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Circle Packing within a Fixed Equilateral Triangle
March 1, 2010
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(As I rewrite this, the challenge will become: can I get a seventhgrader to not only understand this, but to do many of the exercises. The exercises  and the rewrite  are not included here. Let me know if you're interested).
Circle Packing within a Fixed Equilateral Triangle The pattern below is often called the “Sierpinski Triangle”, and indeed, it does resemble the famous gasket of Waclaw Sierpinski. It also resembles Pascal’s Triangle, coloring in all odd integers. Further, it mimics the “chaos game” of Michael Barnsley. But because I enjoy completing this pattern in a rowbyrow sequence, I view this from a cellular automata perspective – algorithmically.
While attempting to enclose these circles, I was asked an interesting question: “How much space inside the equilateral triangle surrounding the circles is NOT covered by a circle, whether white or black?” A good question!
To get a better idea of the problem, let’s take a look at some examples:
There seems to be much more uncovered space when there is just one circle, and this makes logical sense. As the circles get smaller and smaller, the uncovered space should be smaller. Right? Of course, as the circles get smaller and smaller, it’s also the case the number of uncovered spots grows as well. What is the effect of the increasing number of smallerandsmaller uncovered spots? Let’s see.
Circle Packing within a Fixed Equilateral Triangle My plan: if I knew the area of all the circles inside the triangle, and then knew the area of the external triangle, I could subtract the former from the latter and get the “uncovered space”.
The Area of the Inscribed Circles Let’s build some structure into this process, to ensure my calculations are heading in the right direction. My first goal was to get the total area of all the circles inscribed in the triangle. What do I need to make this happen? Since all circles are of equal size, I need only find the area of one circle, and then, coupled with the total number of circles, I will have the total area of all circles. That is:
I know, with a circle or radius r, the area of the circle is given by:
How many circles are there? Well, this depends on the total number in the bottom row. I also see each higher row has one less circle than the lower row, leading up to the top row, which has one circle. 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 area of the circles inscribed in the triangle?
Now I know – and I’m halfway done (maybe)! Stay tuned for Part II!
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where coincidence or luck has worked in your favor, and use this phrase. Report back with the context!

Circle Packing within a Fixed Equilateral Triangle (II)
March 2, 2010
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The Area of the Exterior Triangle After enclosing my pyramid of circles within an equilateral triangle, I have a diagram like this.
All I’ve labeled is the radius of the similar inscribed circles: r. Let’s call the length of each side of the equilateral triangle S. If I can find what S is relative to the radii r, and find the height in a similar manner, then I can find the area of the triangle.
The Area of the Exterior Triangle After enclosing my pyramid of circles within an equilateral triangle, I have the following. As I’m just looking for the length of the side, and because I have an equilateral triangle all sides are equal, I’ll truncate my figure to the relevant section.
A large section of this I can define directly in terms of the circle radius r. However, what happens in the corners of the triangle? How do I account for this distance? Zooming in on the corner section and applying some information I know about triangles, I can break the figure into a triangle including only the line segment for which I lack the distance.
I have basic relations for a “30/60/90” triangle, and can use these as the basis for finding the length of my segment in question. Given this basic relationship, and knowing the radius r of my circle is also the height of my triangle, (which is also the distance of the segment opposite the 30º), I know the length of the segment in question is rÖ3. It would seem I’ve answered the question, because I have all the pieces: Before I get all excited I’ve got the total distance of the side of the triangle:
I need to remember, in this example, I’ve only used 3 circles in the bottom row. What is the formula if I had 1 circle? 10 circles? In counting circles earlier, I derived a general formula for n circles. Can I do the same here? Let’s see. Regardless of how many circles are in the bottom row, the only two of concern are the corner circles, and I now know the distance of the segment to the corner.
My goodness, am I done yet? Not quite. Stay tuned for Part III:
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where trouble or misfortune exists, and use this phrase. Report back with the context!

A Fable
March 3, 2010
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Chapter 1 Forgive me if I seem out of breathe as I tell you my story, but I've just spent a good deal of time removing the boulders that have rolled onto my driveway. "Boulders?", you may ask. It does seem odd, and I wouldn't have believed it myself only four years ago. It wasn't suppose to be this way. And now, most every morning, I awaken to clear the debris that covers my driveway. My address is 14397 Consequence Lane. But let me start at the beginning. Years ago  has it been only five  I lived atop a beautiful mountain overlooking the countryside. The name of the mountain: Principle Mountain. (insert picture of mountain)
Others joined. We laughed. We enjoyed each other's company. We worked hard. We traded. We worked things out. It was hard work to get to the top of Principle Mountain. We had a team of oxen pulling a beautifully outfitted wagon up a trail. It was hard work. The team was led by the driver, Joe Smith. When still others came, we ran into a bit of a problem. They, too, wanted land. Those of us there first, having claimed the land to ourselves, weren't so eager to sell our land. "How can you own the whole mountain?", they asked. They were polite, and they were right. This simple instance showed us we needed some mechanism to formally work things out. A constitution. A governing body. But with certain restrictions. Some residents of Principle Mountain weren't so sure. "If the governing body gets too powerful, what are we to do?" "Vote them out." "What happens if the people who vote continue to vote them in?" "That's what 'democracy' is all about!" (insert cloud)
"But suppose it's just two wolves and one sheep, voting on what to eat for lunch?" "That will never happen." Time passed. More people came. And Joe Smith's team grew. He now had four wagons. It was hard work, but he had a good business Some people complained about access to the mountain. Particularly those who lived on Slippery Slope. (insert picture of Principle Mountain and Slippery Slope)
"Joe's got a monopoly on wagon travel. It's not fair." Joe didn't have a defense. Anyone was free to start a wagon business, and he was the only one who had. For years. Nonetheless, the Governing body passed a law regulating the price of the wagon rides. Further, they mandated Joe allow other's who wanted to start a wagon team but didn't have the money to be able to use Joe's oxen team! Joe normally worked only 9:00  5:00, but now his oxen were forced to pull the heavier and heavier wagons from 7:00  7:00. (insert picture of wagon team)
Chapter 2 It was Sue who found the old cave. Thousands of years old, and pictures on the wall! Surely evidence of some culture long since gone. She took the news to the governing body. "It must be preserved!" "But how do we pay for the preservation?" "We'll charge everybody 1 cent out of every dollar they make." "A tax?" "What's one penny out of a hundred, to preserve such a remarkable bit of history?" There was a vote. It was a landslide victory in favor of the tax. 95% of those living on Slippery Slope voted in favor of the tax. 95% of those living on Consistency West voted against the tax. Unfortunately, most people now living on Principle Mountain lived on Slippery Slope. The cave was preserved. But it needed upkeep. The governing body was consulted. They decided to turn the Cave into a Hotel, with room rates subsidized by the tax revenue collected from the people living on Principle Mountain.. Martha Jones, the hotel owner on Slippery Slope, wasn't happy. "If you turn the Cave into a Hotel, you'll put me out of business. And I'm the one who helped pay for the preservation of the cave? I voted in favor of it?" "We're sorry. The issues were unforeseen. To make matters worse, we're going to have to raise the tax to 2 cents to pay for the renovation." The tax was raised. Martha Jones was put out of business. She moved down to Hypocrisy Court, just off Consequence Lane (which would eventually be my neighborhood).
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where uncertainty exists, and use this phrase! Report back with the context!

