December 1-16 on Part 1 due to System Problem 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
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Finishing Up Part I
December 17, 2008
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A Time-Out To Graph Before going any further, let’s see if our results make sense. We’ve got 6 points (A, B, C, D) on the 400-unit square, and 2 points (G and E) on the 300-unit square. Let’s graph the results and see if we’re on the right track. But right away, I see something is amiss. Everything I’ve done thus far depends on the initial angle I’m bending the 300-unit box. This has been measured in degrees. Excel does not use ‘degrees’; it uses ‘radians’. Let’s make this change – but how?
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.
POINT F
I want to find the coordinates of Point F (xf, yf). What do I know? Well, first my graphic above is misleading, because it seems like it’s on a direct line with a line passing through AB. That’s not necessarily the case, so let’s modify the graphic a bit.
If I draw a line directly from Point A to Point F, I’ve bisected that perpindicular angle, and I know one of the angles already in that corner. This gives me the angle to a triangle created by reaching Point F with only straight lines. With a bit of logic, I can arrive at my angle …
and my hypotenuse …
And carrying out the now obvious logical steps, I can find the coordinates of Point F:
Putting it All Together Applying these new coordinate formulas to Point F should round out two squares. Let’s see. But to make the squares properly connected, I need to do more than plot the points and connect the dots. I need to have the lines come back to a meeting point to move on to the second square. That’s easily done:
POINT H
I’m almost home – two points left: H and I. Like points G, F, and E, I hope it’s as easy as extending a line, dropping an altitude line to a point, using some easy geometry theorems, and applying basic trigonometric formulas. We’ll see. But I see a problem immediately. Where do I extend a line from in moving towards the coordinates of Point H?
Let’s try some things. How about extending a line through G, but parallel to AB. Drop an altitude line from H, and I’ve got my triangle. But do I know anything about the angle involved? I know angle HGB is a right angle, because this is a square shape. But I’m not interested in the whole angle, just a part of it. Do I know anything about the angle α under the new extended horizontal line? I don’t see what. What about the angle I do know something about in all this – the original tilt angle located at BAG? Using everything I know about opposite and interior angles, I don’t see how this helps. I seem stuck. I am stuck. And exhausted. I’ve tried everything, but this one elusive question has me stumped. I can’t continue my programming. My search is over. For now …
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Part II
December 18, 2008
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PART II
Where does a new idea come from? How does one “think outside the box”? I don’t know. However, after struggling for hours to figure out how to find an angle in my triangle to help me find the H-coordinates, I finally hit upon something. I need β. I know something about α and β, because this is a right-angle. Therefore, if I could find α, I’m home free. But how? I’ve been applying the idea of a transversal cutting two parallel lines, but that doesn’t seem to be getting me anywhere.
After banging my head against the wall for hours, it finally came to me: I can’t get α directly by adding angles together like I have earlier (for example, two angles = 180, or a right angle = 90). However, I do know here the ratio of the length of the sides of my triangle is the same of the tangent of α! The arctangent!
Let’s use a concrete example to see how this works: if the tilt is 135˚, above we found the coordinates of G to be (288,1112). Plugging these into the above formula, we have:
This means I now know what β is (90 – α) = 70.89˚. Am I home free yet? Almost. But all I know is the angle. I don’t know anything else about my triangle, do I?
As you can tell, things are really starting to get out of control, there are so many formulas in place. Let’s get some things formalized in a table to make sure we know what we doing, and to document some of the progress.
Putting it All Together OK – let’s see if we can put all of this together, rather than applying formulas here and there, and see if our graph corresponds to our formulas. In a word, have we done things right?
POINT I I leave it to the reader to complete Point I coordinates. Here is the image I used to complete the table above.
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The Dénouement
December 19, 2008
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Plotting VERTICES Tilting the Square from 0 to 360 Degrees
Plotting AREA Tilting the Square from 0 to 360 Degrees
PYTHAGOREAN TRIPLES Plotting Integer Solutions of the Pythagorean Theorem Pythagorean BULLS-EYE Plotting Diagonals in Polar Coordinates For example: the first Pythagorean triple is 3,4,5. Let’s capture the ‘5’, and give it ordinal 1. The second is 10 (6,8,10). It gets ordinal 2. Etc.
Pythagorean FREQUENCIES Plotting Pythagorean-Diagonals
DIAGONAL 5525 has 22 unique x2 + y2 = 55252 combinations
Once we caught our “Second Wind”, it’s clear the sky is the limit as far as what’s possible. How many more questions come to mind – naturally? Is this a lesson plan in how to do all of this? Hardly. In fact, the joy in all of this is doing all of this. And in the process, what was necessary to do all of this “play”? A sampling: Trigonometric Functions – sin, cos, arctan Radian Measure Distance Measurement Angle Theorems Euclidean Theorems Programming Plus a whole lot more!
But most important is the idea I could do all of this – by myself. And, if necessary, I could do it again. Right now. Starting with a blank spreadsheet. And to what end? What good was any of this? What “practical application” is there here? As Faraday properly said, “What good is a newborn baby?” At this point, I’m just playing around. In the process, I’ve likely learned more about math than an introductory algebra/trip college class! Memorization? Hardly. And will there be bumps in the road? You bet. But this path of educational investigation truly is “the path less taken”.
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December 20, 2008
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Macro Challenge #1 One of the items of importance in doing things in Excel is the ability to put data where you want it. We want to do more than simply take prime numbers from an internet site and drop them in our spreadsheet. We want to be able to calculate them ourselves. But that’s not enough. Calculate them? Yes. But organize them in the spreadsheet in an orderly fashion so we can do something with the results? That’s the key. But how? Let’s suppose we want to flip 50 coins, and see how many turn up heads. Of course, we know the expected number is 25. We also know this is laborious. Can we do it in Excel easily? Of course.
We’ve already introduced the “random” function, and in the “conditional cell-formatting” section above, showed how to use the formula to return, randomly, a zero or one.
Let’s do this for 50 coins,
If I push the calculate key [F9], I get a different result. Again and again – with different results.
How can I record all of these results? Said differently, how can I get the macro to record all of the results?
What I want to do is view this “variability” by means of a graph. But this means recording the data. Surely there’s an easier way to do this than pushing “calc” and recording the result.
Of course, we can click on “record macro” to see what happens. Doing this and deleting the nonsense leaves us with the familiar: Sub Macro1() Calculate End Sub
Let’s say we want to do this 25 times:
Sub Macro1() For looping = 1 to 25 Calculate Next looping End Sub
This simulates the 50 coin flips 25 times, but it does not record – or place the results anywhere. How do I do this?
I need to have the macro change the row where the value is being placed.
Another thing: if I copy merely the formula of cell I3, I get a formula. I don’t want the formula. I want the value of the number that’s in the cell. How do I do this? Using the “edit” pull down menu after copying the cell, I use the “paste-special” selection, and select “value”.
Let’s see how it looks, activating the “record macro” feature, in doing all of this just once:
Sub Macro6() Range("I5").Select ActiveCell.FormulaR1C1 = "1" Range("I6").Select Calculate Selection.Copy Range("M3").Select Selection.PasteSpecial Paste:=xlPasteValues End Sub
A lot of this is familiar, and we know how to wrap our looping feature around this. What do we do with the specific “M3” cell reference at the bottom? That’s what we want to vary so we can end up with a column of numbers.
