discovery. The world presented itself to me in a new way – I saw “a new kind of order”. Colorful flowers, a black and yellow bee, a bird soaring overhead, the clouds in the sky, the gentle breeze in the air, the parade of ants on the ground, it all spoke a new language to me.
I stood in front of a small tree. The patterns were everywhere, from the separating of the branches to the growth of the leaves. The tree was no longer just a tree: it was a system, growing in accordance with rules present in the nature of the tiny seed giving rise to the giant tree.
I pulled two leaves from the tree. They were nearly identical. It was beautiful, the symmetry, the order.
While admiring these two leaves, my eyes caught an oblong-shaped leaf from a different part of the tree:
While the similarities were present, the differences were clearly noticeable. How could there be any differences? Looking closer, I, of course, saw these minor differences existed everywhere. How could this be if my understanding above was correct? How could a rule present in the tiny seed give rise to this many differences?
I decided to collect leaves from all four similar trees in the front yard.
Where was all this variability coming from? And coming from different trees, how was it possible for there to be both similarity and differences in the leaves and the trees?
August 26, 2008
I stood aside the basketball court
My bad knee made me lame.
I watched in awe
The game see-saw.
I wish I could play the game.
To play the game meant play it well.
The rules preventing blame.
The court, the ball,
I know it all.
I know how to play the game.
To get to the court I take the bus
I dream of the Hall of Fame.
Practice and drills
It gives me thrills!
The token takes me to the game.
But in school things are different
There I feel much shame.
Math and history?
They're a grand Mystery.
I've no "token" for the game.
On and on the teacher goes
"This helps you in life", they claim!
The bell it tolls
and out I stroll.
I don't even like the game.
Education's the passkey to success in life.
To this I do proclaim!
The foothold's a token,
To make me outspoken!
Please let me play the game.
Chapter 5 (part 1)
August 27, 2008
The Same and the Different
How are things both the same – and different – and the same time? The question nagged at me for days, and I found no means of investigating the idea of “variation”.
What I was looking for was a way to “track” the changes in a model or simulation, to see how change takes place over time, and I’d not found a good way to do it. My “snowflake” investigations, I realized, demonstrated a point back then – how crystals might grow according to a rule – but the model did not afford me a means of following the changes over time.
Now, it was late at night, and I’m watching a summer thunderstorm, the intermixing of brilliant lightning followed by booming thunder. It was a marvelous show indeed!
Watching the path of the lightning, I got an idea. It seemed lightning, in going from the sky to the ground, had to travel through the space between the two. That was obvious. However, the lightning periodically changed directions, as though encountering some obstacle and veering around it. Perhaps it was magnetic. Whatever the cause, it gave me the idea on how to model this phenomenon. With such a model, perhaps I could better see how variation might occur in a simple system.
Let’s start with a model – something resembling the movement of lightning – and see what we can do. My understanding of lightning suggests the current encounters “things” and then moves, one way or the other. Whether that’s right or night is not relevant – right now. What can I create with similar features?
Suppose I encounter a field of boulders. At each boulder I encounter, I have a choice: go to the left or go the right. For example:
OK – I hit the first boulder and veer to one side. Which side? Let’s let my choice be random, and see what happens:
Suppose I continue this for a number of steps. What does my path look like?
A ragged route indeed! Of course, my choices at each point were randomly taken. If I retraced my route, likely the course would be different.
Let’s see how different! Here are 15 such simulations:
What am I to make of these patterns? Most of the time, I end up somewhere in the middle, though infrequently I end up on one of the ends.
Setting the model to simulate not 24 paths but instead 10,000, I add in the feature to keep track of the totals, and hit “run”.
August 28, 2008
A Side Note
If the pattern above seems familiar, it should. That's the nature of a good pattern, after all! This familiar pattern, used often in how I reason, was most prominently used in the Hamlet presentation in Seattle, which since became a book. The power of the pattern in that instance was it allowed me a start when I knew absolutely nothing about Hamlet. With the Santa Fe Trail, the opposite is the case - I "know" a lot about the Trail, but none of it is connected. The "Foothold" Pattern requires a great deal of intellectual effort to "get started" - to "be in the game"!
(more to come)
August 29, 2008
Closing Notes for Today
If you yourself ever actually do these types of logic branch, you'll find you correct them all the time. The logic. The wording. The titles. The order. EVERYTHING!
To Be Enjoyed, It Still Must be Picked!
August 30, 2008
Problem: A goat is tethered to a post at a vertex of an equilateral triangle of area 2π acres. How long should the rope be to ensure the goat can eat at most 1/2 of the area of the triangle.
The "Old Me"
The "old me" is clearly scrambling for points. I leave the problem unsolved, but with a lot of impressive math on my paper. The teacher is unimpressed: Points scored: 3/10
The "Medium Me"
Arranging the facts logically, I breeze through the problem effortlessly, arriving at the answer in 30 seconds. Assured of my correctness, I'm astounded when the grade comes back: 6/10, with a note: square root of 6 what?
The "New Me"
The "new me" rarely leaves "units" behind, with numbers dangling with no meaning. I realize I must convert "acres" somehow, which I do. I arrive at my answer, scoring 10/10.
An easy problem - but just as easy to leave points (fruit) on the tree!
Of course, the "new me" would NOT stop here!
Chapter 5 (part 2)
August 31, 2008
Amazing – I’m not sure what it is telling me about my problem of variation, but I see the “normal distrirbution” appear before my eyes!
This is all fine, I decide, but it’s not getting me any closer to the goal of seeing if, from a simple start, an immense amount of variation can develop.
The lightning analogy, albeit neat, may not be what I’m looking for, I decide. After all, DNA, genes, and chromosomes don’t “choose” different cells from a “menu” of options – they “grow” from one state to another. The model I’m looking for should do the same.
What would happen if I took a combination of two cells and, proceeding in accordance with a rule, “grew” into a third cell.
What might the rules look like? What are the possibilities? There are four possibilities (combinations) of black and white cells. They are:
How might the application of these combinations into a rule look? One example: anytime two black cells are together, the resulting cell is also black. Additionally, any time either cell is black, the resulting cell is black, the resulting cell is black. In fact, what if we assume any time the two cells are white, the resulting cell is black! That is:
Let’s call this the “trivial case”. How does this rule play itself out in our pyramid of cells?
Let’s make one additional observation: in the first row, for example, there is only one cell, but my rule calls for two. Let’s assume the “non-existent” cells are always white.
OK … what do we have? Let’s see:
Nothing exciting seems to be going on here. A trivial rule, and the pyramid is all black. Big deal.
But there are other rules. Let’s apply them all to our pyramid and see what we come up with:
A fantastic array of outcomes! I’m not sure how it helps solve my issue of “variability”, but nonetheless it’s cool.
What I notice about the results of these rules is something very neat. In some instances, there is virtually no pattern created at all. The pyramid becomes all white or all black. Nothing neat there. There are also instances where there are repetitive patterns. Fine. In a couple instances, however, there are “nested” patterns. This is really neat. Three types of outcomes.
Let’s not lose track of why I was doing this: using “lightning” as my analogy, I tried to see what type of variability I could come up with. Many patterns and the normal distribution were the results. This didn’t help any. So I changed the model to something “rule-based”, hoping to see if I could encounter the variability I was seeing in the grass blades in the yard. Neat patterns, but limited variability. Here, I used only “two neighbors”. Would three make a difference? Four?
Was I barking up the wrong tree?