An Open Letter to the Medical Profession
March 4, 2010
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An Open Letter to the Medical Profession
A cruise ship is swamped in the Mediterranean.
Earthquake victims in Haiti and Chili.
A young boy gets hit in the chest with a pitch during a baseball game.
What do these seemingly unrelated topics have in common?
They all give rise to the clarion call: "IS THERE A DOCTOR IN THE HOUSE?"
The doctor always answers.
Thank you.
And after our President has accused your profession of intentionally cutting off feet for profit in the treatment of diabetes, and of removing tonsils simply as a means to make more money, I'm offended.
And I'm equally offended when I see your profession stand behind the President, supposedly speaking for the entirety of your profession.
Yes, there are many things wrong with our current system. But as a physician, you know it's ludicrous to treat the systems of a problem. Unless you treat the core problem, these systems  and likely others  will continue.
You've not helped the patient.
What are the root issues  the core problems  in our current health care system? What are the causal connections between these  and other  entities?
Address the problem  like you would in medicine.
Logically.
And if, after analyzing your patient's symptoms, you come to a diagnosis of cirrhosis, would you recommend as a cure to the problem more alcohol?
What role has government regarding the many problems we face now?
They're significant.
Please don't quit, like Dr. Hendricks did in Atlas Shrugged. Why did he eventually leave the profession?
"I quit when medicine was placed under State control, some years ago," said Dr. Hendricks. "Do you know what it takes to perform a brain operation? Do you know the kind of skill it demands, and the years of passionate, merciless, excruciating devotion that go to acquire that skill? That was what I would not place at the disposal of men whose sole qualification to rule me was their capacity to spout the fraudulent generalities that got them elected to the privilege of enforcing their wishes at the point of a gun.
I would not let them dictate the purpose for which my years of study had been spent, or the conditions of my work, or my choice of patients, or the amount of my reward. I observed that in all the discussions that preceded the enslavement of medicine, men discussed everything—except the desires of the doctors.
Men considered only the 'welfare' of the patients, with no thought for those who were to provide it. That a doctor should have any right, desire or choice in the matter, was regarded as irrelevant selfishness; his is not to choose, they said, only 'to serve.' That a man who's willing to work under compulsion is too dangerous a brute to entrust with a job in the stockyards—never occurred to those who proposed to help the sick by making life impossible for the healthy.
I have often wondered at the smugness with which people assert their right to enslave me, to control my work, to force my will, to violate my conscience, to stifle my mind—yet what is it that they expect to depend on, when they lie on an operating table under my hands?
Their moral code has taught them to believe that it is safe to rely on the virtue of their victims. Well, that is the virtue I have withdrawn. Let them discover the kind of doctors that their system will now produce. Let them discover, in their operating rooms and hospital wards, that it is not safe to place their lives in the hands of a man whose life they have throttled. It is not safe, if he is the sort of man who resents it—and still less safe, if he is the sort who doesn't."
Ayn Rand Atlas Shrugged
I close with two words rarely said during the discussion of what to do with health care in the United States: "Thank you."
Sincerely, Michael Round
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where you see something likely responsible for lack of success, and use this phrase! Report back with the context!

Transforming Data into Information
Prediction?
March 5, 2010
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Let's suppose I asked 50 people to guess a number between 1 and 25. The results might be reported as follows: Should I ask what to expect from the next person, well, the answer is tough: somewhere around 13? Obviously. That's the average. Add in some measure for variability? Sure. Use standard deviation. How much? Let's suppose I wanted to be 90% right? The number of standard deviations to meet this requirement is 1.28. Therefore, my 90% confidence interval is given by: 90% confidence = average +/ (1.28)(stdev) Lots of impressive math. Is it helpful here? Let's look at the data from another perspective: visually: The data is all over the place, so even if I make a prediction based on my impressive math formula, it probably won't be that close. Furthermore ... If I asked 50 more people, the results would look different. After all, we're talking about randomness. Let's take some of that out. Let's say I changed the rules. Here's the new data: What's the next guess? The answer is obvious, but let's consult our good friends, the tabular statistics, to make sure: What's this? They're the same as before, though the data is remarkably different! They provide no help whatsoever in predicting the next data point! Here are some more displays:
These are all radically different graphs. What do the tabular statistics look like?
Yes, the tabular data for all of these graphics is the same! Which means what? The use of these statistics may be relevant in describing what happened in our data. However, it seems meaningless for making predictions  sometimes. What's missing? If we had the visual image of the performance over time, would that help? That is:
Problem solved, it seems: if there is a discernible trend, use it. If there's not, use the underlying tabular statistics. Maybe. I showed my analysis to a friend, who read it carefully, but added: "It's not right." "Why not? It seems to account for everything!" He pointed to this figure and said, "I can imagine a scenario where I can tell you with 100% confidence the next person's results." "Impossible. It's all random." "To you, maybe. But suppose I'm the foreman in charge of the process. Person 1, I say, should guess '9'. Person 2 should guess '16'. And so on. There's no variability at all, in that they're meeting exactly my requirements!" "But I couldn't know that, from the data." "You're right, just as you didn't initially discern trends from the data. Once you accessed the data a new way, the results were obvious. And you had access to my forecasts  if you would only ask me!" "I see ... so, what does our new rule look like?" "You tell me?" "If there is an underlying forecast, use it. If there is a discernible trend, use it. If there's neither, use the underlying tabular statistics." "Maybe." "What do you mean, maybe?" "I mean 'I don't know'. I mean I use whatever is available to help me understand the context, and go from there. I don't have such a hard and fast rule. It's all about analyzing the situation, and everybody's specific situation is different." But often times, I've found the most important element necessary to analyze numbers  to transform raw data into information  is missing. Just some thoughts on "prediction".
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where you see the ridiculous praised while the good is ignored, and use this phrase! Report back with the context!