We modify the formula. Maybe we can tie it to the looping feature. For example, when “looping” = 1 (the first iteration), we want the value of the simulation to be copied into cell “M3”. Therefore, each successive iteration is two rows more than the value of looping. Let’s create a variable recognizing all of this.
Sub Macro7() For looping = 1 to 25 Calculate Newrange = “M”&(looping+2) Range("I3").Select Selection.Copy Range(newrange).Select Selection.PasteSpecial Paste:=xlPasteValues Next looping End Sub
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December 21, 2008
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The 366 Library with a good December, 2008 will become a 2,000 page year!
The autoSOCRATIC Library Growing all the time
The number of books will decrease as I start work on a number of children's short stories in booklet form - ranging from 16-40 pages Two series deal with history "Commerce Connection" and "Somniac Excursion", and one with math/science (On My Mark). The one math series I'm excited about is "The Geometric Mind". There's little structure to the content - it's the method being emphasized that can take any content and get the student quickly to the point where they can "play with it". I'm designing a new form of dictionary I will unveil at the Dictionary Society of North America conference in Indianapolis in 2009. I hope to have a test group of students working with "The Proximate Event", the science novel about understanding how the universe operates computationally. The athletic series "The Arete of Athletic" continues to grow. I'll have "The Arete of Basketball" done by month's end. The goal of this series is to look at sports from the perspective of the detective, rather than simply a fan. One can do both, but it's fun to take sports to a different level. The "Shakespearean Cloud" project will be a fun one. Rather than understanding all of the Shakespearean plays from beginning to end, this will take each only to the point of an initial - but crucial - conflict, and then have the student's play out the dilemma. There will be an interesting comparison between what they did - and why - with what actually happens in the Shakespearean plays. I'll conclude the 6-part =EQUALS= series with a volume #7 as the certification. As always, there's a ton going on with Haiku, with the www.globalhaiku.com project being launched next month. One adult-specific project in progress is a series of brief booklets on understanding statistics. Of course, there are many like this - none succeeding though all popular - and it's illustrative to ask why that's the case. I have, and am approaching the idea from a much different perspective. "about town" is a monthly journal I'd like to make into a weekly column about something of interest right here - "about town". In the inaugural issue, I talk about a violin shop, a cemetery, a friend, a coal train, and the Korean War Memorial. I launch "Bottleneck Notes" next month. More to come on that. A key to it's origin is the asinine idea of "Cliff Notes", where books of 300 pages are "summarized" to 100 pages! How can these be "notes"? What else? A comic book series? A newspaper? A series of plays? A musical? A logic board game? There's probably more I'm missing here, and this list will be growing. However, likely the daily-aspect of 366 will discontinue, but the idea in general lives forever!
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Next Chapter (Tentative)
December 22, 2008
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Macro Challenge #2 Organizing data? Yes. “Macro Challenge #1” is the principal way I use a data to manipulate data in Excel. Likely there are far more efficient ways than that. However, it’s clear in my mind what’s going on, so I adhere to that method.
But data is one thing. Images are another.
Aren’t they?
Let’s suppose I conditionally format a number of randomly generated cells. We know how to do both now, right? Randomization of cells and then conditionally format them, right?
Here’s my output:
But pushing “calculate” [F9], I, of course, get a different result:
How can I capture “lots” of these, without having to do all the work myself? How do I capture them? How can I easily create something like:
My Ambitious Target It seems to consist of two things: 1. how do I capture an image in Excel; 2. how do I place it somewhere specific in Excel.
With these two things, I can do anything. Well, maybe not everything, but quite a lot!
Capturing an Image in Excel How do we get started? As always, we start by asking Excel to Record our macro, which allows us to see what’s going on.
But to do this effectively requires we know all the relevant keystrokes, and with the “Image Capture” routine, we don’t. Here it is. After selecting the range you want now considered an image, you hold down the <alt> and <shift> keys at the same time. While holding these down, select the “Edit” pull-down menu. What’s the difference?
Let’s see how it works in practice. Below, I’ve selected the range “A1:L12”, and then “Alt” “Shift” “Edit”, and Selected “Screen” and “Bitmap” from this selection box:
Sub copying() Range("A1:L12").Select Selection.CopyPicture Appearance:=xlScreen, Format:=xlBitmap Sheets("Sheet7").Select Range("A4").Select ActiveSheet.PasteSpecial Format:="Bitmap", Link:=False, DisplayAsIcon:=False Sheets("Sheet6").Select End Sub
As you can tell, I’ve selected a new sheet (Sheet7) to paste the image, and I’ve put it into cell “A4”.
But this changes everytime I want a new image, right?
We’ll deal with that in a moment. The first part of my ambitious target journey is complete: how do I capture an image. We know now. Now, I need to figure out how to place it where I want
Placing an Image in Excel I’ve got an image, and I want to put it in a definite spot in a spreadsheet so it looks great when a number of images are printed out.
How?
There’s probably lots of ways, and the one shown here is probably more labor-intensive than most. However, since it’s worked well for me for years, I’ll stick to it.
As with “recording macros”, the key is knowing what you want to do. What do you want the format to look like? How big are the pictures? What looks great?
If I had to do it manually, where would the images go?
Above, I decided I wanted 15 images, in 3 columns and 5 rows, like this:
Let’s start by creating a table of all this information: Images 1-15 and where I want them to go.
Easy enough.
Now what?
When I push “Calc” the first time, I want, somehow, a formula, to recognize ‘1’, and, because of this, return the value “a1” in another cell. How do I do this? With our friend, the vlookup function. Play with it.
I need to do this 15 times.
But this begs the question: I thought the macro was doing all this. How do we get the macro to read in “1-15”, and do something with the cell references above?
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Next Chapter (Tentative)
December 23, 2008
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We know how to solve the reading-in problem. We create a loop, and tell the macro to place the value of the “loop” in the appropriate cell. This is done as follows:
Sub copying() For looping = 1 to 15 Range(“n21”).value = looping Range("A1:L12").Select Selection.CopyPicture Appearance:=xlScreen, Format:=xlBitmap Sheets("Sheet7").Select Range("A4").Select ActiveSheet.PasteSpecial Format:="Bitmap", Link:=False, DisplayAsIcon:=False Sheets("Sheet6").Select Next looping End Sub Now my only problem is reading INTO the macro the appropriate cell where I want the image to be placed.
The statement Range(“n21”).value = looping
took a value from our macro and put it into the spreadsheet. Now, we simply reverse it, and call the result a variable to be used in a moment. Let’s call our read-in value NEWRANGE:
newrange = Range(“o21”).value
One last step! We’ve got the new range. Now, let’s get the macro to recognize it. Instead of the hard copied “A4” above, we simply replace it with the name “NEWRANGE”.
Sub copying() For looping = 1 to 15 Range(“n21”).value = looping Newrange = Range(“O21”).value Range("A1:L12").Select Selection.CopyPicture Appearance:=xlScreen, Format:=xlBitmap Sheets("Sheet7").Select Range(NEWRANGE).Select ActiveSheet.PasteSpecial Format:="Bitmap", Link:=False, DisplayAsIcon:=False Sheets("Sheet6").Select Next looping End Sub Wrapping Up Of course, there’s lots of small items I’ve left out here you can resolve yourself once you get in and do this a few times.
