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December 1, 2008
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Part 1 of 3
December 2, 2008
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I awoke in an instant, my eyes watery but alert.
Where was I? Sitting atop a make-shift platform, I saw, looking
across a wide landscape, a huge throng of people. "This isn't
now", I thought. "What is this?" Suddenly, Abraham Lincoln strode upon the platform! "This can't be! This is Gettysburg, and I'm here!"
You can barely see part of me --- look at the man in the front row with a beard. I'm behind him and to his left. Was this a dream? It must be, but it's so real! "Let's play this out," I thought to my dreaming self. "Four score and seven years ago ..." he started. I knew this. I was in quite a favorable position. I knew what was coming. "... Our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition all men are created equal." What made me do it, I don't know. Perhaps the "real-time" me told my dreaming self, "You've nothing to lose. Don't just sit there! Get up!" "YOU'RE WRONG, MR. PRESIDENT!" I yelled, and before anyone could interrupt me, I continued, bravely. "I know you're going to say we've come to dedicate a portion of this field as a final resting place for those who here gave their lives that that nation might live, and that it is altogether fitting and proper that we should do this. I agree with you." President Lincoln looked at his prepared speech, not believing what he was hearing.
I continued. "The reason we're here is 87 years ago, our Fathers declared all men were NOT created equal!" President Lincoln's security force moved in, but he waved them off, and invited me to the podium. Was this really happening? What had I to lose? This was a dream, right? I marched forward, brushing the man with the beard aside. "Yes, Mr. President. It's sadly true. We allowed slavery to continue in this country. We've passed laws dictating what states were to be free and which could be slave states. We've counted slaves as 3/5 of a person for the sake of representation, and now you want to tell everybody here this country was founded on the proposition all men were created equal?" "It's not true!" "What this battlefield teaches us is this: most wounds heal themselves. But a critical wound left untreated is fatal. This battlefield is the result of a critical philosophical wound festering for four-score and seven years." A man from the crowd yelled "His talk belittles our great constitution. Get him." My dream felt more real. What could I do? I held my arms up in a position announcing more was coming. "But let's be fair to our founding fathers. They were in a tough position. They realized a "united" states was necessary, but many states, still distrustful of government, wanted to maintain state sovereignty. As many of those states were southern states where cotton was king and slaves were many, slavery as an institution was allowed to continue." "But, on the other hand, in order to have a "united states" based on certain principles, it was necessary to recognize all people have inalienable rights, among these the right to life, liberty, and the pursuit of happiness. If this were the case, there could not exist the institution of slavery." I tried to diagram my image in the air:
"But we're here because there is no compromise with basic principles. And when we do compromise on basic principles, the bough is bound to break - and it has broken. Right here. On this spot. And the cradle has fallen." President Lincoln looked at me. "But what should the founders have done?" I woke up. The 3/5ths Compromise ... The Missouri Compromise ... The Compromise of 1850. The role of the Supreme Court in the Dred Scott case. The relationship between each branch of government. The relationship between the federal, state, and local governments. The nature of rights. I had work to do! Part 1 of 3 ... stay tuned.
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Tentative Chapter 1
December 3, 2008
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A QUICK START THE SCATTER PLOT
Let’s start with a problem: This is a cool spiral. I’ve seen many like it on the internet. Do a google image search on “spiral” and you’ll see many more.
How is this made?
Let’s try to figure out what’s going, first.
The points being plotted are merely numbers, so they must represent a distance. But what kind of distance? Let’s assume this is the distance from the origin (0,0).
Maybe this.
Since I don’t know, let’s assume it’s something simple: 10˚.
Is this all I need?
Let’s plot the data and see what I get.
But how?
An Example Let’s take the degree shift of 40˚ and a distance of 5. What can I do with these two facts? In my triangle below, I'm looking for distances 'x' and 'y' to plot. If I had these, then I would have the coordinates to plot any of the vectors.
Finding 'x' and 'y' But how do I find 'x' and 'y'? Do I know anything about my triangle? Trigonometry plays a role here. I know cosine and sine have a relationship between the adjacent, opposite, and hypotenuse sides of the triangle. Since I know two of the three facts, I can solve for the third.
Applying these formulas to my data earlier, I have the following coordinates I can plot.
But plugging these into my spreadsheet, I do not get points looking reasonable. The formulas, I’m certain, are correct, yet the results are not. Cos(90) = 0, I know, so something’s wrong - somewhere. What’s going on here? A bit of research reveals Microsoft Excel does not perform trigonometric calculations using angles, but rather by radians! Therefore, to properly use my formulas, I must convert all degree measurements into radians. What are radians – and how do I convert degrees to radians? Let’s find out.
Degrees to Radians I know something about the circumference of a circle, and I know this formula includes the circle radius r. But does this lead me anywhere? I’m still talking about “distance”, while I’m looking for something regarding “angle” or “degree”.
Radian, then, must refer to the angle carved out by the radius along the perimeter of the circle. And if one radian carves out one radius, and there are 2π radii on the circumference, then there are 2π radians in a circle.
I’m closing in on the answer to my question: how do I translate degrees into radians? Above, I gave an expression for one degree, but I don’t have one degree. I have lots of different degrees. Fortunately, the translation is now easy.
Applying these formulas to my data earlier, I have the following coordinates I can plot.
Plotting the Data This is great, but how do I plot these? The most frequent graph I use in this regard is the “scatter plot”. What does it look like when I apply it to these 10 points?
This is a good start, but let’s clean up the graph a bit.
Double-clicking on the outside of the graph activates the “Format Chart Area” option. I usually clear the border and the area (checking “none” on both).
Double-clicking on the inside of the graph activates the “Format Plot Area” option. I usually clear the border and the area (checking “none” on both).
Finally, I usually clear all gridlines by double-clicking on the gridlines and selecting “none”.
What does this leave me?
Awesome!
Plotting More Data Now I’m sure my method works for 10 points, let’s copy the data to include many more points, and see what this looks like
I love it! This was for 10˚. What happens when I change this to different angles? Here is 79˚.
Plotting this Data Differently You’ll notice I’ve plotted merely the points in those two graphs. I have the option to connect the points with lines. What does that look like? Let’s see:
Very cool!
Let’s get ride of the scales, as they seem to be visually “in the way”. I delete the x-and y-axis by merely clicking on them and pushing “delete”.
Automating The Process I notice quickly there are lots of patterns in here, but I tire quickly of typing in numbers, and pushing the “calculate” button. Can I automate any of this, so I can sit back and watch?
Of course.
But the key thing to keep in mind is the hardest part of automating a process is figuring out what you do when you do it manually.
What I currently do when changing something is going to cell “L8”, enter a new number, and then press “calculate” so all the new values are plotted.
This is what I want my “macro” to do. A macro is a set of commands telling the program what to do.
How do I program a macro? Fortunately, you don’t have to. A lot of the time, I turn on the macro-record button, and do what I normally do. The macro captures all of this.
Afterwards, I go in a tweak the macro a bit and out comes my custom macro!
Sitback Macro: Initial Recording Let’s see. I’ll use “44” as my angle-shift, for this example. What does the macro record? Let’s call our macro “sitback”, because that’s what we want to do:
Sub sitback() ' ' sitback Macro ' Macro recorded 12/2/2008 by Mike Round '
' Range("L8").Select ActiveCell.FormulaR1C1 = "44" Range("L9").Select Calculate End Sub
Sitback Macro: Try #1: Initial Cleanup Most of this is nonsense, and I first delete the unnecessary stuff:
Sub sitback() Range("L8").Select ActiveCell.FormulaR1C1 = "44" Calculate End Sub
Sitback Macro: Try #2 There are two steps here talking about cell “L8”. All I want it to do is recognize the value “44”. Let’s do this in one step:
Sub sitback() Range("L8").value="44" Calculate End Sub
Sitback Macro: Try #3 Nice and compact. However, this doesn’t solve my problem. I want it to enter a lot of numbers, not just “44”. To do this requires me to put a loop around all of the programming. Let’s call our loop “easy”, and let it run from 0 degrees to 360.
Sub sitback() For easy = 0 to 360 Range("L8").value="44" Calculate Next easy End Sub
Sitback Macro: Try #4 I run this and see my graphs never change. Why not? My loop goes from 0 to 360? But why doesn’t cell “L8” change? I see. I haven’t told the macro to put the changing value there! Easy enough to change:
Sub sitback() For easy = 0 to 360 Range("L8").value=easy Calculate Next easy End Sub
The Results? Here are 48 randomly chosen graphs … pretty neat!
A Thought I started out trying to mimic a spiral, and ended up creating something entirely unexpected. This happens 999 out of 1000 times, once you actually get into a spreadsheet and do the programming yourself.
You’ll notice something else as well. The “How To” – the formatting of the graphs, the creation of the macro – takes little time. The “What to” – the formulas – takes 95% of the time. “What to put into a cell – what formula” is the heavy lifting here!
“What good is all this?” You’re probably not thinking that right now – caught up in the thrill of creating something neat yourself. “What good is a newborn baby?”, the great scientist Michael Faraday once replied, regarding his work on electricity and magnetism.