A Systems Dynamics Narrative on WorldChangers
March 6, 2010
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Wise Man: And the rain kept falling, for many days. It fell until the world was covered with a great flood. And when there were no more mountains left to stand upon, no dry land anywhere, Man above knew there was only one creature who could save the world.
And he called upon the creature to dive down deep below the water, and scoop up sticks and mud with his paws to make a great mound.
The creature worked hard and long, until he had made a new world  a world with mounds and forests and plains, for all the animals.
And only one creature could have saved the world then, one creature alone who can change the world he lives in.
The wisest of all animals on earth.
The beaver.
Son 1: Is it true?
Son 2: It's a story.
Pasquinel: Think about the river and the streams. The beaver still creates a world of its own.
Son 1: Dams?
Pasquinel: Where the fast water stops. It becomes a pond. And the insects breed. And the trout feed on the insects. And the sawbilled duck dives for the trout. Muskrats come, and the mink come. And the deer comes to eat the meadowgrass. And the moose comes to eat the watergrass.
It's a new world  and only the beaver can make it.
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where times are hard, yet you know they will improve, and use this phrase! Report back with the context!

A Fable
March 7, 2010
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Chapter 3 The next year was a hard year on Principle Mountain. Tourism was down because of a harsh winter that really hit the agricultural community hard. Fortunately, Joe Smith had a good stockpile of food set aside for his teams of oxen. "We're sorry, Joe", said the governing body, but Elmer doesn't have enough food for his cattle, so we're going to have to take some of your food to give to Elmer. We'll give you a good price." "$100 dollars? It's worth 10 times that! And besides, what about our Constitution. It doesn't say you can take things from one person and give to another." "These are extraordinary times, Joe. You know that. But I tell you what, we'll put it to a vote. After all, we are a democracy." "A vote? Wait a minute? You're putting my food up to a vote?" The resolution passed. Joe Smith's oxen team, now working harder, ate less. In the meantime, Mary's cafe, the "Do Drop Inn", was continuing to do good business. Sitting atop Principle Mountain, Mary had incorporated evening Bingo into her cafe, which drew people from all over the mountain. The price to play bingo was relatively small, but it was an attractive draw, and the mountain people enjoyed Mary's great cooking. Unfortunately, the governing body took notice of Mary's business as well, and, seeing tax revenues in decline, decided to award Elmer the right to run a "compliant" bingo parlor. Mary's restaurant, out of compliance, was forced to shut her bingo parlor down.
(More to come)
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where hatred, discrimination, or bigotry exists, and use this phrase! Report back with the context!

An Open Letter to the Lady at the Museum
March 8, 2010
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Madam: After dropping my daughter off at the University for Symphony practice, I drove around looking for a place to park and sit outside, the temperature a beautiful 62˚. The huge outdoor area outside the NelsonAtkins Museum provided an opportunity. The shuttlecocks, however, were less inviting. "Art"? I don't think so.
I've never really understood "art". But lately, I've been thinking about it a lot. From different perspectives. "Give it a try", I told myself. I found a parking spot, and in I went. Not to look at a lot of things, mind you. I think that's always been one of my problems with something like this: there's so much stuff to see I don't see anything. "Stay focused, Mike", I told myself. I went to the front desk. "Impressionism, please?" I asked, in the form of a question / plea. Section 31. Off I went. On the way, I did what I told myself I wasn't going to do. I looked in other displays. I didn't understand any of it. And that which I did, well, it was often just paintings of people. Why is that interesting to me? I arrived. To a huge painting on the wall. I had no idea what it was. Lots of colors. FOCUS, MIKE! My inside was yelling at me. I approached the little card seemingly written in 6point font. "Water Lilies" by Claude Monet. "Water lilies," I told myself. I can see that  now. While I continued to read, you walked up, and to your partner I heard you say, "This is whatever you want it to be." Instead of saying, "This is Water Lilies", by Claude Monet, I let you walk away. Really, I didn't know what to say, because when I first looked, I didn't know what it was either. Why is that? Is that what the artist intends? My thought is Monet would be offended if he heard what you just said. If there's that amount of ambiguity in the work, does that not suggest the problem may lie at the hand of the artist? Maybe. On the other hand, if the slightest bit of reading allows me to see the Water Lilies, does that not suggest the viewer needs to invest a bit of time? Likely both. So, Madam, I'd like to share with you a bit of research I've done on Monet and Water Lilies. When I got home, the first thing I did was a Google Image search on "Water Lilies". You wouldn't believe what I saw. It wasn't the picture we saw at the museum. In fact, what I saw was a lot of pictures! Here's one screen shot: "How can this be?" I thought. A bit of research on Monet helped here.
Monet, it seems, moved to Giverny, France, and was amazed at the variety of things taking place in his pond. Colors. Life. Variety! This was the majority of his work for the last 30 years of his life. 250 paintings of water lilies! TWO HUNDRED AND FIFTY! What we saw was merely one! Something else that made this possible, as simple as it may seem, was the En Plein Air movement taking place in the 1800s. "In the open air". What made this possible? The box easel and paint in tubes. It took the artist outside. Having said that, watch the following video, imagining Monet sitting at his pond, easel and tubes, looking at his pond  really looking at his pond  for a new view:
I now can't wait to go back to the Nelson to see this painting. But let's not stop there. Let's invest a bit more in the process so special to this series of paintings: really looking at reality. This is what Monet did. Let's try it, with something very simple. For example, I just went outside to look at my lawn. My simple grass. What did I see? Blades of grass? I can do better than that. The insects? The individual blades? The brownness mixed in with the green? The compactness of the grass recovering from a hard winter? And what don't I see? The roots under the ground? The sunlight transforming the brown grass to green? The chemical reactions taking place in the grass? The absorption of water? If I take the time, it really is right in front of me  if I only look. What do I see? Now I really want to go back to the NelsonAtkins and look at this huge painting. What did he see? What did he paint? And if you decide to now go back, one thing I'm certain: you won't say, "This is whatever you want it to be." Sincerely, Mike Round
Ps: You may object to all of this "analysis". You analyzing nature and reality in this method, do we not take away from the wonder of the moment? Can’t we marvel at the beauty of the water lily without breaking it down into scientific analysis? Richard Feynman, the great physicist, I believe addressed this point wonderfully in “The Pleasure of Finding Things Out”: “I have a friend who’s an artist and he’s sometimes taken a view which I don’t agree very well. He’ll hold up a flower and say, “Look how beautiful it is,” and I’ll agree, I think. And he says  “you see, I as an artist can see how beautiful this is, but you as a scientist, oh, take this all apart and it becomes a dull thing.” And I think that he’s kind of nutty. First of all, the beauty that he sees is available to other people and to me too …
At the same time, I see much more about the flower than he sees. I could imagine the cells in there, the complicated actions inside which also have a beauty. I mean it’s not just beauty at this dimension of one centimeter; there is also a beauty at smaller dimensions, the inner structures. Also the processes, the fact that the colors in the flower evolved in order to attract insects to pollinate it is interesting  it means that insects can see the color. It adds a question: Does this aesthetic sense also exist in the lower forms? Why is it aesthetic? All kinds of interesting questions which shows that the science knowledge only adds to the excitement and mystery and the awe of a flower. It only adds; I don’t understand how it subtracts.”