For example, in the previous macro, I’m coping from “Sheet6” to “Sheet7”. This will be different for you.
When you copy an image into a new sheet, likely you’ll resize it, or resize the columns and rows, or both! This becomes second-nature and there’s a thousand ways to do all of this. How do you get the macro to resize images? RECORD A MACRO AND SEE!
That’s the key to all of this. Unsure what to do in a macro? Get the macro recording what you’d like it to do, and see what it actually records.
“Image copying to a new spreadsheet”! A great way to document “change”, “variability”, “movement”, etc.
Wrapping Up: Part 2 Applying this “Image Capture” method to our statistical project above provides us quickly with 21 different pictures of the system:
Or allows us to simulate the creation of a lot of triangles …:
But why limit this to math? What would a spinning head look like, if we tracked its rotation from 0 to 360 degrees?
Easy enough.
How did I get the face to spin? There is a “rotate” feature when you click on the object options. How did I get the program to do this? RECORD MACRO and see what it does, and then read in to the relevant code my looping number.
A recurring theme! Modeling the “Big Bang” Theory We can play for a while, and in the process of playing, see how to do a lot of neat things. We can also come upon a lot of neat questions.
For example: suppose we attempt to model “The Big Bang”. Is this how the Universe came into being?
How would we model such a beast?
Implications of our Model
The mere process of doing this ourselves leads to a number of questions:
1. if “Frame 1” contains “everything”, what is all of that “outside” everything? 2. if the matter in the universe is expanding, does there reach a point where the distance between the matter continue to grow? In our model, it did! 3. in our model, the matter disperses outward circularly (we programmed that). Is this true? Is the dispersion random?
But this is all material for Volume #3 of THE GEOMETRIC MIND
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Poetic Communication in the 21st Century
December 24, 2008
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I often think about "Kansas", my home state, in terms of how to describe it to others. What "is" Kansas? Surely, this is right:
But does that say anything about "Kansas"? Not much. What about the role Kansas played in period leading up to the Civil War? John Brown? The border wars? Quantrill's Raiders? We were known as "Bleeding Kansas". Does this go into a description of what Kansas "is"? What of the name itself? It's Native American, and this land belonged to the Native Americans for who knows how long. How is this included in a summary of what "Kansas" is? Kansas became the nation's 34th state in 1861, though the territory itself became US territory as part of the Louisiana Purchase. When I describe Kansas, is this a part? Is the Spanish and French ownership to be included? Agriculture is king in Kansas. Surely this belongs in any description of what Kansas "is", though it in itself does not explain they "why" behind the agricultural phenomenon in Kansas. In the 19th century, the Santa Fe Trail found itself winding from Missouri through Kansas on the way to Santa Fe. The Commerce Connection. Surely this is an important aspect of Kansas history. Is it included? History. Culture. Geography. Politics. Economics. It all plays a part.
Visual Explanations Let's go one step further. Rather than explain in words what Kansas "is", let's restrict our explanation to an image. How would we do it? Given the toughness of the challenge, let's first look at what others have done in this regard. After all, isn't this what a flag is - a visual representation of what something really means?
Let's look at the Kansas flag? What have our flag-makers said Kansas "is"? What's important about us? What do we notice? Purple mountains. A sunflower. A blue and gold bar. 34 stars. Native Americans chasing buffalo. A farmer and plow. A prairie schooner. A sun appearing behind the mountains. A ship. The Latin phrase Ad astra per aspera.
All of these are there for a reason. But what reason? What events took place by which these visual items are intended to summarize a massive amount of information?
Let's look at the Prairie Schooner as an example. The Prairie Schooner? What on earth is it, and why was it in Kansas?
At this point, there is a ton of research to be done - about this one simple element of the flag. The Prairie Schooner. The Santa Fe Trail. Mexican independence from Spain. Commerce. The why and the how.
But why stop at the logical explanation of "the Prairie Schooner". Let's incorporate the "logical haiku" process to describe this element of our Flag.
A formal explanation of the "logical haiku" process can be found here ...
The Goal of It All I've learned a lot about Kansas, merely starting with my flag and investigating one element on the flag. What else will I learn about my state while researching what's on our flag? What can I learn from others doing similar research on their own state flags? Their country flags?
This is the goal of GLOBAL HAIKU - to understand the wonder of the globe as explained by people living there.
The Goal of It All How does this differ from what exists on Wikipedia or a number of other sites? Can't we simply consult an almanac for this information? Is my logical explanation above anywhere in an almanac? Does an almanac include the thoughtful creation of the beautiful Japanese Haiku? Does an almanac include descriptions from many perspectives from many children?
The goal of Global Haiku: the logical-haiku description of a country's flag. Many elements. Many children.
This is just the start.
What common elements do we see when investigating what's important on flags? Independence? Commerce? Agriculture? Religion? How are these depicted?
What do the colors mean? The shapes?
For example, above the Louisiana Purchase was mentioned as an important feature in the history of Kansas. Why was this encoded in our flag as a twisted blue and gold bar? What significance does the bar have? The "twistedness"? The colors?
So much to learn - and so much time!
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A Helping Hand in Assisting in the Understanding of that which is Unknown via that which is
December 25, 2008
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Flipping through the October 2008 North American Actuarial Journal, I came upon an article titled "Securitization of Longevity Risk in Reverse Mortgages". The article wasn't important to me. One image was.
What was it about this image? It was eerily similar to one I created in this TOC article on profitability in a multiple-product environment.
A Profitability Image from the TOC P-Q Game
I'd never even seen my image above, in fact, until I ran the various product scenarios in the famous TOC P-Q game through my model to simulate profitability. The March 25th article - and model - demonstrated the tragic consequences of "common-sense" business practices - which were wrong - having significant impacts on the bottom line - while counter-intuitive practices resulted in profitable outcomes.
And now the same graphic appears - but in a different context!
What am I to make of this?
I have a framework with which to read this otherwise technical article. I don't know if the framework is right or not, but it's a starting point!
Our common theme returns. Mere exposure to a variety of things in reality allows one to have possible insight into other aspects of reality. See Erathosthenes and Einstein in this regard.
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On The Brink of ...
December 26, 2008
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Researchers led by Oliver A. Hampton and Aleksandar Milosavljevic at the Baylor College of Medicine in Houston have now compared the genome of a type of breast cancer cell with that of normal cells. They find 157 rearrangements, they report in the current issue of Genome Research. The graphic summarizes their results. Round the outer ring are shown the 23 chromosomes of the human genome. The lines in blue, in the third ring, show internal rearrangements, in which a stretch of DNA has been moved from one site to another within the same chromosome. The red lines, in the bull's eye, designate switches of DNA from one chromosome to another. One of the rearrangements disrupts a gene called RAD51C which is involved in mending serious chromosome breaks, those in which both strands in the DNA are disrupted. The impairment of double strand break repair could be a major cause of all the other rearrangements, the researchers suggest.