“What good is it?” In the process of doing this, we used trigonometric functions, radian / degree conversion, and demonstrated two forms of graphing – Cartesian and Polar coordinates! Not bad for a day’s work!
Your Assignment We plotted positive consecutive integers above. What would the graph look like if we plotted even numbers only? Odd numbers? Multiples of 5?
How would you modify your spreadsheet to do this?
Your Assignment Part 2 Instead of plotting just integers, look at the series below. These are the first 10 prime numbers. What does our spiral look like when we plot prime numbers?
Find a website with the first 10,000 primes and bring them into your spreadsheet. What does the graph look like?
You're (hopefully) thinking, if you found such a site, how did they calculate all of these primes? You're probably wondering how you yourself could write a program to do the same - not just checking to see if a number is prime or not - but gathering a lot of primes?
How would you write a program to find the prime numbers yourself?
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An Introduction to the Syllogistic Dictionary
An Abstract Submitted to the 2009 Dictionary Society of North America Conference
December 4, 2008
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LOGICIONARY-1 Dictionary Society of North America Michael Round The Center for autoSocratic Excellence (913) 515-3911
Coming upon the word “abhor” recently, my son and I looked up the definition. “To regard with extreme repugnance – loathe”. Does “abhor” really mean “loathe”? If so, why not simply say “loathe”. If not, how does it differ? Why do we say “burgeoning” if “growing” suffices? Is the latter synonymous with the former? If not, under what context are the terms appropriate? A chair, for example, might be defined as a piece of furniture used for sitting. This simple definition defines both a common characteristic (genus) and the characteristic differentiating it from other furniture (differentia). How might this criteria be used in searching for applied definitions in all cases? Let’s suppose such a structure were viable. Is this sufficient for improved dictionary use? Can a visual syllogistic structure be used to improve readability, understanding, and use? What other characteristics might be added in building our own “syllogistic dictionary”? This presentation introduces the Logicionary, a genus/differentia-based dictionary, incorporating corpus-based examples of actual concept-usage.
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Part 2 of 3
December 5, 2008
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I awoke with a feeling of excitement. My dream
- I think - the night before had energized me to a part of history I had
thought little of in the past. The Gettysburg Address? Yes,
I could recite it with the best. But had I ever really thought
about the substance? Of the contradictions in it? No! My intellectual journey was on, now. I stepped outside to get the paper. There was no paper. There was no "Kansas"! Where was I? The city pulsed with horse-drawn carriages and well-dressed towns-people walking up and down the brick-layed streets. Where was this? I looked at myself - yes, I too was dressed like one of them! Was I still in my dream? Was this a different dream? I stepped outside, tentatively, and walked down the street, making careful note of where I'd started. Seeing a newspaper stand, I went to get straightened out. Philadelphia. 1787. June 4. This can't be! The Continental Convention had officially started May 25 when a quorum of seven states was secured. Others had arrived thereafter. Of course, this was a secret meeting. These delegates were meeting not to amend the Articles of Confederation, but to establish an entirely new constitution. Was this even legal? They were meeting in secret so as to not be distracted by the public. What kind of delegates are they, not wanting to be distracted by the very people they represent? I went to the Pennsylvania State House, where the Convention was taking place. It was guarded. Only state delegates were allowed in. How could I get in? "I represent the Spirit of the Boston Tea Party," I announced, and, building my credibility, asked, "Have Franklin and Revere arrived yet?" "Mr. Franklin is here already, but Mr. Revere of Virginia is not coming at all. He fears too much power will be taken from the states and given to the federal government, so he refused to partake in these historic undertakings." I walked in. The delegates were in a heated discussion, debating the dilemma of "large states" versus "small states", and the ideas of equal vs. proportional representation.. I stayed in the background, and found a chair as deliberations continued. No one noticed me. You can almost see me in this picture. I'm sitting beside the window in the back left, to the left of the three men standing against the wall.
The discussion was on slavery. Was this a coincidence, my jumping from place to place, with slavery the central issue. Northern states argued, for the purposes of taxation, slaves should be counted as people. Fine, said southern states' representatives. But if they're people, then we get to count them in determining representation in the House of Representatives. Nope. Fine. If slaves are not to be counted as people for the purposes of representation, then they must be property, and our states are not subject to taxation. Nope. Both sides clearly wanted to have it both ways: the north: count slaves as people for the purposes of taxation, but as property for the purposes of representation. the south: count slaves as people for the purposes of representation, but as property for the purposes of taxation! This was going nowhere. And then James Wilson of Pennsylvania stood up, announcing a compromise: the famous 3/5 compromise.
Gentlemen. "This discussion is going nowhere. Is there a middle road somewhere? Can we count a slave as a 'fraction of a person'. Both sides win: the issues of taxation and representation are both addressed." The state representations talked among themselves, and reached agreement this was an equitable solution. I sat quietly in the back, my heart racing. I knew where this would lead, because I saw where it had led. This discussion right now would eventually lead to the Dred Scott decision, and the Civil War. But could I tell them that? Not only would they not believe me, but I would be likely be kicked out and institutionalized! But what could I do? My mind raced. When I looked back at the chain of events leading to the civil war, I realized this was the turning point. Why can't I reverse the logic? If the chain of events is so certain, I ought to be able to convince these men of the logical consequences of the events that are taking place right now. It was worth a shot. "I object", I yelled, and stood up. "I came here to establish a United States of America - not destroy it." "And you are ..." "Yes I am," I quickly interrupted, not wanting to expose my identity, particularly when I really had no identity in that room. This was a dream, right?
"11 years ago, our Founding Fathers met to establish 'a new way'. Many of those men are in this room right now. Thomas Jefferson was the principal architect of our Declaration of Independence. We all know the words.
I went over to a copy of the Declaration, and read my relevant passage: We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security.
"Listen to what you all are saying. Are slaves property or people, and your answer is NEITHER? They're 3/5 of a person?" "What are we talking about?" "It's no wonder Paul Revere did not come to these proceedings! But since he's not, I'll invoke his words - changing just one: Give EVERYONE liberty or give me death!"
"But sir," Mr. Wilson continued, "if we free the slaves, no southern state will ratify this constitution. We will have no country. Is it better to give a little and have something, than have nothing at all?" "I'm saying your compromise here today does two things: most importantly, it denies freedom and rights to a group of people, which is contrary to the words in the Declaration; secondly, it ensures that this country will be at war with itself, not today and maybe not next year. Certainly, within 3 or 4 generations, the bough will break and the cradle will fall." I was treading on thin ice here. Stick to the logic of the situation, and avoid mention of any particulars, I kept telling myself. "What can you mean by this, sir? How do you know this? Prove it?" (part 3 to follow) |
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Tentative Chapter 2
December 6, 2008
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THE SCATTER PLOT Continued
2,500 Years Too Late Cleaning Up the Mess of Zeno
“THE PARADOX” PARADOXZeno of Elea is well known from ancient times for formulating interesting paradoxes regarding motion. Perhaps his most famous paradox is the “Tortoise and the Hare”, where he purportedly demonstrates a slow-moving tortoise, if given a head start, can never be overcome by a speedy hare.How can this be?Well, we’re told, surely the hare, in pursuing the tortoise, must move half the distance to the tortoise.But in the time it takes the hare to move this distance, the tortoise itself has moved. Hence, when the hare again attempts to overtake the tortoise, it must again move halfway to the tortoise. Clearly, every time the hare moves halfway, the tortoise has moved, albeit slightly.Hence, we’re told, the always-moving tortoise will never be overtaken by the rapidly-approaching hare, which must infinitely make up “half-distances”.Of course, we know in reality the hare does overtake the tortoise, just as a fast-moving runner overtakes the plodding jogger. Why did Zeno himself not recognize his logic did not conform with reality, and wonder himself where he went wrong?Richard Feynman, the great physicist, verbalized this wonderfully in “Surely You’re Joking, Mr. Feynman!”. While at Princeton pursing his graduate degree, Feynman was talking with the mathematicians, who claimed you could cut up an orange into a finite number of pieces, and, putting it back together, arrive at something as big as the sun.“Impossible”, claimed Feynman.When given the mathematical explanation about cutting the orange, Feynman interjected: “But you said an orange! You can’t cut an orange peel any thinner than the atoms.”When given further mathematical justification about being able to cut continuously, Feynman concluded, “No, you said an orange, so I assumed that you meant a real orange.”Indeed – dealing with reality.A GEOMETRICAL PARADOXICAL PERSPECTIVERather than deal with this specific paradox, let’s modify the behavior of the tortoise, and say he doesn’t move at all. What of the course of action of the hare? How can we visualize it? With the ending point stable, we need only graph the halfway point between the ever-changing starting point and the stable ending point.Let’s use our scatter plot just introduced to look at this problem. If I was going from 0 to 100, I want to plot 0, 50, 75, 87.5, etc. – that is, halfway to the ending point.But with a scatter plot, I can’t just plot 0, 50, 75, 87.5, etc. I need a second coordinate – a ‘y’ coordinate. Fine. I’ll make one up.What does the data look like? |
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December 7, 2008
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Take a guess on where my graph came from ...