Circle Packing within a Fixed Equilateral Triangle (Part III)
March 9, 2010
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Where did we leave off? I was trying to find the area of a triangle and the area of the circles embedded in the triangle. It's proving more difficult than I thought. But I also realize I can work my way through the process, if I only take my time, and organize the thoughts  logically. That's what I've done. I'm now trying to find the area of the external triangle. This is where I ended up  the length of a side of the equilateral triangle:
That being said, I know I can now find the base of the triangle (1/2 the length of the side). Given that, and given the general relationships of a 30/60/90 triangle earlier noted, I can find the height of our triangle, as follows: Fine. We're almost done! We've got the distance of the base of the triangle, and the distance of the height: Therefore: And the last touch: this gave me the area of the triangle above with the altitude based on ½ the length of the equilateral triangle. Therefore, the area formula for my equilateral triangle with ninscribed circles on the bottom row, each with radius r, is:
Circle Packing within a Fixed Equilateral Triangle: the process Given n circles on the bottom row, each with radius r, the area of all circles is:
and the area of the external triangle is:
Let's try a few and see what we get:
This decreasing ratio tells me the amount of uncovered space is diminishing as the number of circles increases. This makes sense, as more and more circles are crammed into a fixed space. Visually, this makes sense as well. I wonder, if I increase the number of circles dramatically, if the amount of uncovered space is removed altogether. Let’s see.
Extending the Number of Circles Inscribed in a Fixed Equilateral Triangle: Exponential Circle Growth The uncovered area does continue to decrease, though it seems to tail off when the number of circles is increased. Let’s see what happens when the number of circles grows exponentially, rather than linearly:
The uncovered area approaches – but does not exceed – approximately 1.10266 – it appears. If it exists, it’s called a horizontal asymptote. Using limiting theories from calculus, let’s see what happens not when n=10 or n=100, but rather when n→∞:
which is
approximately 1.102657790844, as we predicted earlier! Circle Packing within a Fixed Equilateral Triangle
When someone's says they're happy to see the sun, be it after a long winter or a series of cloudy days, a good response: "And what's Mr. Sunshine saying to you?" When you get the likely "huh" look, respond with: "How do ya do! Doris Day? Calamity Jane? A Woman's Touch?"

An Open Letter  Responding to a Response to My Open Letter
March 10, 2010
comments? 
THE PLEASURE OF FINDING THINGS OUT
Dear Sir: You like what I've written above about art, but have reservations. "I've offered a prescription on how to appreciate art", you claim. "Who am I to dictate how to appreciate art? Why don't I recognize the different learning styles?" Let's check some premises. Decades past, I was taken on field trips to the museum, shown "great paintings", and attended "art appreciation" classes. I've been asked, "What do you see, Mike?" I've never seen anything. Worse than that, I've been ridiculed for admitting I don't see anything. "You don't see it?" Nobody  except me  seems to have cared that many people were seeing many different things in the paintings, not caring whether that was the intent of the artist. Subjectivity is praised, but ignorance is condemned. I object to this, but that's another story. Let's get back on topic. I, like most people (I suspect) were thrown in to do the work of a carpenter in building a house with neither tools nor knowledge of how to use them. And here I am  46 years old, unappreciative of this "cultured" realm. The professional educational art community abandoned me long ago, just as the professional mathematical community has abandoned many students  for decades. Until now. Now, I've given myself tools. To help me. And they work. And you object? A whole new arena of thought has cascaded about me! Van Gogh! Renoir! Degas! Picasso! Everything has come to life! I've brought it to life! My understanding goes deep! When I look at paintings now, I have an unbelievably beautiful understanding of what I am seeing. And herein lies a very interesting part of the story – perhaps it is the story. Previously, I would look at a painting, and thought I was looking merely at a painting. Upon close examination via logical structures, I realized there was so much more to the painting than I was seeing, and a rigorous but joyful process of investigation led me to an unbelievably deep understanding of the painting, the artist, the background, the context! It’s tempting to think the scientific mind has taken over – that I would suddenly start looking at the structure of things, trying to understand them, and this approach may take away from “the beauty of art”. Interestingly – perhaps amazingly – with this continued analysis, my experience of the painting returns to “it’s just a water lily”, though I am not seeing “the same water lily”; I look deeply into the logical nature of the water lily:
And with this, I’m reminded of a quote from the great martial artist Bruce Lee: “Before I studied the art, a punch to me was just a punch, a kick was just a kick. After I’d studied the art, a punch was no longer a punch, a kick was no longer a kick. Now that I understand the art, a punch is just a punch, a kick is just a kick.”
But the above, I believe, is a description of a Zen state! Is that what’s going on here?
“The Painting is the Painting” If the experience is rewarding, start there and look to understand it better. Let's call this "dynamic quality". It's grabbed your attention. Great! If that doesn't trip your trigger, start with the understanding, as I've done with water lilies, which allows me to go back to the painting with great admiration! Let's call this "static quality". And you see, now, the "iterative" nature of what's going on above. There's no prescription on how to enjoy art  merely multiple entry points on how to enjoy the piece of work!
But why stop here, kind sir. You may still be unconvinced of anything I say. Let's put it to the test: you take 10 kids at random and go to the Joslyn Museum in Omaha. You explain the art anyway you like to the kids. I'll take 10 kids at random and go to the NelsonAtkins, investigating works of art as described above. Let's report back and see what we've got. I eagerly await your reply. Sincerely, Michael Round
How many times do you hear a person who wants to tell you a story or something interesting say, "I don't know where to start". A good response: "As they said in The Sound of Music, 'Start at the very beginning  it's a very good place to start!"

The Visual Display of Information
"Coal and Commerce Through Olathe, KS"
March 11, 2010
comments?

Maps are funny things. They provide a great visual layout of the terrain. But in doing so, they may provide too much of a visual display. Here's one example: A Rail Map of the United States. There's clearly a lot going on here, and there's a lot to be understood. For example, two cities more than any stand out due to the confluence of rail lines: Chicago and Kansas City. What else stands out? The density differences? There's probably a lot of small railroad lines included here. What if we look at just the major rail lines  the "Class I" Railroads? This simplifies it a bit, and gives us an idea of where the major carriers operate  I think. The graphic is still congested. What's another way to look at this data? Let's break it down by primary carriers: The Burlington Northern / Santa Fe (BNSF)
The Union Pacific
Kansas City Southern \
NorfolkSouthern
CSX Transportation
The geographic differences are striking! Having said all of this, I'd like to embark on two different journeys, and then revisit our maps above. To be continued ...
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where you see intentional deception for a purpose, and use this phrase! Report back with the context!