I'm cautiously excited when I read stories about this, "cautious" because we've been on the brink many times, it seems, to solving this killer called cancer; "excited" because of the introduction of the science of chaos theory into the medical profession. Maybe. The science of medicine, like most sciences, deals with prediction and rationality. If I raise my arm and it hurts, likely raising my arm again will hurt - again. Raise it higher and it will hurt more. Raise it not so high and it hurts less. Why shouldn't things be this way? When one gets within a certain range of behavior, one expects results to be "as predicted". Is this the case?
The Logistic Map Let's not bother with the details of the map, of iteration, of population growth. Let's just look at the results of the map for certain seed cells. If I set the seed at "2.00", I get absolutely stable behavior. Increase the seed to "3.00", and I start to get erratic behavior. At "4.00", as you can see, the system behavior is completely random:
What we typically take away from such a graphic is the common notion: as the seed increases, do does randomness. Pretty simple. Pretty obvious.
But is it right? Let's check: if instead of viewing the above images for a couple seed values, I capture the results for many. Let's plot seed value along the x-axis, and the behavior along the y-axis, and see what we get. What I'm trying to capture is the behavior after the system has run for some time. For example, the seed "2.00", resulted in behavior ultimately stabilizing at "0.50", so that's all I graph. The "3.00" seed seems to oscillate between 066 and 0.68, so I capture all of these points. What does the graph look like?
Is this what we expected? The seed "1.00" settles down to "0.00" behavior. Fine. "2.00", as we saw above, settles down to 0.50" behavior. Even "3.20" seems to settle down to behavior that oscillates between two points, like bouncing back and forth between two walls. But what's happening after that? And what are those bands of white? What is going on here? Let's zoom in and see.
How can the behavior act like this? It's completely random about 3.800. It's completely random slightly after 3.900. But inbetween, there's both complete randomness and no randomness at all? Let's see what the behavior actually looks like to confirm this:
On the Brink ... This seems ominous. Cancer research, based on the theories of normal science and predictability, may be more difficult than we thought. After all, if behavior is not governed by the simple laws we're used to, then the search for cures seems to be for complex solutions. After all, if we're to describe the complex behavior we see above, it must be found in complex inputs, right?
Another "generally-held" conviction.
But the simple iterated maps above suggest this may be a wrong conviction, in this instance. That the solution, if not found by now despite billions of dollars in research, may be because we're looking in the wrong place. My hope is researchers do not see the image as evidence the cure is more difficult than originally thought, but rather focus on the closing paragraph:
One of the rearrangements disrupts a gene called RAD51C which is involved in mending serious chromosome breaks, those in which both strands in the DNA are disrupted. The impairment of double strand break repair could be a major cause of all the other rearrangements, the researchers suggest.
An Article Dedicated to the Memory of a Great Man Who died because of this disease, but likely knew more about the philosophy of the cure than most scientists ...
A Follow-Up Chapter 6 of "THE PROXIMATE EVENT" In search of how to model what I see in reality, my “two-neighbor” project flopped. In reality, I see a ton of variability. In my model, however, I saw little. Neat patterns? You bet! A possible algorithm for how things work? Hardly. I’ll extend the process to three neighbors and see what happens. I don’t hold out much hope. More possibilities? Sure. But variation? Unlikely. But it’s a good programming opportunity, I decide, so let’s go forward! The structure I’m toying with is this:
There are eight possible combinations I’ll encounter. They are:
Let’s just choose “rules” at random, and see what happens: The first couple are not promising: no patterns and simple patterns.
My third try returns the same “nested” pattern I saw earlier. At least there’s some interesting stuff going on!
On and on I go, my hopes evaporating. Lots of neat things. Cool patterns. But all patterns, and I’m looking for “variation”!
And then I hit on one. What is this? Is this a pattern? Is it random? It seems impossible to classify what this is!
I let it “run” for a lot more steps, seeing what will happen with my “anomaly”:
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December 27, 2008
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The Geometric Mind (Part III) is a series of 9 3-page articles as a wrap-up to The Geometric Mind series. The goal here is to provide little - if any - documentation to a process. At this point in the series, the student should have a great idea of how I created the graphics and how they might do this themselves. Here are several of the 3-pagers ...
THE EQUILATERAL ENIGMA I read in a book if you take a triangle – any triangle – and trisect each angle, you will form an equilateral triangle in the middle.
I don’t believe it.
Heathy skepticism is good in modeling.
Because first off, it requires me to make sure I know what it is that is being said.
The Claim
My Ambitious Target
To make sure our claim is valid not just for a chosen triangle, but for triangles in general, let’s randomize the three vertices of our triangle and see what happens:
Additional Thoughts The claim seems valid, though in actually creating the triangles and trisecting segments, I see there is more than one set of points in the middle that can form an equiliateral triangle. I wonder if people making this claim have ever actually done the calculations!
I also see an opportunity to export some of the images to Microsoft Paint, and play around a bit with the shading of some of the triangles. Some examples:
The Origins of the Word “Fractal” and the Fabulous Mandelbrot Set The Mandelbrot Set. The origins of the word “Fractal”. Literally, billions of calculations to arrive at the answer to a question: what – mathematically – goes to infinity and what does not. The formula: Z2 ← Z + c Iteration. Complex numbers. A lot of checking – within a macro. Once I’ve checked, though, I need a means to record the results. I’ll record them in columns. Once I’ve done this, it’s simple to graph the results and see what I get. This is what Benoit Mandelbrot essentially did.
The Results: The Mandelbrot Set
It’s been said Mandelbrot’s theory started with a gentleman named Julia. Via a similar algorithm, here are several “Julia Sets”:
The Meaning of It All What is it that’s going on, you might ask. We’re checking for the behavior of points via a certain algorithm, not unlike our logistic map above. Behavior via a process. Below is essentially the process taken: take a number and run it through an algorithm. If the distance from a certain point at a certain time heads off to infinity, toss it out. If not, keep the point.
Keep all these points. Graph them. The Mandelbrot Set. Further investigation might reveal some relation to the operation of the universe. Who knows.
LINE DESIGNS AND PROBABILITY
A common type of question in probability classes goes like this: If one urn has balls labeled A-E and a second urn contains balls labeled 1-5, and you draw one urn from each, how many combinations of balls can be drawn? The means of communicating the possibilities looks like this:
How would I draw such a beast?
My Ambitious Target What do I need? I could use the scatter-plot graph, but that lacks the flexibility I’m looking for here. However, I do know something about the line-draw method.
Taking it to the Next Step Drawing designs? Easy enough. The freedom to do just about anything allows me to do just about anything!
A Single Urn A related “urn” question: instead of two urns, there’s only one – containing eight balls a-h. You draw two. What are the possibilities? Let’s see. And once we see, we also see a ton of other possibilities ...