DIALOGUE FROM CARL SAGAN'S CONTACT
KENT CLARK Shhh. You hear that?
ELLIE I hear it.
KITZ Hear what?
ELLIE Harmonics?
KENT CLARK Bingo. Retune to 8.9247 Ghz. There’s a lot more here folks.
ELLIE Alright Fish, let’s get on the negative side band.
FISHER On it.
KITZ What is going on?
ELLIE We’re tracking the signal at double the frequency, it looks like... somebody get a TV monitor.
WHERE TO START? As we were told in The Sound of Music regarding where to start, let's start at the very beginning. Why? It's a very good place to start!
In Volume #1 of =EQUALS=, the Mandelbrot Set was discussed. In the course of creating the Mandelbrot Set ourselves, we realized the idea of "escape value" was central to the calculations.
Briefly, this "escape value" told us how long to check a point to see if it's in the Mandelbrot Set. The idea was many patterns of numbers go like this: 1,2,4,8, and off to infinity quickly. Other sequences cycle through a pattern like this: 1, 1.5, 1, 1.5, etc. In both of these cases, we have an idea quickly about what's going to happen.
Other times, we don't have such a quick idea: what about this one:
It's tough to tell what's going to happen, because this thing seems to be bouncing all over the place. But we have to put a limit on how long we'll check, and we called this the "escape value". If our process escapes within a certain period, great. If it doesn't, we assume it never will.
Well, not exactly. We just mark the result as "not escaping".
In Volume #1 of =EQUALS=, we captured the images of certain escape values.
But is there a way to put all these images together into one consolidated image? Let's just color the image and see:
Maybe nothing to those of you who have used many wonderful fractal programs on the internet, but additionally wonderful to me since it was done in Excel. What else can Excel tell us? As the color changes, it means a process was at one point considered "under control" (meaning it didn't escape before a certain number of iterations), but when that threshold (escape value) was increased, it was found that behavior did change - and exceed our threshold distance. So there's some behavior issues going on. Let's see how the points change over time: Let's extract 62,500 points from our grid, equally spaced, from the following range:
and set our escape value to 1000. That is, if the sequence does not escape by then, we stop. We don't necessarily say it will never escape, but merely that it didn't by this point. We run the program. What's the distribution of points by 'escape value'? But first, what do we expect? Looking at the above graph (iterations 1-9), we see a lot of color changing. Each point within a color change means changing the escape value threshold changed the graph. So we expect a lot of early changing. Let's see:
Indeed, a lot of changing. Not included here are 15,140 points that made it all the way to the 1000th calculation, and were still within the threshold. But there's additionally a lot where the change took place above 25. What can be going on here? Fortunately, because we've created all of this ourselves, we can see. For example, point (0.03, -0.63) generates a sequence where the escape distance is breached only after 965 iterations! 965 iterations? What does the behavior look like? You may already be guessing the answer:
THE MARVELOUS MANDELBROT SET So, it's been said the mystery of the Mandelbrot Set may hold the secrets of the Universe. If so, I claim, as in Contact, there's a SECOND MESSAGE AT DOUBLE THE FREQUENCY!
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Part 3 of 3
December 8, 2008
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Chapter 3 How to Cause the Change? “Prove it?” All eyes were on me. How could I prove this without revealing my identity? I thought of the Civil War that was the inevitable result of the issue of Slavery. Why was it inevitable? I’ll focus on that to get me started. “Right now, we’re trying to resolve an important issue. My northern friends want to recognize all individuals have inalienable rights. If that’s the case, surely we must abolish slavery.”
The southern representatives started to clamor, but I raised my hands indicating I wanted to continue: “Let’s not forget how we freed ourselves from British Rule. We wanted representation. We wanted self-rule. Where does this take place? If it’s at the state level, then the decision on slavery is to be made at the state level. If that’s the case, souther states choosing to allow slavery must be given this right.”
And you think this compromise – of counting slaves as 3/5th a person will solve anything? Looking at my northern friends, I said: “Are you happy? Your principle has been violated! And you southerners, are you happy? Entirely?”
“But sir,” spoke one gentleman. “Isn’t compromise necessary in situations like this? What choice do we have?” “I think you’re right – sometimes. I think there are some instances – many instances – where you give a little, and things work out. But I think in other instances there are principles involved that cannot be compromised.” “And this is one of them?” I had to be careful here. I could alienate both parties if I told them this was one where general principles were at stake, and no compromise was possible. I don’t think this would help anyone. I had to change my plan. Arguing about the philosophical merits or demerits of slavery obviously wasn’t going to change anyone’s minds here. Tie all of this to the Civil War. That was the key. “It may or not be – I’m not a judge or jury. Will you grant me this: if I can show that the logical consequences of this compromise are dire, then this compromise is not a good one,?” “Granted!”, asserted Benjamin Franklin. It was the first time he had spoke since I entered the room, and his resonating voice carried much weight.
A general concern of everybody here is regarding “equal representation” in the legislative bodies, right?” “That’s right”, spoke up Mr. Wilson. “And we’ve agreed on two legislative bodies, right?” “Two bodies?”, responded Mr. Wilson. What are those? Oops. I must have jumped the gun. They must not have talked about this yet. I acted as if I had heard the comment somewhere else. “I’m sorry – I must have heard of this idea on my way here to Philadelphia.” I turned towards the representatives of Delaware. “You, being one of the smallest states, believe there should be equal representation.” I faced the contingent from Massachusetts. “And you, one of the largest states, believe there should be proportionate representation. A seemingly impassable gap. But the thought I heard discussed were two branches of equal strength: one branch (the Senate) where each state is represented by two persons. Another branch (the House of Representatives) is filled with representatives based on the population of the state. This way, each state is represented equally in one branch, and proportionately in another.
Franklin winked at me. Maybe this was the idea he was going to propose. “A great idea,” Mr. Wilson said, excitedly. “But let me get back to my point,” I said, “regarding representation. This 3/5th compromise makes you all happy right now, right?” “What does this compromise hold for you – in the future?” Franklin was intrigued. Maybe it was a thought he hadn’t considered. The future. What are the logical consequences of this discussion. Of course he had. He’s a scientist, for goodness sakes! “We know our colonies – states – right now are a small part of this continent, right? We know there’s land to the west – a lot of it – unexplored.” “What happens when we start expanding westward. What happens when Georgians, for example, move westward, and create a new state? It’s bound to happen? A northern representative shouted up. “The balance of power would be upset! I’m not going to sign any constitution where the south would have an unequal share of the power!” “That’s exactly my point,” I said, confidently. “The solution seems simple,” said Mr. Wilson. He was a talkative gentlemen for sure. I’ve since learned he spoke more than anyone else at the meetings. “We’ll simply not allow one state into the union at a time. We’ll make sure, if a free state wants admission to the union, we’ll not grant the request until we have a slave-state ready for admission, and vice-versa.”
“Wonderful!” It was not Mr. Franklin. I was liking him more and more every moment. “Here’s a simple example,” I said. “Right now, the south produces a huge amount of cotton, but to remove the cotton fibers from the seedpod takes a lot of time – and a lot of people. Many slaves do this, and it’s a laborious and time-consuming process. Let’s suppose a machine were invented to do this in place of slaves --- call it a ‘cotton-engine’. What do you think will happen?” “It would seem there would be less need for slaves, then. Does such a machine solve this dispute for us?” It was a gentlemen from the north talking. “Maybe – especially if there were a limited amount of land. However, we know that’s not the case. There is, to us at least, an unlimited amount of land to the west. The easier it is to harvest and produce the cotton, the more land will be seized. I don’t see how this will affect the slave population at all, perhaps only to increase the need.” It was Franklin who spoke up. “What a viscous cycle we’re creating, gentlemen Do you understand what he’s saying? I see it in an instant now. We’re going to expand. We know this. Compromises? Where will they lead us, but to generations of compromises.” I smiled at Franklin. I always liked him. Now I know why. I continued with other hypothesis: “Right now, Spain and France and England own a large part of this continent. Is it to far a reach to expect they’ll recognize they, being across the Atlantic Ocean, will want to cede this property to us – for a price? This land stretches north and south across the plains. You have a happy divide now, an agreement regarding the north being “free” and the south being “slave”. What happens when a northern state requests to be a slave state. Do you allow this? It’s a state’s issue, you’ve already decided, but only if the state resides in a particular region of the country. “At some point, I guess we will have to allow the states to decide for themselves.” Again, it was Mr. Wilson. He was a talkative gentleman! I recognized the validity of his claim. “I think you’re right. You’ll probably, as a federal legislature, open up territories for expansion, and allow the people themselves to vote. What do you think will happen.” It, of course, was Franklin, who spoke up. “Both sides with flood that area with folks sympathetic to their cause. These won’t be people who live there – just people who want to vote. Can you imagine? A territory crowded with pro and anti-slavery people – there just to vote? Clashes are inevitable!”