March 12, 2010
comments? 
Some Background I’ve always liked designs, particularly ones drawn with rulers. Now that I can program them, I do, but they still look the same as when I once used a ruler. Here are a few:
There are many more, of course. And, in the course of programming these designs, I’ve stumbled upon other types of mathematical designs, algorithms, etc. Pascal’s Triangle
The Chaos Game
Sierpinski’s Gasket
New Kind of Science Cellular Automata
A Common Theme Of all these designs, there’s a common image popping up lots of places:
Before we start to explore this image, I was asked an interesting question, as a friend watched me draw this diagram: “How much space inside the equilateral triangle surrounding the circles is NOT covered by a circle, whether white or black?” A good question!
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where you feel hatred directed at a person in charge, and use this phrase! Report back with the context!

March 13, 2010
comments? 
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where you are stuck in discussions with a good person, and need to "break the ice", and and use this phrase! Report back with the context!

A Key to Creativity?
March 14, 2010
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The Great Pyramids of Egypt: Their magnificence is matched only by their intrigue. How were they built? I've never bought any of the theories on their construction. Take the "ramp" idea:
Ridiculous. Which isn't to say I ever had anything to offer by way of alternative theories. I didn't. Time out for a moment. We all know the story of Einstein and relativity. Lessknown is Eratosthenes, the man who "measured the earth" (among other things). What do these two have in common? Einstein worked in the patent office. Eratosthenes worked in the Great Library at Alexandria.
Exposure. Time in. A while back, I wrote a brief article on the marvelous ASB bridge in Kansas City, a verticallift bridge spanning the Missouri.
The counterweights descend, pulling the lower span up. Think about that  several times. And now think of how the Great Pyramids might have been constructed.
Your task, should you decide to accept it (from Mission Impossible): Identify a circumstance where you have a good idea not accepted, and and use this phrase! Report back with the context!

March 15, 2010
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Beware the Ides of March 
An Introductory Chapter (Part II)
March 16, 2010
comments? 
Before we start to explore this image, I was asked an interesting question, as a friend watched me draw this diagram: “How much space inside the equilateral triangle surrounding the circles is NOT covered by a circle, whether white or black?” A good question!
A Starting Point To get a better idea of the problem, let’s take a look at some examples:
(here is inserted 6 examples where the student must draw and color the above images themselves to get a feel for the problem  and an intuitive introduction to a possible answer) ... Do you see what’s going on? There seems to be much more uncovered space when there is just one circle, and this makes sense. As the circles get smaller and smaller, the uncovered space should be smaller. Right? Of course, as the circles get smaller and smaller, it’s also the case the number of uncovered spots grows as well. What is the effect of the increasing number of smallerandsmaller uncovered spots? Let’s see. But “to see”, we need to formulate a plan.
My Plan If I knew the area of all the circles inside the triangle, and then knew the area of the external triangle, I could subtract the former from the latter and get the “uncovered space”. For example:
Or, said differently:
How do I do all of this? Let’s make sure we know what we’re doing by starting with a single circle:
A Starting Point If I know the area of the equilateral triangle, And if I know the area of the circle, Then I can figure out how much area is left over. Let’s get started. What do I know? Not a whole lot. Let’s draw our diagram: Still unsure what to do, let’s label what we know about triangles to see if this helps:
Now I’ve got a bit of a starting point, because I know how to find the shaded area. I also know, this being an equilateral triangle, I can now find the total area of the triangle. That is:
But what now? I seem stuck. To find the area of the triangle, I need to know something about the length of a side, and I don’t. Or do I? What else do I know about my diagram? Can I add the radius of the circle? I don’t know it, but I can still draw it. And in drawing it, I’ve got a new triangle. An interesting triangle. What do I know about anything in my new triangle? Quite a bit. Fill in the blanks.
My New Triangle
I know all three angles inside the darker triangle. I know the length of one of the sides (the radius). Can I find b from this? I don’t see how. But let’s try.
When someone asks what you're so happy about, tell them "You love to laugh  loud and long and clear".
(Perhaps remind them "it's getting worse every year"!)

An Introductory Chapter (Part III)
March 17, 2010
comments? 
An Introduction to Trigonometry For example, suppose my base angle is 45˚. If I measure out this triangle, I see it’s the same as going up one, and over one, up one over one, etc. That is:
The total rise (4) equals the total run (4). If this is the case, my base angle is 45˚. The reverse is true as well: it’s also the case if my base angle is 45˚, then the rise equals the run. What other angles can I add to my table? How about 90˚? There is never any “run”!
What other angles can I add to my table? How about 0˚? There is never any “rise”!
Let’s record all of this information:
Now we have an idea what’s going on, and how to calculate “rise” and “run”, let’s do this for a series of angles. (Approximate the “rise” and “run” the best you can). THE TANGENT GRAPH
Compare your approximations with exact calculations. How’d you do? (We’ll talk about the other columns in a bit).
To be continued ...

March 18, 2010
comments? 
The first rail bridge across the Missouri  in Kansas City, Missouri .. THE HANNIBAL BRIDGE.
The first rail bridge across the Mississippi  in St. Louis, Missouri ... THE EADS BRIDGE
In being the first, these bridges determined the fate of these towns  and others.

An Introductory Chapter (Part IV)
March 19, 2010
comments? 
Let’s remember where we were. We were trying to find b below, and all we knew were the angles, and the radius of the circle. Now we know what to do, because we know there’s a relationship between “the rise (1) and the run (b)”! I have b. So what? Does this help me find the area of the triangle? After all, that’s the goal of all this. Yes. But now I need h! How do I do this? The same way we did earlier in finding b. Do this yourself now, keeping in mind what constitutes “rise” and “run” – these are from the perspective of the angle you’re considering! Solve for h below: And FINALLY, I can return to my initial equation, now with both values b and h. I also know, this being an equilateral triangle, I can now find the total area of the triangle. That is:
I think we’re almost done – with part 1!

An Introductory Chapter (Part V)
March 20, 2010
comments? 
Let’s remember what we’re doing here. First finding the area of the
triangle, we then wanted to find the area of the inscribed circle, so we
could discover how much area is not covered. My Single Circle I’ve got my single circle with radius 1. I know how to find the area of this circle: And finally tabulate all of our data: Let's take a rest, because you've got a lot of work to do before we continue ... To be continued ...