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Isaac Asimov: The First TOC Writer
December 28, 2008
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THE HEIGHT OF UP by: Isaac Asimov Most of us would consider the surface of the sun to be pretty hot. Its temperature, as judged by the type of radiation it emits, is about 60000 K. (with “K.” standing for the Kelvin scale of temperature). However, Homo sapiens, with his own hot little hands, can do better than that. He has put together nuclear fission bombs which can easily reach temperatures well beyond 100,0000 K. To be sure, though, nature isn’t through. The sun’s corona has an estimated temperature of about 1,000,0000 degrees K., and the center of the sun is estimated to have a temperature of about 20,000,000 degrees K. Ah, but man can top that, too. The hydrogen bomb develops temperatures of about 100,000,0000 degrees K. And yet nature still beats us, since it is estimated that the central regions of the very hottest stars (the sun itself is only a middling warm one) may reach as high as 2,000,000,0000 degrees K. Now two billion degrees is a tidy amount of heat even when compared to a muggy day in New York or Tampa, but the questions arise: How long can this go on? Is there any limit to how hot a thing can be? Or to put it another way, How hot is hot? That sounds like asking, How high is up? and I wouldn’t do such a thing except that our twentieth century has seen the height of upness scrupulously defined in some respects. For instance, in the good old days of Newtonian physics there was no recognized limit to velocity. The question, How fast is fast? had no answer. Then along came Einstein, he advanced the notion, now generally accepted, that the maximum possible velocity is that of light, which is equal to 186,274 miles per second, or, in the metric system, 299,776 kilometers per second. That is the fastness of fast. So why not consider the hotness of hot? One of the reasons I would like to do just that is to take up the question of the various temperature scales and their interconversion for the general edification of the readers. The subject now under discussion affords an excellent opportunity for just that. For instance, why did I specify the Kelvin scale of temperature in giving the figures above? Would there have hen a difference if I had used Fahrenheit? How much and why? Well, let’s see. The measurement of temperature is a modern notion, not more than 350 years old. In order to measure temperature, one must first grasp the idea that there are easily observed physical characteristics which vary more or less uniformly with change in the subjective feeling of “hotness” and “coldness.” Once such a characteristic is observed and reduced to quantitative measurement, we can exchange a subjective, “Boy, it’s getting hotter,” to an objective, “The thermometer has gone up another three degrees.” One applicable physical characteristic, which must have been casually observed by countless people, is the fact that substances expand when warmed and contract when cooled. The first of all those countless people, however, who tried to make use of this fact to measure temperature was the Italian physicist Galileo Galilei. In 1603 he inverted a tube of heated air into a bowl of water. As the air cooled to room temperature, it contracted and drew the water up into the tube. Now Galileo was ready. The water level kept on changing as room temperature changed, being pushed down when it warmed and expanded the trapped air, and being pulled up when it coaled and contracted the trapped air. Galileo had a thermometer (which, in Greek, means “heat measure”). The only trouble was that the basin of water was open to the air and air pressure kept changing. That also showed the water level up and down, independently of temperature, and queered the results. By 1654, the Grand Duke of Tuscany, Ferdinand II, evolved a thermometer that was independent of air pressure. It contained a liquid sealed into a tube, and the contraction and expansion of the liquid itself was used as an indication of temperature change. The volume change in liquids is much smaller than in gases, but by using a sizable reservoir of liquid which was filled so that further expansion could only take place up a very narrow tube, the rise and fall within that tube, for even tiny volume changes, was considerable. This was the first reasonably accurate thermometer, and was also one of the few occasions on which the nobility contributed to scientific advance. With the development of a desire for precision, there slowly arose the notion that, instead of just watching the liquid rise and fall in the tube, one ought to mark off the tube at periodic intervals so that an actual quantitative measure could be made. In 1701, Isaac Newton suggested that the thermometer be thrust into melting ice and that the liquid level so obtained be marked as 0, while the level attained at body temperature be marked off as 12, and the interval divided into twelve equal parts. The use of a twelve-degree scale for this temperature range was logical. The English had a special fondness for the duodecimal system (and need I say that Newton was English?). There are twelve inches to the foot, twelve ounces to the Troy pound, twelve shillings to the pound, twelve units to a dozen and twelve dozen to a gross. Why not twelve degrees to a temperature range? To try to divide the range into a multiple of twelve degrees-say into twenty-four. or thirty-six degrees-would carry the accuracy beyond that which the instrument was then capable of. But then, in 1714, a German physicist named Gabriel Daniel Fahrenheit made a major step forward. The liquid that had been used in the early thermometers was either water or alcohol. Water, however, froze and became useless at temperatures that were not very cold, while alcohol boiled and became useless at temperatures that were not very hot. What Fahrenheit did was to substitute mercury. Mercury stayed liquid well below the freezing point of water and well above the boiling point of alcohol. Furthermore, mercury expanded and contracted more uniformly with temperature than did either water or alcohol. Using mercury, Fahrenheit constructed the best thermometers the world had yet seen. With his mercury thermometer, Fahrenheit was now ready to use Newton’s suggestion; but in doing so, he made a number of modifications. He didn’t use the freezing point of water for his zero (perhaps because winter temperatures below that point were common enough in Germany and Fahrenheit wanted to avoid the complication of negative temperatures). Instead, he set zero at the very lowest temperature he could get in his laboratory, and that he attained by mixing salt and melting ice. Then he set human body temperature at 12, following Newton, but that didn’t last either. Fahrenheit’s thermometer was so good that a division into twelve degrees was unnecessarily coarse. Fahrenheit could do eight times as well, so he set body temperature at 96. On this scale, the freezing point of water stood at a little under 32, and the boiling point at a little under 212. It must have struck him as fortunate that the difference between the two should be about 180 degrees, since 180 was a number that could be divided evenly by a large variety of integers including 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60 and 90. Therefore, keeping the zero point as was, Fahrenheit set the freezing point of water at exactly 32 and the boiling point at exactly 212. That made body temperature come out (on the average) at 98.60, which was an uneven value, but this was a minor point. Thus was born the Fahrenheit scale, which we, in the United States, use for ordinary purposes to this day. We speak of “degrees Fahrenheit” and symbolize it as “degrees F.” so that the normal body temperature is written 98.6 degrees F. In 1742, however, the Swedish astronomer Anders Celsius, working with a mercury thermometer, made use of a different scale. He worked downward, setting the boiling point of water equal to zero and the freezing point at 100. The next year this was reversed because of what seems a natural tendency to let numbers increase with increasing heat and not with increasing cold. Because of the hundredfold division of the temperature range in which water was liquid, this is called the Centigrade scale from Latin words meaning “hundred steps.” It is still common to speak of measurements on this scale as “degrees Centigrade,” symbolized as “degrees C.” However, a couple of years back, it was decided, at an international conference, to call this scale after the inventor, following the Fahrenheit precedent. ‘Officially, then, one should speak of the “Celsius scale” and of “degrees Celsius.” The symbol remains “degrees C.” The Celsius scale won out in most of the civilized