He was right, of course. This is exactly what happened with my home state of Kansas. The Lawrence Massacre. John Brown. The Border Ruffians. Quantrill’s Raiders. Bleeding Kansas. All inevitable, as Mr. Franklin said. I was on a roll. “What happens when a slave goes to a free state? Is he still a slave, or is he free? You’ve said here today he’s neither! He’s 3/5th a person, so maybe he’s property, but he’s definitely not a citizen.” I, of course, was thinking of the Landmark Dred Scott.
“If you write this into this Constitution, the Supreme Court will have no choice but to assume, if he’s not a citizen, he’s property. What happens when this slave shows up at your house and asks for protection?” I was talking to a northern representative. “You’re against slavery. Are you going to return him – or help him?” “I never thought of that.” “Well you all better think of it, because these are all things …” I caught myself almost saying, ‘that will happen’. I amended my comment: ‘that are likely to happen.’” “And finally, I said, “after generations of compromises and eventual fighting, what do you think will happen?” Again, it was Franklin. “We will surely come to a Civil War.” Predictably, it was Mr. Wilson who spoke again: “Are you saying as a result of us passing a 3/5th vote, we’re going to eventually have a Civil War?” “I’m not saying it, sir. The logic of the analysis reads as a chain of events, and I’m saying when the compromise-bough breaks, the cradle will fall. What I’m telling you all here today is let’s address this issue BEFORE THE BOUGH BREAKS!’ My eyes blurred. I rubbed them. I was awake. At least I think I was. Where was I? I was scared to look. I recognized my house. At last, this was now! As I walked to the kitchen to fix breakfast, I thought about my dreams. Were they dreams? They were so real! I wondered what the Continental Congress had done in 1787. Had I changed their minds? If so, what solution had they come up with? I started to open the blinds, but then thought: “Let’s suppose I had actually gone back in time. That all of this had actually happened. I had changed their minds. They addressed the issue of slavery properly. How would the world be different? I opened the blinds to look outside.
QUESTIONS 1. What were the “Articles of Confederation?” 2. Why was it necessary to amend / change the Articles to our new Constitution? 3. What was the Declaration of Independence? 4. What was the Boston Tea Party? 5. What was the relationship between the Boston Tea Party and the Declaration of Independence? 6. What states were “slave” and which were “free” states? 7. Where did all the southern slaves come from? 8. When was the Civil War fought? 9. What is the “cotton-engine” mentioned, who invented it, and how did it work? 10. What was the first great compromise regarding the alignment of free/slave states in 1820? 11. How did we get the Louisiana Territory? 12. Who championed the next great compromise in 1850, and what was it called? 13. What is “popular sovereignty”? 14. Who is John Brown? 15. What are our branches of government? How many senators and representatives are in each? 16. What are the “Checks and Balances” built into our system? 17. If the Constitution declared a slave property, would the Supreme Court be able to overturn this decision? 18. When and why was slavery introduced to the colonies? 19. Draw a timeline of the “USA” from 1600 to 1865, including as many of the above events as possible.
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Tentative Chapter 3
December 9, 2008
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CELL FORMATTING THE SECOND STEP The Scatter Plot – with or without points connected – is one of three powerful tools I use in Excel to display graphics. This relies on creating data, and then graphing it. The second looks at creating data, with the visual output remaining right in the cell. This is done with “Conditional Formatting”. Let’s start with an example.
AN EXAMPLELet’s randomize cells a1:e5 with our “random” formula used earlier. We want out output to be either zeros or ones, with equal probability. We know how to do that as well:
if(rand()>.5,1,0)
This gives us a 5 x 5 grid. If we push “calculate” [f9 key], this changes.
Instead of zeros or ones, however, I want to color these cells to white cells or black cells.
CONDITIONAL FORMATTING: How do I do this? Using the “Format” pull-down menu, there is an option “Conditional Formatting”. This allows us to format the cell based on what is in the cell. In my case, I wanted everything (font and pattern) to be black if the cell value was zero; otherwise, if the cell value was one, I set the font and pattern to white. Setting the font color to be the same as the pattern (background) color makes it look like there’s nothing in the cell. We know there is. Just us. To allow you to see how each cell is impacted, I’ve left the grid in tact. That is, I kept the border black at all times, but this can be changed if you’d like.
PUSHING [CALC = F9] What kind of variety does this simple experiment generate? Here are some examples:
SOME ORDER TO THE CHAOS That’s all fine and dandy, but can I order this somehow, now that I now how to format the cells. Let’s assume, for example, we want these grids to be symmetric. How would we do this? Let’s start with a simple 3 x 3 grid:
How do I populate it to ensure I have some symmetry? Which cells do I need to fill? Let’s start with an even simpler square grid:
By putting something in ½ the grid, and transposing it for symmetry, I get the following:
But why stop at putting something in ½ the grid? What happens if I populate ¼ of the grid, and then transpose it two ways for symmetry?
Why stop at ¼ the grid? What happens if I populate just an eighth of the grid, and then transpose it for symmetry across the grid?
I could go on forever, I thought, but here was a starting point I could use, thinking back to my tiny grid. If the following three cells were filled in, all the other cells could be filled in:
With this pattern and symmetry, the rest of the grid is filled:
POPULATING OUR GRID 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 were eight possibilities. Further, I could list the rules with 1s and 0s. What happens if, instead of two open columns, I use three?
Three columns yielded 64 distinct symmetric patterns. Likening the results to ancient hieroglyphics, I coined the phrase “Binary Symmetric Hieroglyphics”, and liked the fact I had “discovered” something! Pretty as they were, this was not the answer to my question “Are all snowflakes different?”, though I was 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 OCTILLION! How can this be? This small grid below contains more different illustrations than any number I’ve ever encountered!
Here are more!
Your Assignment We’ve chosen our colors such that there is a 50/50 chance of each being white or black. Modify this to see what happens when the percentages change: 90/10, 75/25, 60/40, etc.
Your Assignment Part 2 We’ve made the whole grid symmetric. What happens if we break it into small grids, each individually symmetric but different from the whole. For example:
I call this a “digital turkish rug”, but it’s really our simple conditionally formatted 0 and 1 grid, with a couple of modifications. Can you create something similar?
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Science Fiction - or the Reality of Today?
December 10, 2008
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Mr. Bartholomew: Jonathan, let's think this through together. You know how the game serves us. It has a definite social purpose. Nations are bankrupt, gone. None of that tribal warfare any more. Even the corporate wars are a thing of the past.
So now we have the majors and their executives. Transport, food, communication, housing, luxury, energy. A few of us making decisions on a global basis for the common good.
The team is a unit. It plays with certain rhythms. So does an executive team, Jonathan. Now everyone has all the comforts, you know that. No poverty, no sickness. No needs and many luxuries, which you enjoy just as if you were in the executive class. Corporate society takes care of everything.
And all it asks of anyone, all it has ever asked of anyone ever, is not to interfere with management decisions.
You know I've always considered your situation, Jonathan, and your needs. Now you have to consider mine - and ours.
No player is greater than the game itself. It's a significant game in a number of ways. The velocities of the ball, the awful physics of the track. And in the middle of it all, men playing by an odd set of rules. It's not a game a man is supposed to grow strong in, Jonathan. You appreciate that, don't you?
You must take good advice. You're not to play against Tokyo. You're not to play again.
Jonathan E: It might be I won't ever find out why I've been asked to leave the game, but I do know I can get some concessions and I want 'em.
Mr. Bartholomew: Specifically, you're bargaining for the right to stay in a horrible social spectacle. It has its purposes. You've served those purposes brilliantly. Why argue when you can quit?
And you say you want to know why decisions are made. Your future comfort is assured. You don't need to know. Why argue about decisions you're not powerful enough to make for yourself? Energy will treat you well, you know that.
Jonathan E: I'll see you in Tokyo.
Mr. Bartholomew: You can be made to quit! You can be forced!
Jonathan E: You can't make me quit!
Mr. Bartholomew: Don't tell me I can't. Don't ever say that! I can! You can be stopped!
Mr. Bartholomew: In my opinion we are confronted here with something of a situation. Otherwise I would not have presumed to take up your time. Once again it concerns the case of Jonathan E. We don't want anything extraordinary to happen to Jonathan. We've already agreed on that. No accidents, nothing unnatural.
The game was created to demonstrate the futility of individual effort. Let the game do its work. The Energy Corporation has done all it can. If a champion defeats the meaning for which the game was designed, then he must lose. I hope you agree with my reasoning.
Thank you all.
JONATHAN E - TALKING WITH HIS EX-WIFE ON THE NATURE OF CIVILIZATION
Ella: You know, Johnny, all they want is a kind of incidental control over just a part of our lives. They have control economically and politically, but they also provide.
Jonathan E: Provide, huh?
I've been thinkin', Ella. Thinkin' a lot. I've been watching. It's like people had a choice a long time ago between ...well, having all them nice things or freedom.
Of course, they chose comfort.
Ella: But comfort is freedom. It always has been. The history of civilisation is the struggle against poverty.
Jonathan E: No! No, that's not it. That's never been it. I mean, them privileges just buy us off.