The Tall Office Building  Artistically Considered
March 21, 2010
comments? 
the tall office building artistically considered by Louis H. Sullivan March, 1896
It is not my purpose to discuss the social conditions; I accept them as the fact, and say at once that the design of the tall office building must be recognized and confronted at the outset as a problem to be solved a vital problem, pressing for a true solution. Let us state the conditions in the plainest manner. Briefly, they are these: offices are necessary for the transaction of business; the invention and perfection of the high speed elevators make vertical travel, that was once tedious and painful, now easy and comfortable; development of steel manufacture has shown the way to safe, rigid, economical constructions rising to a great height; continued growth of population in the great cities, consequent congestion of centers and rise in value of ground, stimulate an increase in number of stories; these successfully piled one upon another, react on ground values and so on, by action and reaction, interaction and inter reaction. Thus has come about that form of lofty construction called the "modern office building". It has come in answer to a call, for in it a new grouping of social conditions has found a habitation and a name. Up to this point all in evidence is materialistic, an exhibition of force, of resolution, of brains in the keen sense of the word. It is the joint product of the speculator, the engineer, the builder. Problem: How shall we impart to this sterile pile, this crude, harsh, brutal agglomeration, this stark, staring exclamation of eternal strife, the graciousness of these higher forms of sensibility and culture that rest on the lower and fiercer passions? How shall we proclaim from the dizzy height of this strange, weird, modern housetop the peaceful evangel of sentiment, of beauty, the cult of a higher life? This is the problem; and we must seek the solution of it in a process analogous to its own evolution indeed, a continuation of it namely, by proceeding step by step from general to special aspects, from coarser to finer considerations. It is my belief that it is of the very essence of every problem that is contains and suggests its own solution. This I believe to be natural law. Let us examine, then, carefully the elements, let us search out this contained suggestion, this essence of the problem. The practical
conditions are, broadly speaking, these: This tabulation is, in the main, characteristic of every tall office building in the country. As to the necessary arrangements for light courts, these are not germane to the problem, and as will become soon evident, I trust need not be considered here. These things, and such others as the arrangement of elevators, for example, have to do strictly with the economics of the building, and I assume them to have been fully considered and disposed of to the satisfaction of purely utilitarian and pecuniary demands. Only in rare instances does the plan or floor arrangement of the tall office building take on an aesthetic value, and thus usually when the lighting court is external or becomes an internal feature of great importance. As I am here seeking not for an individual or special solution, but for a true normal type, the attention must be confined to those conditions that, in the main, are constant in all tall office buildings, and every mere incidental and accidental variation eliminated from the consideration, as harmful to the clearness of the main inquiry. The practical horizontal and vertical division or office unit is naturally based on a room of comfortable area and height, and the size of this standard office room as naturally predetermines the standard structural unit, and, approximately, the size of window openings. In turn, these purely arbitrary units of structure form in an equally natural way the true basis of the artistic development of the exterior. Of course the structural spacings and openings in the first or mercantile story are required to be the largest of all; those in the second or quasi mercantile story are of a some what similar nature. The spacings and openings in the attic are of no importance whatsoever the windows have no actual value, for light may be taken from the top, and no recognition of a cellular division is necessary in the structural spacing. Hence it follow
inevitably, and in the simplest possible way, that if we follow our
natural instincts without thought of books, rules, precedents, or any
such educational impediments to a spontaneous and "sensible" result, we
will in the following manner design the exterior of our tall office
building to wit: This may perhaps seem a bald result and a heartless, pessimistic way of stating it, but even so we certainly have advanced a most characteristic stage beyond the imagined sinister building of the speculator engineer builder combination. For the hand of the architect is now definitely felt in the decisive position at once taken, and the suggestion of a thoroughly sound, logical, coherent expression of the conditions is becoming apparent. When I say the hand of the architect, I do not mean necessarily the accomplished and trained architect. I mean only a man with a strong, natural liking for buildings, and a disposition to shape them in what seems to his unaffected nature a direct and simple way. He will probably tread an innocent path from his problem to its solution, and therein he will show an enviable gift of logic. If we have some gift for form in detail, some feeling for form purely and simply as form, some love for that, his result in addition to it simple straightforward naturalness and completeness in general statement, will have something of the charm of sentiment. However, thus far the results are only partial and tentative at best relatively true, they are but superficial. We are doubtless right in our instinct but we must seek a fuller justification, a finer sanction, for it. I assume now that in the study of our problem we have passed through the various stages of inquiry, as follows: 1st, the social basis of the demand for tall buildings; 2nd, its literal material satisfaction; 3rd, the elevation of the question from considerations of literal planning, construction, and equipment, to the plane of elementary architecture as a direct outgrowth of sound, sensible building; 4th, the question again elevated from an elementary architecture to the beginnings of true architectural expression, through the addition of a certain quality and quantity of sentiment. But our building may have all these in a considerable degree and yet be far from that adequate solution of the problem I am attempting to define. We must now heed quality and quantity of sentiment. It demands of us, what is the chief characteristic of the tall office building? And at once we answer, it is lofty. This loftiness is to the artist nature its thrilling aspect. It is the very open organ tone in its appeal. It must be in turn the dominant chard in his expression of it, the true excitant of his imagination. It must be tall, every inch of it tall. The force and power of altitude must be in it, the glory and pride of exaltation must be in it. It must be every inch a proud and soaring thing, rising in sheer exultation that from bottom to top it is a unit without a single dissenting line that it is the new, the unexpected, the eloquent peroration of most bald, most sinister, most forbidding conditions. The man who designs in the spirit and with the sense of responsibility to the generation he lives in must be no coward, no denier, no bookworm, no dilettante. He must live of his life and for his life in the fullest, most consummate sense. He must realize at once and with the grasp of inspiration that the problem of the tall office building is one of the most stupendous, one of the most magnificent opportunities that the Lord of Nature in His beneficence has ever offered to the proud spirit of man. That this has not been perceived indeed, has been flatly denied is an exhibition of human perversity that must give us pause. One more consideration. Let us now lift this question into the region of calm, philosophic observation. Let us seek a comprehensive, a final solution: let the problem indeed dissolve. Certain critics, and very thoughtful ones, have advanced the theory that the true prototype of the tall office building is the classical column, consisting of base, shaft and capital the molded base of the column typical of the lower stories of our building, the plain or fluted shaft suggesting the monotonous, uninterrupted series of office tiers, and the capital the completing power and luxuriance of the attic. Other theorizers, assuming a mystical symbolism as a guide, quite the many trinities in nature and art, and the beauty and conclusiveness of such trinity in unity. They aver the beauty of prime numbers, the mysticism of the number three, the beauty of all things that are in three parts to wit, the day, subdividing into morning, noon, and night; the limbs, the thorax, and the head, constituting the body. So they say, should the building be in three parts vertically, substantially as before, but for different motives. Others, of purely intellectual temperament, hold that such a design should be in the nature of a logical statement; it should have a beginning, a middle, and an ending, each clearly defined therefore again a building, as above, in three parts vertically. Others, seeking their examples and justification in the vegetable kingdom, urge that such a design shall above all things be organic. They quote the suitable flower with its bunch of leaves at the earth, its long graceful stem, carrying the gorgeous single flower. They point to the pine tree, its massy roots, its lith, uninterrupted trunk, its tuft of green high in the air. Thus, they say, should be the design of the tall office building; again in three parts vertically. Others still, more susceptible to the power of a unit than to the grace of a trinity, say that such a design should be struck out at a blow, as though by a blacksmith or mighty Jove, or should by thought born, as was Minerva, full grown. They accept the notion of a triple division as permissible and welcome, but non essential. With them it is a subdivision of their unit: The unit does not come from the alliance of the three; they accept it without murmur, provided the subdivision does not disturb the sense of singleness and repose. All of these critics and theorists agree, however, positively, unequivocally, in this, that the tall office building should not, must not, be made a held for the display of architectural knowledge in the encyclopedic sense; that too much learning in this instance is fully as dangerous, as obnoxious, as too little learning; that miscellany is abhorrent to their sense; that the sixteen story building must not consist of sixteen separate, distinct and unrelated buildings piled one upon the other until the top of the pile is reached. To this latter folly I would not refer were it not the fact that nine out of every ten tall office buildings are designed in precisely this way in effect, not by the ignorant, but by the educated. It would seen indeed, as though the "trained" architect, when facing this problem, were beset at every story, or at most, every third or fourth story, by the hysterical dread lest he be in "bad form"; lest he be not bedecking his building in some other land and some other time; lest he be not copious enough in the display of his wares; lest he betray, in short, a lack of resource. To loosen up the touch of this cramped and fidgety hand, to allow the nerves to calm, the brain to cool, to reflect equably, to reason naturally, seems beyond him; he lives, as it were, in a waking nightmare filled with the disjecta membra of architecture. The spectacle is not inspiriting. As to the former and serious views held by discerning and thoughtful critics, I shall, with however much of regret, dissent from them for the purpose of this demonstration, for I regard them as secondary only, non essential, and as touching not at all upon the vital spot, upon the quick of the entire matter, upon the true, the immovable philosophy of the architectural art. This view let
me now state, for it brings to the solution of the problem a final,
comprehensive formula. Yet to the steadfast eye of one standing upon the shore of things, looking chiefly and most lovingly upon that side on which the sun shines and that we feel joyously to be life, the heart is ever gladdened by the beauty, the exquisite spontaneity, with which life seeks and takes on its forms in an accord perfectly responsive to its needs. It seems ever as though the life and the form were absolutely one and inseparable so adequate is the sense of fulfillment. Whether it be the sweeping eagle in his flight or the open apple blossom the toiling work horse, the blithe swan, the branching oak, the winding stream at its base, the drifting clouds, over all the coursing sun, form ever follows function, and this is the law. Where function does not change form does not change. The granite rocks, the ever brooding hills, remain for ages; the lightning lives, comes into shape, and dies in a twinkling. It is the pervading law of all things organic and inorganic, of all things physical and metaphysical, of all things human and all things superhuman, of all true manifestations of the head, of the heart, of the soul, that the life is recognizable in its expression, that form ever follows function. This is the law. Shall we, then, daily violate this law in our art? Are we so decadent, so imbecile, so utterly weak of eyesight, that we cannot perceive this truth so simple, so very simple? Is it indeed a truth so transparent that we see through it but do not see it? Is it really then, a very marvelous thing, or is it rather so commonplace, so everyday, so near a thing to us, that we cannot perceive that the shape, form, outward expression, design or whatever we may choose, of the tall office building should in the very nature of things follow the functions of the building, and that where the function does not change, the form is not to change? Does this not readily, clearly, and conclusively show that the lower one or two stories will take on a special character suited to the special needs, that the tiers of typical offices, having the same unchanging function, shall continue in the same unchanging form, and that as to the attic, specific and conclusive as it is in its very nature, its function shall equally be so in force, in significance, in continuity, in conclusiveness of outward expression? From this results, naturally, spontaneously, unwittingly, a three part division, not form any theory, symbol, or fancied logic. And thus the design of the tall office building takes its place with all other architectural types made when architecture, as has happened once in many years, was a living art. Witness the Greek temple, the Gothic cathedral, the medieval fortress. And thus, when native instinct and sensibility shall govern the exercise of our beloved art; when the known law, the respected law, shall be that form ever follows function; when our architects shall cease struggling and prattling handcuffed and vainglorious in the asylum of a foreign school; when it is truly felt, cheerfully accepted, that this law opens up the airy sunshine of green fields, and gives to us a freedom that the very beauty and sumptuousness of the outworking of the law itself as exhibited in nature will deter any sane, any sensitive man from changing into license, when it becomes evident that we are merely speaking a foreign language with a noticeable American accent, whereas each and every architect in the land might, under the benign influence of this law, express in the simples, most modes, most natural way that which it is in him to say; that he might really and would surely develop his own characteristic individuality, and that the architectural art with him would certainly become a living form of speech, a natural form of utterance, giving surcease to him and adding treasures small and great to the growing art of his land; when we know and feel that Nature is our friend, not our implacable enemy that an afternoon in the country, an hour by the sea, a full open view of one single day, through dawn, high noon, and twilight, will suggest to us so much that is rhythmical, deep, and eternal in the vast art of architecture, something so deep, so true, that all the narrow formalities, hand and fast rules, and strangling bonds of the schools cannot stifle it in us then it may be proclaimed that we are on the high road to a natural and satisfying art, an architecture that will live because it will be of the people, for the people, and by the people."
Questions to followup with: why did the steel structure come into being at that place and time? what was the status quo regarding "tall buildings"? what was the lighting environment like in buildings? how was water raised? how did elevators work?