world. Scientists, particularly, found it convenient to bound the liquid range of water by 0 degrees at the freezing end and 100 degrees at the boiling end. Most chemical experiments are conducted in water, and a great many physical experiments, involving heat, make use of water. The liquid range of water is therefore the working range, and as scientists were getting used to forcing measurements into line with the decimal system (soon they were to adopt the metric system which is decimal throughout), 0 and 100 were just right. To divide the range between 0 and 10 would have made the divisions too coarse, and division between 0 and 1000 would have been too fine. But the boundaries of 0 and 100 were just right. However, the English had adopted the Fahrenheit scale. They stuck with it and passed it on to the colonies which, after becoming the United States of America, stuck with it also. Of course, part of the English loyalty was the result of their traditional traditionalism, but there was a sensible reason, too. The Fahrenheit scale is peculiarly adapted to meteorology. The extremes of 0 and 100 on the Fahrenheit scale are reasonable extremes of the air temperature in western Europe. To experience temperatures in the shade of less than 0 degrees F. or more than 100 degrees F. would be unusual indeed. The same temperature range is covered on the Celsius scale by the limits -18 degrees C. and 38 degrees C. These are not only uneven figures but include the inconvenience of negative- values as well. So now the Fahrenheit scale is used in English-speaking countries and the Celsius scale everywhere else (including those English-speaking countries that are usually not considered “Anglo-Saxon”). What’s more, scientists everywhere, even in England and the United States, use the Celsius scale. If an American is going to get his weather data thrown at him in degrees Fahrenheit and his scientific information in degrees Celsius, it would be nice if he could convert one into the other at will. There are tables and graphs that will do it for him, but one doesn’t always carry a little table or graph on one’s person. Fortunately, a little arithmetic is all that is really required. In the first place, the temperature range of liquid water is covered by 180 equal Fahrenheit degrees and also by 100 equal Celsius degrees. From this, we can say at once that 9 Fahrenheit degrees equal 5 Celsius degrees. As a first approximation, we can then say that a number of Celsius degrees multiplied by 9/5 will give the equivalent number of Fahrenheit degrees. (After all, 5 Celsius degrees multiplied by 9/5 does indeed give 9 Fahrenheit degrees.) Now how does this work out in practice? Suppose we are speaking of a temperature of 2O degrees C., meaning by that a temperature that is 20 Celsius degrees above the freezing point of water. If we multiply 20 by 5/9 we get 36, which is the number of Fahrenheit degrees covering the same range; the range, that is, above the freezing point of water. But the freezing point of water on the Fahrenheit scale is 32 degrees. To say that a temperature is 36 Fahrenheit degrees above the freezing point of water is the same as saying it is 36 plus 32 or 68 Fahrenheit degrees above the Fahrenheit zero; and it is degrees above zero that is signified by the Fahrenheit reading. What we have proved by all this is that 20 degrees C. Is the same as 68 degrees F. and vice versa. This may sound appalling, but you don’t have to go through the reasoning each time. All that we have done can be represented in the following equation, where F represents the Fahrenheit reading and C the Celsius reading: F = 9/5 C + 32 (Equation 15) To get an equation that will help you convert a Fahrenheit reading into Celsius with a minimum of thought, it is only necessary to solve Equation 1 for C, and that will give you: C = 9/5 (F — 32) (Equation 16) To give an example of the use of these equations, suppose, for instance, that you know that the boiling point of ethyl alcohol is 78.5 C. at atmospheric pressure and wish to know what the boiling point is on the Fahrenheit scale. You need only substitute 78.5 for C in Equation 15. A little arithmetic and you find your answer to be 173.3 degrees F. And if you happen to know that normal body temperature is 98.6 degrees F. and want to know the equivalent in Celsius, it is only necessary to substitute 98.6 for F in Equation 16. A little arithmetic again, and the answer is 37.0 degrees C. But we are not through. In 1787, the French chemist Jacques Alexandre César Charles discovered that when a gas heated, its volume expanded at a regular rate, and that when it was cooled, its volume contracted at the same rate. This rate was 1/ 273 of its volume at 0 degrees C. for each Celsius degree change in temperature.The expansion of the gas with heat raises no problems, but the contraction gives rise to a curious thought. Suppose a gas has the volume of 273 cubic centimeters at 0 degrees C. and it is cooled. At - 1 degree C. it has lost 1/ 273 of its original volume, which comes to 1 cubic centimeter, so that only 272 cubic centimeters are left. At -2 degrees C. it has lost another 1/ 273 of its original volume and is down to 271 cubic centimeters. The perceptive reader will see that if this loss of 1 cubic centimeter per degree continues, then at -273 degrees C., the gas will have shrunk to zero volume and will have disappeared from the face of the earth. Undoubtedly, Charles and those after him realized this, but didn’t worry. Gases on cooling do not, in actual fact, follow Charles’s law (as this discovery is now called) exactly. The amount of decrease slowly falls off and before the -273 degrees C., point is reached, all gases (as was guessed then and as is known now) turn to liquids, anyway; and Charles’s law does not apply to liquids. Of course, a “perfect gas” may be defined as one for which Charles’s law works perfectly. A perfect gas would indeed contract steadily and evenly, would never turn to liquid, and would disappear at -273 degrees. However, since a perfect gas is only a chemist’s abstraction and can have no real existence, why worry? Slowly, through the first half of the nineteenth century, however, gases came to be looked upon as composed of discrete particles called molecules, all of which were in rapid and random motion. The various particles therefore possessed kinetic energy (i.e. “energy of motion”), and temperature came to be looked upon as a measure of the kinetic energy of the molecules of a substance under given conditions. Temperature and kinetic energy rise and fall together. Two substances are at the same temperature when the molecules of each have the same kinetic energy. In fact, it is the equality of kinetic energy which our human senses (and our nonhuman thermometers) register as “being of equal temperature.” The individual molecules in a sample of gas do not all possess the same energies, by any means, at any given temperature. There is a large range of energies which are produced by the effect of random collisions that happen to give some molecules large temporary supplies of energy, leaving others with correspondingly little. Over a period of time and distributed among all the molecules present, however, there is an “average kinetic energy” for every temperature, and this is the same for molecules of all substances. In 1860, the Scottish mathematician Clerk Maxwell worked out equations which expressed the energy distribution of gas molecules at any temperature and gave means of calculating the average kinetic energy. Shortly after, a British scientist named William Thomson (who had just been raised to the ranks of the nobility with the title of Baron Kelvin) suggested that the kinetic energy of molecules be used to establish a temperature scale. At 0 degrees C. the average kinetic energy per molecule of any substance is some particular value. For each Celsius degree that the temperature is lowered, the molecules lose 1/ 273 of their kinetic energy. (This is like Charles’s law, but whereas the decrease of gas volume is not perfectly regular, the decrease in molecular energies-of which the decrease in volume is only an unavoidable and imperfect consequence - is perfectly regular.) This means that at -273 degrees C., or, more exactly, at -273.16 degrees C., the molecules have zero kinetic energy. The substance-any substance-can be cooled no further, since negative kinetic energy is inconceivable. The temperature of -273.16 degrees C. can therefore be considered an “absolute zero.” If a new scale is now invented in which absolute zero is set equal to 0 degrees and the size of the degree is set equal to that of the ordinary Celsius degree, then any Celsius reading could be converted to a corresponding reading on the new scale by the addition of 273.16 (The new scale is referred to as