One of my favorite movies - one of my favorite actors.
=================================================== Beatty: "You like baseball, don't you, Montag?"
Montag: "Baseball's a fine game."
Beatty: "You like bowling, don't you, Montag?"
Montag: "Bowling, yes."
Beatty: "And golf?"
Montag: "Golf is a fine game."
Beatty: "Basketball?"
Montag: "A fine game."
Beatty: "Billiards, pool? Football?"
Montag: "Fine games, all of them."
Beatty: “More sports for everyone, group spirit, fun, and you don’t have to think, eh? Organize and organize and super organize super-super sports. More cartoons in books. More pictures. The mind drinks less and less. Impatience. Highways full of crowds going somewhere, somewhere, somewhere, nowhere. The gasoline refugee. Towns turn into motels, people in nomadic surges from place to place, following the moon tides, living tonight in the room where you slept this noon and I the night before."
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A Statistical Look at the Draft Over the Years
December 11, 2008
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The 1986 draft, with the tragic death of Len Bias and the troubles of Chris Wasburn and Roy Tarpley, has been tagged the worst draft in NBA history. Was it? How would we know? What's the best draft? How would we tabulate and compare the data? A simple comparison of career statistics over the history of the NBA seems the likely route, until one realizes the ABA was in existence until 1976. Therefore, any draft is tainted to the effect not all players were drafted by the NBA, and not all statistics were achieved in the NBA. The draft, in fact, went 10 rounds in the early days! Looking at the data, however, one rarely sees any player in the lower rounds who actually played. What a cruel hoax perpetrated on those players. It's no wonder they reduced the number of rounds to 2, and allowed other players to try out with any team they wanted. The draft, over the years:
To simplify things and get the statistical analysis going, let's take a look at the draft from 1977 - 2008, and record career statistics of each of the top 20 players drafted. What is the data? Let's get something on the table:
Viewing Career Numbers How do we compare "career" numbers when players just drafted still have their career in front of them? Ideally, our analysis would look at "first 3 years", or "first 8 years" in the league, or some comparison like this. Unfortunately, I don't have this data. Therefore, I'll pick 1995 as an arbitrary point where most all players drafted before that point have retired. There are exceptions, of course, but let's get started anyways.
But what do we graph? Let's chart the career statistics of not just the draftees, but total the data by "Top 5", "Top 20", and "Top 50" draftees.
Indeed, our 1986 draft shows a marked decline in all data!
Viewing Career Numbers Individually Of course, data aggregated is susceptible to outliers. One great statistical career can carry the load for a number of mediocre careers. Let's plot the individual data and see what this look likes.
Viewing Career Numbers By Draft Pick Up to this point, our analysis has focused on "Year" of the Pick. What happens when we compare the picks themselves? How do first-picks compare over the year? What is the difference between the first pick and the 20th pick? Let's see:
A General Question You may notice the graphs above are heavily dependent on the "greatest performance". Assists is a good example above. One data point far beyond all others causes all others to be bunched together.
But this leads to an interesting question: we have all this data and we've been plotting lots of points. Who - statistically - is the best player since the 1977 draft?
How do we compare a 20-point, 8-rebound, 3-assist player with a 15-point, 12-rebound, 2-assist player with a 10-point, 5-rebound, 9-assist player?
Does length of career matter? Is a 20/8/3 player over 8 years the same as a 20/8/3 player over 15 years?
That's the topic of the next issue regarding the NBA draft.
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Tentative Chapter 4
December 12, 2008
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CELL FORMATTING THE SECOND STEP When we last talked with our good friend, Pascal, we created an impressive lineage spanning a 1000 years. For now, let's focus on the triangle of coefficients he's widely known for discovering:
It all starts with a ‘1’ … Of course, in Excel, our grid doesn’t exactly look like this. We’re working with a square grid, so we’ll have to modify the triangle a bit – for now. To get started, let’s create the triangle:
Fine. I’ve got a triangle of numbers. Pascal used this for the sake of probability, with these numbers representing the coefficients of the binomial expanstion. Let’s not be so aggressive here. I simply want to know which of these numbers are even – or odd. But how do I do this? Even, I know, is a number “evenly” divided by two. There’s nothing left over. Odd numbers have ‘1’ left over after dividing by two. But how do I record just the remainder? There are many ways, but a powerful one is using the “mod” function, which returns just the remainder of a division problem. A couple of examples: mod(6,3) = 0 since 6 / 3 = 2 remainder 0 mod(31,7) = 3 since 31 / 7 = 4 remainder 3 Therefore, my new triangle, dividing everything by ‘2’ to find even and odd numbers, gives me the following:
I’m almost there! Lots of ones – and lots of zeros. In the previous chapter, we discussed “conditionally formatting” cells to provide a visual sense of what’s going on. Let’s do that again, coloring ones white and zeros black, and see what happens:
Carrying this out a few more steps, I get the following image:
Carrying this out for more rows, however, there’s a problem. The numbers get so big I can’t see them in the cells. Even if seeing the values weren’t important (as it’s the calculation I care about), eventually, the numbers will be too big for the spreadsheet to even recognize. Is there an alternative? The thing I care about, eventually, is the remainder of a division problem. For example: one of cells is the sum of ‘6’ and ‘4’ equals ‘10’. Dividing this by 2, I get 5 remainder 0, and it’s the ‘0’ I care about. That is:
What if, instead of finding the remainder after a long chain of calculations, I carry along the remainders in the initial triangle I create? That is:
Another! Let’s say my numbers are 61 and 33, and I’m dividing by 7.
Now we’re talking! And the amount of calculations has been cut in half, because I don’t need the second triangle of calculations. Of course, this is for calculations involving ‘2’. What happens when I divide by 3? By 4? Let’s see:
Your Assignment As usual, you see pretty quickly you don’t want the ‘2’ to be in each formula. You want it in a cell so if the cell-value changes, the whole spreadsheet updates. Likely you’ve already done that. Write a macro to cycle through numbers 2 – 25, as I have above. If you’re having trouble getting started, that’s because you’re not sure what you really want done. If you’re sure what you want done, simply turn on the “Macro Record” button, and see what it records (as we did in Chapter 1 with our ‘sitback’ macro).
Your Assignment Part 2 To make the plots above, we used a number-trick-shortcut regarding remainders. We said, if we add two numbers x and y and divide by n, we’ll get the same remainder as if we took x/n and y/n, added those remainders, and the took the remainder of all that divided by n. That’s a long-winded explanation of what we saw naturally. Mathematically, it looks like this:
Is this true? How would we prove it, beyond the intuitive proof we’ve already provided?
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A Statistical Look at the Draft Over the Years
December 13, 2008
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A General Question You may notice the graphs above are heavily dependent on the "greatest performance". Assists is a good example above. One data point far beyond all others causes all others to be bunched together.
But this leads to an interesting question: we have all this data and we've been plotting lots of points. Who - statistically - is the best player since the 1977 draft?
How do we compare a 20-point, 8-rebound, 3-assist player with a 15-point, 12-rebound, 2-assist player with a 10-point, 5-rebound, 9-assist player?
Does length of career matter? Is a 20/8/3 player over 8 years the same as a 20/8/3 player over 15 years?
Let's, to start, get a lay of the land. What are the statistics? Let's limit our search of "the best" to drafted players scoring 5,000+ career points.
The performance of Jordan is obvious. He has the highest lifetime scoring average (30.1 ppg) of players drafted since 1977. However, Karl Malone stands apart from the rest on two respects: his career average and career total. What impact should "longevity" have on the search for the "best" player. Is a player scoring 10 points a game for 20 years equal to one averaging 20 points for 10 seasons? There's no right answer - only an operational one. Let's assume "average" is the metric affording us the best peek at the "best" player. I've displayed points, rebounds, and assists above. Obviously, there are more. Let's focus on these. But how? Add them together? That gives more weight to points. Is that right? A Starting Point What if we just rank averages for the three main categories. I player first in scoring is brought down by finishing 40 in rebounding, and 66 in assists, for example, while a player ranking 30th in all three categories is rewarded. And this has just introduced a new variable into our analysis of "the best". In combining all three categories, we're really looking for the best "all-around" player. What do the rankings tell us?
And combining all three "ranks" leads to the following "top-20 all-around post 1977 NBA draft picks":
Objections Raised Of course, simple rankings have "cons" as well as pros. Most notably, a great-great performance is not rewarded by simply ranking it '1' and the next achievement '2'. Order is important, but what about degree of difference? But what alternative do we have? What if the order of magnitude is recognized, with all three metrics contributing to the calculation of "all around best player". For example: with points, compare everyone's performance with Jordan's 30.1. Iverson gets a .9136, because his 27.5 ppg average is 91.36% of the best in that category. Do the same for rebounds and assists, and add the three together to get a ranking of "goodness". The optimum total is, obviously, 3. I'm the best in all three categories. Let's see:
So, Bird is '1' via Method #1, and Magic #1 via Method #2. Which is the better method? Who is the better all-around player? Good water-cooler discussions, for sure!