(Another) Letter to the Lady at the Museum
March 22, 2010
comments? 
Good morning, Madam ... When I last wrote, I was providing information regarding Monet, in response to the painting at the NelsonAtkins, Water Lilies. I'd like to followup with some research done while recently in St. Louis. "St. Louis?", you may be thinking? Bear with me. This is the home of Scott Joplin, one of the founders of Ragtime  "ragged time". They challenged the status quo  and the status quo fought back. Why? We might hypothesize "people are resistant to change". It seems reasonable. That is:
The Scott Joplin House  in St. Louis:
However ... Below is the Wainwright Building  the remarkable "skyscraper". Really the first steelstructure skyscraper in the world.
This represented a radical departure in office building architecture. No longer was height restricted by weight. The steel structure allowed us to soar. Louis Sullivan did. A paradigm shift in office building architecture. If our theory above was right  regarding change, what we'd expect to see was a resistance to the office building. That is:
But there wasn't! It was embraced  enthusiastically! "Resistance to change"? Nonsense. Maybe. Maybe sometimes. Maybe sometimes not. This, too, is in St. Louis ...
Here is a status of Lewis and Clark, residing at the starting point of the expedition extending from the Louisiana Purchase. Remarkable. Change. It opened "the west" ... This, too, is in St. Louis ...
This is the Eads Bridge  the first bridge across the Mississippi. Lewis and Clark may have opened the west, but this bridge gave rail access to the west. A remarkable change. There was resistance here  from the ferry industry! This, too, is in St. Louis ...
You might be asking at this point, what this has to do with Monet? Why all this talk of paradigm shifts, change, etc.? We go into the museum and look at a painting. Big deal. But if one considers, at this time, the artistic paradigm was painting "realistic" pictures, the idea of painting "impressions", rather than exact images, was revolutionary.
And a hard sell. Huge resistance to change. And the Impressionists fought on, a sailboat against the wind  moving upstream! Think of it! Monet, Degas, Renoir  think of them as revolutionaries. Now, included in our earned knowledge of Monet, imagine him painting at the pond after a lifetime of battling the status quo, a wartorn veteran still manning the frontlines of artistic creativity. With his shield, rather than on it.
NOW what do you think of Water Lilies?
Sincerely, Michael Round