the absolute scale or, more appropriately in view of the convention that names scales after the inventors, the Kelvin scale, and degrees on this scale can be symbolized as either “degrees A.” or ‘degrees K.”) Thus, the freezing point of water is 273.1 6 degrees K. and the boiling point of water is 373.16 degrees K. In general: K = C + 273.16 (Equation 17) C = K - 273.16 (Equation 18) You might wonder why anyone would need the Kelvin scale. What difference does it make just to add 273.16 to every Celsius reading? What have we gained? Well, a great many physical and chemical properties of matter vary with temperature. To take a simple case, there is the volume of a perfect gas (which is dealt with by Charles’s law). The volume of such a gas, at constant pressure, varies with temperature. It would be convenient if we could say that the variation was direct; that is, if doubling the temperature meant doubling the volume. If, however, we use the Celsius scale, we cannot say this. If we double the temper-ature from, say, 20 degrees C. to 40 degrees C., the volume of the perfect gas does not double. It increases by merely one-eleventh of its original volume. If we use the Kelvin scale, on the other hand, a doubling of temperature does indeed mean a doubling of volume. Raising the temperature from 20 degrees K. to 40 degrees K., then to 80 degrees K., then to 160 degrees 0 K., and so on, will double the volume each time. In short, the Kelvin scale allows us to describe more conveniently the manner in which the universe behaves as temperature is varied-more conveniently than the Celsius scale, or any scale with a zero point anywhere but at absolute zero, can. Another point I can make here is that in cooling any substance, the physicist is withdrawing kinetic energy from its molecules. Any device ever invented to do this only succeeds in withdrawing a fraction of the kinetic energy present, however little the amount present may be. Less and less energy is left as the withdrawal step is repeated over and over, but the amount left is never zero. For this reason, scientists have not reached absolute zero and do not expect to, although they have done wonders and reached a temperature of 0.00001 degree K. At any rate, here is another limit established, and the question: How cold is cold? is answered. But the limit of cold is a kind of “depth of down” as far as temperature is concerned, and I’m after the “height of up,” the question of whether there is a limit to hotness and, if so, where it might be. Let’s take another look at the kinetic energy of molecules. Elementary physics tells us that the kinetic energy (E) of a moving particle is equal to ½mv2, where “m” represents the mass of a particle and “v” its velocity. If we solve the equation E = ½mv2 for “v”, we get:
But the kinetic energy content is measured by the temperature (T), as I've already said. Consequently, we can substitute "T" for "E" in Equation 19 (and I will also change the numerical constant to allow the figures to come out correctly in the units I will use). We can say that:
Now then, if in Equation 20 the temperature (T) is given in degrees Kelvin, and the mass (in) of the particle is given in atomic units, then the average velocity (v) of the particles will come out in kilometers per second. (If the numerical constant were changed from 0.158 to 0.098, the answer would come out in miles per second.) For instance, consider a sample of helium gas. It is composed of individual helium atoms, each with a mass of 4, in atomic units. Suppose the temperature of the sample is the freezing point of water (273 degrees K.). We can therefore substitute 273 for ‘T’ and 4 for “in” in Equation 20. Working out the arithmetic, we find that the average velocity of helium atoms at the freezing point of water is 1.31 kilo- meters per second (0.81 miles per second). This will work out for other values of “T” and “in.” The velocity of oxygen molecules (with a mass of 32) at room temperature (300 degrees K.) works out as 0.158V 300/32 or 0.48 kilometers per second. The velocity of carbon dioxide molecules (with a mass of 44) at the boiling point of water (373 degrees K.) is 0.46 kilometers per second, and so on. Equation 6 tells us that at any given temperature, the lighter the particle the faster it moves. It also tells us that at absolute zero (where T = 0) the velocity of any atom or molecule, whatever its mass, is zero. This is another way of looking at the absoluteness of absolute zero. It is the point of absolute (well, almost absolute) atomic or molecular rest. But if a velocity of zero is a lower limit, is there not an upper limit to velocity as well? Isn't this upper limit the velocity of light, as I mentioned at the beginning of the article? When the temperature goes so high that "v" in Equation 6 reaches the speed of light and can go no higher, have we not reached the absolute height of up, the ultimate hotness of hot? Let's suppose all that is so, and see where it leads us. Let's begin by solving Equation 6 for "T". It comes out: T = 40mv2 (Equation 7) The factor, 40, only holds when we use units of degrees Kelvin, and kilometers per second. Let's set the value of "v" (the molecular velocity) equal to the maximum possible, or the 299,776 kilometers per second which is the velocity of light. When we do that, we get what would seem to be the maximum possible temperature (Tmax): Tmax = 3,600,000,000,000 m (Equation 8) But not we must know the value of "m" (the mass of the particles involved). The higher the value of "m", the higher the maximum temperature. Well, at temperatures in the millions all molecules and atoms have broken down to bare nuclei. At temperatures of hundreds of millions and into the low billions, fusion reactions between simple nuclei are possible so that complicated nuclei can be built up. At still higher temperatures, this must be reversed and all nuclei must break apart into simple protons and neutrons. Let's suppose, then, that in the neighborhood of our maximum possible temperature, which is certainly over a trillion degrees, only protons and neutrons can exist. These have a mass of 1 on the atomic scale. Consequently, from Equation 8, we must conclude that the maximum possible temperature is 3,600,000,000,000˚ K. Or must we? For alas, I must confess that in all my reasoning from Equation 5 on there has been a fallacy. I have assumed that the value of "m" is constant; that if a helium atom has a mass of 4, it has a mass of 4 under all conceivable circumstances. This would be so, as a matter of fact, if the Newtonian view of the universe were correct, but in the Newtonian universe there is no such thing as a maximum velocity and therefore no upper limit to temperature. On the other hand, the Einsteinian view of the universe, which gives an upper limit of velocity and therefore seems to offer the hope of an upper limit of heat, does not consider mass a constant. The mass of any object (however small under ordinary conditions, as long as it is greater than zero) increases as its velocity increases, becoming indefinitely large as one gets closer and closer to the velocity of light. (A shorthand way of putting this is: "Mass becomes infinite at the velocity of light.") At ordinary velocities, say of no more than a few thousand kilometers per second, the increase in mass is quite small and need not be worried about except in the most refined calculations. However, when we are working near the velocity of light or even at it as I was trying to do in Equation 8, "m" becomes very large and reaches toward the infinite regardless of the particle being considered, and so consequently does "Tmax". There is no maximum possible temperature in the Einsteinian universe any more than in the Newtonian. In this particular case, there is no definite height to up. SOURCE: View from a Height by Isaac Asimov
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December 28, 2008
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Piracy along the coast of Africa is in the news, ships and crews routinely seized and held for ransom. Most countries comply. What does such compliance mean over time? What does one gain by continually paying off such bandits - such terrorists. This President decided "enough was enough", and decided, after a ship and crew were held hostage for ransom, the US military was going to do something about it. You may think I'm talking about the east coast of Africa and the Somalia pirates. I'm not. I'm talking about the north coast of Africa along the Mediterranean Sea - the Barbary Coast. The President? Thomas Jefferson. But could a naval bombardment succeed? Would the prisoners be harmed? What was needed was a ground-force attack. But this was the United States 1805. Overseas ground forces? There was no such thing. Except for the leadership of William Eaton.