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Tentative Chapter 5
December 14, 2008
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LINE and OBJECT DRAWING THE THIRD STEP Scatter plots (with and without connecting the points) and conditional formatting. Two powerful methods of displaying things in Excel. I want to briefly touch on a third I use a lot, and that’s the “line draw / object draw” method. A simple example:
Getting Started What is going on here? How do I even get started? Like many things already mentioned, to get an idea of what is going on, simply turn on “Macro Record”, draw a line, and see what Excel is actually doing. I draw a line …
And Excel records this: Sub Macro12() ' ' Macro12 Macro ' Macro recorded 12/10/2008 by Mike Round '
' ActiveSheet.Shapes.AddLine(56.25, 27#, 201#, 104.25).Select End Sub
First, get rid of all the junk: Sub Macro12() ActiveSheet.Shapes.AddLine(56.25, 27#, 201#, 104.25).Select End Sub
What do these numbers mean? Let’s draw another one and see if we can figure it out:
Sub Macro14() ActiveSheet.Shapes.AddLine(56.25, 26.25, 171.75, 13.5).Select End Sub
As the first two hardly changed (except for my unsteady mouse), they refer to the starting coordinates of the line. The last two must be the ending coordinates. As the numbers are lower in the upper left section of the grid, the layout of the grid – the coordinate system, if you will – must look as follows:
and the “addline” method contains four points, relating to the start and end of the line. The syntax is: addline(start x, start y, end x, end y)
Let’s check to see if our intuition is right. I want to draw a line from the upper left to the lower right, with coordinates: (0,0) to (200,200). Sub Macro14() ActiveSheet.Shapes.AddLine(0,0, 200, 200).Select End Sub
And it works! What would 100 random lines look like? A note: in Excel programming language, the “random” function is a bit different than when using the spreadsheet. In the programming language, the randomize function is: RND() I want the starting and ending points to all randomly be between 0 and 100: Sub Macroxx() For xxx = 1 To 100 ActiveSheet.Shapes.AddLine(200 * Rnd(), 200 * Rnd(), 200 * Rnd(), 200 * Rnd()).Select Next xxx End Sub
OK … now let’s structure some things. But before we get started, there’s a little problem of removing all the lines drawn. I’ve found only one way. If you find another, let me know: Using the pull down menus: Select [edit] Select [goto] Click on [special] Click on [objects] This selects all of the drawn objects at one time. Then push the [delete] key.
Getting Started What is going on here? How do I even get started? Like many things already mentioned, to get an idea of what is going on, simply turn on “Macro Record”, draw Sub Macrotry1() ActiveSheet.Shapes.AddLine(50, 50, 250, 50).Select ActiveSheet.Shapes.AddLine(50, 50, 250, 100).Select ActiveSheet.Shapes.AddLine(50, 50, 250, 150).Select ActiveSheet.Shapes.AddLine(50, 50, 250, 200).Select ActiveSheet.Shapes.AddLine(50, 50, 250, 250).Select End Sub
A Second Try Though the macro works, it also looks a bit cumbersome. Let’s clean it up a bit. Instead of repeating all of the lines, let’s look at what’s changing in each: just one number. And we see how it’s changing. It’s changing by 50. We see it happens 5 times. With all of that information, let’s modify our macro: Sub Macrotry2() For yyy = 1 To 5 endy = 50 * yyy ActiveSheet.Shapes.AddLine(50, 50, 250, endy).Select Next yyy End Sub
A Third Try The same result - just as I hoped - but this time with a better looping macro! We’re almost there! After we connect these fives lines, what happens next? I drop down one point and repeat the process. That is, my “starting y” changes. So, we want to wrap another “for / next” loop around our current loop. Let’s see: Sub Macrotry3() For xxx = 1 To 5 For yyy = 1 To 5 starty = 50 * xxx endy = 50 * yyy ActiveSheet.Shapes.AddLine(50, starty, 250, endy).Select Next yyy Next xxx End Sub
A Fourth Try It seems like we’re done. In fact, we are! Of course, you can play around with this lots of ways – more points, bigger, smaller, etc. Let’s suppose, however, I’m having trouble with the for/next loops. I’m having trouble writing the macro. I know all the coordinates, but the program won’t work. What to do? First, do I know all the points? In a spreadsheet, this is a test. In this case, it’s easy. 25 lines in the above graphic, and here are the coordinates for all of them:
If I could just get the macro to recognize these points, I’d be home-free. Let’s do it. Let’s create a separate section, called our “lookup section”, for which we want our macro to concentrate on. Above, we talked about the “vlookup” function. What I want it to do is take a given number, and find all the coordinates for that “number”.
How do we do this? How do we read numbers into cell ab7? How do we get the macro to recognize the values in cells AC7:AF7? Cell “AB7” is the key. What I want the macro to do is put a number here, my “looping number”. My loop now goes from 1 to 25. Sub Macrotry5() For zzz = 1 To 25 Range("ab7").Value = zzz Calculate startx = Range("ac7").Value starty = Range("ad7").Value endx = Range("ae7").Value endy = Range("af7").Value ActiveSheet.Shapes.AddLine(startx, starty, endx, endy).Select Next zzz End Sub I find this method particularly helpful when the coordinates may not be as simple to calculate in a macro, but I can easily calculate them in the spreadsheet itself.
A Fifth Try There are many ways to skin a cat. A macro. A spreadsheet that inputs data to the macro. What about our scatter plot? Will that work? Of course, when you realize this is just points being connected by lines. Also, it takes a bit more work to create the graph in one series, because once the line is drawn, it has to go back to where it started to draw a new line.
Bothersome? Hardly. Exciting, because I have another method to visualize data.
Your Assignment Create the following three graphics:
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An Unbelievable Story
December 15, 2008
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THE EDUCATION OF THE ELEVEN YEAR OLD BOY WHO LECTURED BEFORE THE
HARVARD PROFESSORS ON THE FOURTH DIMENSION
BY HAROLD ADDINGTON BRUCE ILLUSTRATED WITH A PORTRAIT The American Magazine, 1910, #69, 690 – 695. Contributed by Ann Hulbert
from the Sidis Archives Created and Hosted by Dan Mahony
Two years ago Prof. William James, in one of the most remarkable articles ever published in this or any other periodical, formulated for the readers of THE AMERICAN MAGAZINE his startling psychological doctrine of the hidden energies of man. “Everyone knows,” Professor James wrote, “what it is to start a piece of work, either intellectual or muscular, feeling stale―or cold, as an Adirondack guide one put it to me. And everybody knows what it is to ‘warm up’ to his job. The process of warming up gets particularly striking as the phenomenon known as ‘second wind’. On usual occasions we make a practice of stopping an occupation as soon as we meet the first effective layer (so to call it) of fatigue. We have then walked, played, or worked ‘enough’, so we desist. . . . But if an unusual necessity forces us to press onward, a surprising thing occurs. The fatigue gets worse up to a critical point, when gradually or suddenly it passes away, and we are fresher than before. “We have evidently tapped a level of new energy, masked until then by the fatigue-obstacle usually obeyed. There may be layer after layer of this experience. A third and a fourth ‘wind’ may supervene. Mental activity shows the phenomenon as well as the physical, and in exceptional cases we may find, beyond the very extremity of fatigue-distress, amounts of ease and power that we never dreamed ourselves to own, sources of strength habitually not taxed at all, because habitually we never push through the obstruction, never pass those early critical points . . . “It is evident that our organism has stored-up reserves of energy that are ordinarily not called upon, but that may be called upon: deeper and deeper strata of combustible or explosible material, discontinuously arranged, but ready for use by anyone who probes so deep, and repairing themselves by rest as well as do the superficial strata. Most of us continue living unnecessarily near our surface.” The controversy which these views of Professor James provoked still waxes warm. For the most part his scientific colleagues are at odds with him. Yet all the time, while his critics have been criticising him, facts have been coming to light tending to prove that Professor James’s theory, far from being a gospel of overstrain, is a gospel of hope, opening up to the human race vistas of possibilities and achievement unreached in any epoch of the history of the world.
A Marvel―and Still a Child There is at Harvard University today a student who has caused much astonishment, perplexity, and debate among the members of the faculty. He is only eleven years old. At an age when most boys are struggling desperately with the elementals of education, this lad is specializing in advanced mathematics, and, since admission at the beginning of the college year last September, has easily held his own with fellow students in most cases more than twice his age. Even before coming to Harvard he had progressed far on the road towards mastery in the science of mathematics. Algebra, trigonometry, geometry, differential and integral calculus―all these he had at his fingers’ ends by the time he was nine or ten. He has even written a treatise on the properties of the hypothetical “fourth dimension.”