or: How to win $10,000?
March 23, 2010
comments? 
March madness. Brackets. Pools. ESPN runs one online  winner: $10,000. Enter as many times as you like. A thought came to mind: how many possibilities are there? Couldn't I submit all possibilities, and ensure myself a winner? Where would we start? What's the general formula for this problem? Using our standard template, let's get started:
Step 1: My Problem: How many possibilities are there with 65 teams? (Remember there is one playin game)
Step 2: Get SOMETHING On The Table: Let's start with a couple of the simplest examples. Let's suppose I have only two teams. Therefore, I have one game. The number of possible outcomes? 2  Team1 wins, or Team2 wins. Fine. What about four teams? There are three games  and I'm already seeing a pattern emerging. The number of games = number of teams  1. The number of possibilities? I could have Team1 and Team3 winning in the first round, and then Team1 winning the championship. or, I could have Team1 and Team3 winning in the first round, and then Team3 winning the championship. or, and on and on. Eight possibilities, with four teams  or three games.
Step 3: Write a Note on What to do: With two teams, there was one game, 2 possible outcomes With four teams, there were three games, and 8 possible outcomes. A general rule may be: the number of possibilities is 2^games (or 2^(teams  1)).
Step 4: Solve the Problem: If the general rule is # possibilities = 2^(teams1) and if I have 65 teams then # possibilities is 18,446,744,073,709,551,616.
Let's put this into our diagram:
Good luck! Bonus question: Say that number  in words!

The Next Chapter: Circle Packing
March 24, 2010
comments? 
The Next Step We’ve figured out the area of the triangle and the area of a circle inscribed in the triangle. But, remembering the original problem, there were many circles embedded in the triangle.
What we need now is to figure out what the total area is of all the circles in the triangle. A method:
Before, we said the radius of the circle was 1. Let’s keep that assumption, which means as the number of circles increases, the size of the triangle will change also. That’s OK. The ultimate problem is to find the relationship between covered and uncovered space. I know the area of the circle, then: A = pr^{2}. What I don’t know is how many circles there are. This depends. Let’s look at some examples:
"Given a certain number n, what is the sum of all positive integers from 1 to n?” How do we figure this out? Do we need to add all the circles up every time n changes? It seems so. Gauss’ teacher thought so, too, and to keep the young students busy one day, the teacher told the students to add all the numbers up from 1 to 100. Gauss paused a moment and blurted out, 5,050! How did Gauss, who became one of the greatest mathematicians of all time, do it?
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 are all equal. That is:
What do I have? The summation “11” 10 times, or a total of 110. Of course, I’ve added the series twice, so to get the sum of the integers from 1 to 10, I need to divide this total by 2, getting 55. That is:
Back to our question: “Given a certain number n, what is the sum of all positive integers from 1 to n?, we now have the answer:
All of this work has been in answer to the question: What is the total area of the circles inscribed in the triangle?
What about the area of the triangle? This is dependent on the number of circles (as we’ve held constant each circle’s radius). This is the next step. Next time.

The Next Chapter: Circle Packing
March 25, 2010
comments? 
Closing In! Let’s get some data on the table. Let’s figure out the answers when there is one circle, and two circles (on the bottom row):
The ratio of the covered area (is increasing, is decreasing) (circle one). That is, it seems:
But there is a lot of work, in using a calculator, to figure this out. Let’s put this into a spreadsheet and see what we get:
Extending the Number of Circles Inscribed in a Fixed Equilateral Triangle: Linear Circle Growth
Extending the Number of Circles Inscribed in a Fixed Equilateral Triangle: Exponential Circle Growth The uncovered area does continue to decrease, though it seems to tail off when the number of circles is increased. Let’s see what happens when the number of circles grows exponentially, rather than linearly:
Fine. We’ve got lots of data. Let’s plot it and see what we’ve got:
Graphing the Results
The uncovered area approaches – but does not exceed – approximately __________ – it appears. If it exists, it’s called a horizontal asymptote. Using limiting theories from calculus, let’s see what happens not when n = 10 or n = 100, but rather when .
What do you think will happen? _____________________________________________________
We shall see!

The Next Chapter: Circle Packing
March 26, 2010
comments? 
BACKTRACKING … It’s been a long road, but before we leave, I want to revisit an item: tangent.
The tangent of an angle q, we said, was equal to rise over run:
Part of the reasoning behind this was as “run” gets bigger, “rise” gets higher. There’s a relationship.
If we knew the “rise” and the “run”, we could find the angle. If we knew the angle and the “rise”, we could find the “run”. If we knew the angle and the “run”, we could find the “rise”. The three work hand in hand. But we’re forgetting something, aren’t we? The third side of the triangle: the hypotenuse!
If there’s a relationship between the angle, the “rise”, the “run”, surely the hypotenuse is affected as well, right? So, if tangent of an angle is “rise” over “run”, what can we do to incorporate the hypotenuse? There’s two ways: use rise / hypotenuse and run / hypotenuse:
Let’s work a few quick problems.