What makes the case of this child-undergraduate still more amazing is the fact that, unlike almost every other “infant prodigy” of whom history gives any account, his marvelous precocity is far from being confined to a single department of knowledge. He is almost as good an astronomer as he is a mathematician, and for the past few months has been industriously charting the heavens according to a new system of his own. He has invented a universal language which, he clams, is free from the objections that have been raised against Esperanto. He has studied anatomy, physiology, physics, geography, history, and political science. Withal, he has remained essentially a child. He is as truly a boy as is the barefoot urchin playing ball in the street. He is no bulging-browed, bespectacled, anaemic freak. His cheeks have a ruddy glow, his eyes sparkle, he has a ringing laugh, and is fairly bubbling over with animal spirits. He is, in fact, so much of a boy that when, at the age of eight, his parents entered him in a high school, the school authorities, at the end of three months, were glad to see the last of him, so damaging to the discipline of the classroom were his pranks and antics. In some respects he is more childlike than the average youngster of his years and has not yet outgrown his fondness for the toys of the nursery. Of this, as of his wonderful intellectual attainments, I can speak from long personal observation, as I have known him since he was seven.
The Father of the Boy―and His Ideas How to account for him is a problem that is puzzling the savants of Harvard. One man, however, the boy’s father, feels absolutely certain that he can give the true and only adequate information. His son’s mental growth, he declares, is the result not of heredity, not of exceptional native talent, but of a special education he has received, an education having as its chief purpose the training of the child to make facile, habitual, and profitable use of his hidden energies. The father is himself a psychologist with a reputation on two continents. His name is Boris Sidis. Although best known in the scientific world as a medical psychologist, he has for years been making a special study of educational psychology. Like Professor James, with whom he is a co-discoverer of the law of latent energy, a subject on which Dr. Sidis has been working and experimenting for years, he is firmly convinced that most of us “live unnecessarily near the surface,” and he throws the blame for this largely on our educational system. In particular, he condemns the custom of delaying any attempt at formal education of the child until he arrives at “school age.” “The notion that the young child’s mind should be allowed to lie fallow,” is the way Dr. Sidis put it to me, “is utterly wrong and pernicious. The child is essentially a thinking animal. No power on earth can keep him from thinking, from using his mind. From the moment his inquiring eyes first take in the details of his surroundings he begins the mental processes which education is intended to guide and develop. He observes, he draws inferences from everything he sees and hears, he seeks to give expression to his thoughts. “Left to himself, however, he is certain to observe inaccurately and to make many erroneous inferences. Unless he is taught how to think he is sure to think incorrectly, and to acquire wrong thought habits, causing him to form bad judgments respecting matters not only vital to his own welfare but also important to the welfare of society. In fact, in order to get the best results, his training in the principles of correct thinking should begin as soon as, or even before, he starts to talk. There need be no fear of over-taxing his mind. On the contrary, the effect will be to develop and strengthen it, by accustoming him to make habitual use of the latent energy which most people never utilize at all.”
Learning to Spell and to Read Before Three Years Old Holding these views, Dr. Sidis, upon the birth of his son―who was named William James Sidis, after Professor James―resolved to put them to the test of experiment. To realize his great aim of energizing and rationalizing the child, he began to train him in the use of his observational and reasoning faculties before he was two years old, and, with the aid of a box of alphabet blocks, actually succeeded in teaching him how to spell and read before he was three. He did this by playing with the boy, shifting the alphabet blocks around to spell different words, pointing to the objects spelt, and naming them aloud. The effect of this was not simply to teach the child spelling and reading, but also to give him a thorough grounding in the principles of sound reasoning. Moreover, the method employed by Dr. Sidis seemed to impart to his son a power of mental concentration seldom seen in children. All children, as every parent is aware, are eager “tom know about things,” but as a rule their inquisitiveness is easily satisfied, and they flit, like so many butterflies, from one subject to another without giving much thought to anything. Not so with little William James Sidis. Once his attention was arrested, his interest aroused, he was not content until he had learned the exact nature of whatever had excited his curiosity. At the age of three and a half, for example, he chanced one day to wander into his father’s office while Dr. Sidis was writing a letter on a typewriter. He watched the movement of the carriage back and forth, he heard the clicking of the types, the ringing of the bell, and forthwith tugging at his father’s coat. What was that machine for, he demanded, how did it work, and many other questions. Then, climbing into his father’s lap, he pressed his little fingers on the keys, and exultantly read the words his father showed him how to form. This first lesson was followed by others, until within six months―when he was only four years old―he was typewriting with considerable dexterity. He had already learned to write with a pencil. When he was six―his parents having in the meantime removed from New York, where he was born, to Brookline, Mass.―he was sent to a public school. His career there was brief but spectacular. In half a year he passed through seven grades, leaving behind him a succession of bewildered, wide-eyed teachers, aghast at the precocity he displayed. An interval of two years of study at home was followed by three months of attendance at the Brookline High School. Then two years more of study at home, and now, as has been said, he is a special student at Harvard, toying with vector analysis and other forms of higher mathematics. At Harvard, as may be imagined, his career is being watched with the liveliest interest. Aside from the surprise occasioned by his proficiency in the difficult field of study which he has selected, those who have come into contact with him are most deeply impressed by the manner in which he, so to speak, takes himself for granted. He does not seem to regard his precocity as anything out of the usual, and enters as a matter of course into the new life opened up to him by his admission to the university. He is a regular attendant at the Harvard Mathematical Club, and enters freely into the discussion of the various papers read, his criticisms commanding as respectful a hearing as though coming from a man of mature years.
Lecturing to Professors and Others Indeed, not long ago he read a paper of his own before the Mathematical Club, taking as his subject the theme, “Four-Dimensional Bodes.” As may be imagined, the attendance at that meeting of the club was the largest of the year. More than 75 men were present―professors, assistant professors, instructors, students, and some specially invited guests. Not a few came in a profoundly skeptical frame of mind, having heard about the boy, but believing that his powers had been greatly overrated. Before the evening was at an end they were listening to him with the most intense interest and evident astonishment. Many of them were quite unable to follow his complicated calculations, which he made with assurance and ease. As he explained, in opening his lecture, the “space” with which we are acquainted is of three dimensions, but it is quite conceivable that there may be space with more than three dimensions―with four, five, or any number of dimensions. In four-dimensional space it would be possible to construct mathematical figures of very different form from our ordinary three-dimensional figures. The explanation of how many of these figures there might be, how they could be constructed, and what they looked like, was the subject of his lecture. For upward of an hour and a half this little lad in knickerbockers held the closest attention of his auditors, now speaking directly to them, now reading from a carefully prepared paper, with not a little oratorical effect, and now, in a childish scrawl, demonstrating on a blackboard the mathematical proof of the theories he was advancing. As he explained it, figures in the fourth dimension could be of the most remarkable shapes, having even as many as six hundred sides. A six-hundred-sided four-dimensional figure he called a “sextacosiahedragon,” a bit of original terminology which he surpassed when he referred to another many-sided four-dimensional figure as a “hecatonicosahedragon.” In conclusion, this youngest lecturer in the annals of Harvard insisted that it was a great mistake to suppose, as do many non-mathematicians, that the hypothesis of the fourth dimension is of no practical value. On the contrary, it is of the greatest usefulness to mathematicians, who by its aid are enabled to solve many problems that would otherwise baffle them, and more particularly geometrical problems. And all this, be it remembered, is, according to his father, the result of special education, having as its principal object the training of the boy to utilize those hidden energies which, as Professor James pointed out in his AMERICAN MAGAZINE article, the vast majority of people never make any use of whatever.
How the Father and Mother Managed the Boy’s Education To attain this object Dr. Sidis has, in the main, relied on the familiar educational principle of teaching a child through appealing to his interest, but he has made the appeal to interest in an unusual way―namely, by systematic application of the influence of that little understood but tremendously powerful psychological factor, “suggestion.” Now, suggestion is no mysterious or uncanny force, operable only under exceptional conditions. Everybody knows what is meant by a “suggestive teacher,” a “suggestive book,” a “suggestive picture.” By suggestion is meant nothing more than the intrusion of an idea into the mind with such skill and power that it dominates and, for the moment, disarms or excludes all other ideas which might prevent its realization. In dealing with little children, as many educators have long recognized, the one sure way of implanting in their minds the ideas which one wished to make dominant is by arousing their curiosity and stimulating their interest. This has led to the method of education through play, as exemplified in the kindergarten. But Dr. Sidis believed that, if properly manipulated, the method of education through play might be extended to subjects not taught in the kindergarten―that, in fact, a child might be led to undertake and continue the study of any subject provided it were made sufficiently interesting to him. Today, as we have seen, his son excels in mathematics. There was a time, however―while he was at the grammar school―when no subject could possibly have been more distasteful to him, and he seemed totally unable, or at all events unwilling, to apply himself to it. Discovering this, Dr. Sidis did not attempt to drive him to the study of mathematics. Instead he purchased some toys―dominoes, marbles, etc.―with which he invented games requiring more or less knowledge of addition, subtraction, multiplication, and division. Every evening, for an hour or more, he played these games with his little son, deftly managing matters so that his interest in time shifted from the toys to the principles underlying their use. In the boy’s presence, too, he continually discussed with Mrs. Sidis―who has throughout loyally co-operated with her husband in his unique educational experiment―questions involving the practical application of arithmetic and “suggesting” its importance in the affairs of every-day life. This process proved so effectual that the boy spontaneously, and with the greatest enthusiasm, took up the study of mathematics, progressing in it so rapidly that in a couple of years his mathematical knowledge was superior to that of his father. The same method has been followed by Dr. Sidis in stimulating him to the study of other subjects to which he first showed indifference or positive dislike. And the result has invariably been the same. Once really interested he has gone at every subject with eagerness and enthusiasm, grasping and mastering its principles with amazing ease. Nor is this the only way in which Dr. Sidis has made use of suggestion to stimulate his son’s intellectual development. Everything about us, as is now beginning to be pretty generally appreciated, is of suggestive. From our friends, our books, the very pictures on our walls, from everything in our environment, we constantly receive suggestions which influence us to a varying but nonetheless unmistakable extent. This is particularly true of the plastic period of childhood. Recent psychological investigation has made it certain that everything the child sees or hears, no matter whether he is consciously aware of it or not, leaves a more or less profound impression, is “subconsciously” remembered by him, and may at times exercise a determining influence upon the whole course of his life.
A Story About Helen Keller One impressive bit of testimony as to the permanence of the impressions of childhood and their influence upon the child’s later development is afforded by an experience in the life of Miss Helen Keller, who, as is well known, was left by illness deaf, dumb, and blind when less than two years old. Among the many accomplishments she has acquired not the least astonishing is her power for appreciating music, which she “hears” by placing her hand lightly on the instrument and receiving its vibrations. It occurred to Dr. Louis Waldstein, an authority on the “subconscious,” that quite possibly her appreciation of music was connected with subconscious memories of music she had heard before her illness. To test this theory he obtained from her mother copies of two songs which nhad often been sung to Miss Keller as an infant in Alabama, but which she had not heard since. These he played in Miss Keller’s presence, with remarkable effect. She became greatly excited, clapped her hands, laughed, and communicated: “Father carrying baby up and down, swinging her on his knee! Black Crow! Black Crow.” It was evident to all present that she had been drawn back in memory to the surroundings of her infancy. But no one knew what she meant by the words “black crow” until her mother explained that that was the title of a third song which her father used to sing to her. She had not heard it since her nineteenth month, when she lost all sense of hearing, but now, many years afterwards and although dependent solely on the sense of touch, she was able not merely to remember it, but even to recall its name! As a psychologist―and, for that matter, as the author of a standard textbook on “The Psychology of Suggestion”―Dr. Sidis was well aware of the possibility of so arranging his son’s environment as to cause it to radiate upon him suggestions quickening and enlarging his intellectual capacities.
With the Boy in His Study Room While the boy was still a mere infant, he set aside a room for him, a bright, cheery, well-lighted apartment, hung with a few attractive pictures. A little writing table was placed in one corner of the room, with pad and pencil. Opposite the child’s bed a small bookcase was placed. It was filled in part with the ordinary books of childhood―volumes of nursery rhymes, fairy tales, picture books. But it also held books of serious interest, simple tales of travel, of history, of science, and the like, most of them illustrated. As the child grew older, books of a more advanced character were added to his little library, studies in literature and biography, mathematical and scientific text-books. A large revolving globe, showing the countries of the world in bright colors, was placed near the window. Toys having a scientific basis also found a way to his room, which thus became a sort of educational museum, inspiring him with a love for knowledge. “And,” says Dr. Sidis, emphatically, “it is because he has been inspired with such an interest, such a genuine enthusiasm, that he has made the progress which people regard as surprising. Any normal child would make as good a showing if he were given the same training. The trouble is that parents neglect their children―allow them to fritter away their energies, to acquire habits of loose and incorrect thinking, at the very time when they stand most in need of careful education. It is the first years that count for most. Then it is that the child should be taught to observe accurately, to think correctly. “I do not mean by this that the child should be deprived of play. My boy plays―plays with his toys, and plays with his books. And that is the key to the whole situation. Get the child so interested in study that study will truly be play. Don’t tell me it can’t be done. I have done it.” Dr. Sidis would probably speak with less assurance were it not that this is by no means his only experiment in the development of latent energy.
Story of Another Boy Some years ago he made the acquaintance of a young foreigner, a boy of fifteen, who was desperately anxious to secure an education that would fit him for a professional career. His parents, who had but recently arrived in the United States, were very poor, and were bitterly opposed to his “ambitious notions,” believing that instead of going to college he should set to work to earn his living. He had had no schooling in his native land, knew scarcely a word of English, and was ignorant of even the elementary knowledge possessed by the youngest primary-school child. Nevertheless, with a confidence that was pathetic, he applied for admission to a high school. “No,” he was told, “we cannot admit you. You do not know enough. You must go first to a primary school and then to the grammar school before you can enter here.” He was in despair when Dr. Sidis sent for him. “You wish to get into the high school, I hear,” said he. “Very well, you shall. Go and find out exactly what they require you to know before they will admit you, and then come back to me.” For hours daily he labored with the boy, teaching him first of all the rudiments of spelling, reading, and arithmetic by methods which “trained him to use his mental faculties correctly and to use them fully.” The result was much the same as achieved in his son’s case. At the end of eight months the young foreigner passed with flying colors an examination for admission to the high school. He completed the high-school course with phenomenal rapidity, graduating with the highest honors. Then he entered college, where he again distinguished himself, and, passing to a medical school, won further laurels there. Today he is holding a responsible public position. In another case the subject of experiment was a man of forty, a tailor by trade. Dr. Sidis became interested in him on learning that, in a dim, vague, inchoate way, he had longings not merely to better himself but to be of some service to humanity. He talked with the man and found that, although rather stupid and uneducated, being scarcely able to read, he was really stirred by altruistic ambitions. “Then I took him in hand. I began to educate and energize him. He came to me every day, and when he was not with me he was studying the text-books I gave him to read. I kept him at work, with his mind set on the distinct goal of helping his fellow man. “Before long, he displayed an intellectual ability that amazed those who had known him before the process of energizing began. He seemed, as some of his friends said to me, to be a new man. Whereas before he had been timid and diffident he became self-assertive and masterful. He attended and even organized workingmen’s clubs, he developed a marked gift as a public speaker, and before his death, which occurred a few years ago, won considerable reputation as a labor leader. “But I could have done much more with him had I had him much earlier. It is by beginning in early childhood that the best results can be obtained. You know the old saying―’As the twig is bent the tree’s inclined.’ Parents cannot too soon begin the work of bending the minds of their children in the right direction, of training them so that they shall grow up complete, efficient, really rational men and women.”
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Part 1 of 3 on the Myth of Exhaustion
December 16, 2008
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The Pythagorean Theorem not only tells us if the central angle is a right angle (90˚), then the relationship a2 + b2 = c2 holds, but the reverse as well. That is, if I have a triangle where a2 + b2 = c2, then the central angle is a right angle.
But these simple thoughts led to a simple – and profound – question: what happens to the diagram when the angle is not 90˚? How would I graph all of this?
Our Situation On A Graph Before I get started, let’s label all the points on our Pythagorean Theorem Image with points A – I.
Points A, B, C, and D should be easy to find. This square does not change at all, and I know the length of each side. For simplicity sake, I’ll make a length ‘4’ equal 400 on my scatter plot diagram, and center this square with the following coordinates:
Let’s get started on our square with length 3 (now 300). How can I find the coordinates for this square? First off, let’s tip the square to get a realistic look at what we’re confronting. After all, that’s the question that got this train of thought going in the first place. Instead of this square being at 90˚, let’s tip it to 135˚. What does this look like?
POINT G
I want to find the coordinates of Point G (xg, yg). What do I know about anything? I’m assuming I know the angle I’ve tilted the 300-unit box, so that’s a start. I can extend a line from B past A, and, dropping an altitude line down from G, create a triangle within my 300-unit box
But do I know anything about the angle within the triangle? Do I know anything about the distances of the segments of the triangle? Just one right now: the hypotenuse has length 300. What about the other items? I know if I extend the line from B to A, I will have 180 degrees on one side of the line. This is bisected by one angle of 135 degrees, meaning the remaining angle is 45 degrees.
Knowing this, the fact the length of my box of length ‘3’ is really 300 according to my scatter plot, and a bit of trigonometry, I may be on my way. The two basic trigonometric functions I’m interested in are as follows:
Let’s put all of this information together now:
What do I know now? I know xg and yg, but what are they? Do they allow me to plot point G? They help, yes. However, they are not points. They are merely distances. I must add these distances to a point I already know to be able to plot Point G. I know the coordinates of Point A. Let’s use it:
POINT E
I next want to find the coordinates of Point E (xe, ye). What do I know about anything? I’ve already determined angle α above, and I know angles α + β together form a right angle. Therefore, I can find β. Knowing this and the same trigonometry from above, likely I can find the distances needed to plot the coordinates for Point E. Let’s see.
Can I first find the angle β?
What about the lengths of the sides of the triangle?
What do I know now? I know xe and ye, but what are they? Do they allow me to plot point E? They help, yes. However, they are not points. They are merely distances. I must add these distances to a point I already know to be able to plot Point E. I know the coordinates of Point A. Let’s use it:
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