Who was William Eaton? He was the former Consul to Tunis, and had returned to the Mediterranean with the title of "Naval Agent to the Barbary States" in 1804. He had been granted permission from the United States government to back the claim of Hamet Karamanli, the rightful heir to the throne of Tripoli, who had been deposed by his brother Yussif. (wikipedia entry)
So Eaton, in Egypt, seeks out Hamet and tells him of the plan to restore Hamet as the rightful heir to the throne, and thus end the piracy along the coast. Hamet agreed. But they were 500 miles from Derne (in Tripoli - which was then a nation), with no army. Eaton recruited about 500 Arab, Greek and Berber mercenaries, and marched across the Libyan desert, to the port city of Derne. Can you imagine? They linked up, via agreement, with the naval forces in Bomba, where he coordinated efforts with the US naval forces. You think you know the rest of the story. You likely know a lot of it, but there's a detail that is truly amazing I heard about while watching a documentary on the Sahara Desert. The forces were coordinated. Eaton's ground forces were not making headway on attacking the city of Derne, and he sensed if progress was not imminent, his recruits would lose hope. He came up with an idea: "When I say 'Charge', CHARGE! They will get one shot off with their muskets, panic, and retreat through the city." What daring! Of course, now-a-days, such a command would be met with mutiny. Most semi-automatic handguns can fire many shots in a few seconds. But this was 1805, and the musket was the distance firearm of choice. What was it like to fire a musket? How accurate was it? According to wikipedia ... Loading and Firing a Musket The 18th century musket, as typified by the Brown Bess, was loaded and fired in the following way: Upon the command "Prime and load", the soldier would make a quarter turn to the right at the same time bringing the musket to the priming position. The pan would be open following the discharge of the previous shot, meaning that the frizzen would already be up. Upon the command "Handle Cartridge", the soldier would draw a cartridge. Cartridges consisted of a spherical lead bullet wrapped in a paper cartridge which also held the gunpowder propellant. The other end of the cartridge away from the ball would be sealed with a twist of paper. The soldier then ripped off the paper end of the cartridge and threw it away, keeping the main end with the bullet in his right hand. (The idea that the ball itself was somehow bitten off the top of the cartridge and held in the mouth is a myth invented by modern historical novels). Upon the command "Prime", the soldier then pulled the dogshead back to half-cock and poured a small pinch of the powder from the cartridge into the priming pan. He then closed the frizzen so that the priming powder was trapped. Upon the command "About", the butt of the musket was then dropped to the ground and the soldier poured the rest of the powder from the cartridge, followed by the ball and paper cartridge case into the barrel. This paper acted as wadding to stop the ball and powder from falling out if the muzzle was declined. (The myth of spitting the ball into the end of the barrel from the mouth is easily disproved - as soon as it is fired, the barrel becomes extremely hot; it would be extremely painful to place the lips anywhere near the hot metal.) Upon the command "Draw ramrods", the soldier drew his ramrod from below the barrel. First forcing it half out before seizing it backhanded in the middle, followed by drawing it entirely out simultaneously turning it to the front and placing it one inch into the barrel. Upon the command "Ram down the cartridge", he then used the ramrod to firmly ram the wadding, bullet, and powder down to the bottom followed by tamping it down with two quick strokes. The ramrod was then returned to its hoops under the barrel. Upon the command "Present", the butt was brought back up to the shoulder. The soldier pulled the cock back and the musket was ready to fire, which he would do on hearing the command "Fire". When the men fired they usually didn't hit a specific target, but the volume of fire was deadly within 20 meters.
They charged, and the Derne' force? One shot and a hasty retreat. The city? Taken. Hamet? Returned to his throne. The problem of piracy? Dealt with. Where? FROM THE HALLS OF MONTEZUMA TO THE SHORES OF TRIPOLI ...
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December 30, 2008
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THE GAME OF LIFE In the 1970s, John Conway invented “The Game of Life” to simulate “life” under certain circumstances. If I have too many neighbors, for example, I will die of overcrowding. Too few and I also die, this time of lonliness. Where does life come from? Death? The status quo? The game had rules which one could play if you had a checkerboard. The spreadsheet is a “checkerboard”! My Ambitious Target
Some Results Using random starting conditions, let’s capture some results of various scenarious and see what our “Game of Life” produces:
Let the Program Run Lots of scenarios – lots of results. Let’s see what happens when we simulate random initial conditions and capture the results as we move further into the simulation:
Stephen Wolfram published the remarkable book “A New Kind of Science” regarding cellular automata. The book starts with the idea of the possibilities when we consider three cells and what they might evolve into. There are eight possible combinations I’ll encounter.
Let’s just choose “rules” at random, and see what happens: The first couple are not promising: no patterns and simple patterns.
On and on I go, my hopes evaporating. Lots of neat things. Cool patterns. But all patterns, and I’m looking for “variation”!
And then I hit on one. What is this? Is this a pattern? Is it random? It seems impossible to classify what this is!
I let the program “run” for a lot more steps, seeing what will happen with my “anomaly”:
What is going on here? How can chaotic behavior be the result of simple rules? What else can I do in my spreadsheet? All sorts of things!
THE AIR WE BREATHE
I’ve heard it said so many times “hot air rises” it’s become a cliché with me. The hot air baloon rises, of course. If it were the other way around, we might call it a cold-air baloon.
But why does hot air rise? What’s going on here? Let’s suppose we try to create a model that tells us what’s going on.
My Ambitious Target
Before I Get Started … Let’s try to gain some intuition about what’s going on with these “air molecules”. For example, merely creating the graphic above, lots of questions come to mind. What is it, for example, that is “expanding”? Is it the molecule itself – in size, or is the heat causing the molecule to “bounce around” more, taking up more space but itself not changing in actual size – or is it something else? And what is it that’s outside the space taken up by the air molecules? What is this “empty” space? Therefore, to aquire some intuition about this, lets graph a couple of relevant items: temperature and air pressure.
My Logic and My Data Not only do I have an idea of what’s going on, but I can also infer some things. For example, if heated air molecules “spread out”, the space occupied by these air molecules becomes less dense than the surrounding space. Therefore, this warm air rises. If it rises, it’s replaced by colder air molecules “sinking”. What can I infer from this? There should be a relationship between temperature and air pressure. When it’s colder, the pressure is greater, and vice versa. What does the data tell us?
Something’s wrong – and that’s a good thing! Now it’s time to get to work fixing the model, the logic, etc. The real work begins!
In Search of …
So what exactly am I calling “The Geometric Mind”? Is it the use of visual logic tools to understand the flow of the argument? Yes. Is it the use of math in – and out – of these visual logic tools to understand the mathematical flow of the argument? Yes. Is it the use of ambitious targets to diagram and understand what needs to be done to solve a problem? Yes. Is it programming, animation, simulation, graphics? Yes. Is it the use of all of these tools – plus many more – to better understand and solve problems? Yes! What’s possible when any one of these intellectual “pistons” is firing? An example:
Where Do I Start? But where do I start? Which is best? Whatever works! Anywhere! Build your own model from scratch. Try to recreate what someone else has done. There is no best answer – only one imperative: GET STARTED! For more information, see:
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December 31, 2008
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What a Year! Unofficially: 2,050 pages bound, approximately 1300 images (traditional images and TOC logic devices) and all in the 366 archives. A note about these archives: though most of these daily articles have been proofed a couple of times, there's still a lot of work to be done in them. A lot of that work was done in the "official" copy, the "archives", if you will.
What does 2009 have in store for us? Some goals are listed above. It's just the start ...
Jan 01: George Washington Carver In The 21st Century Jan 02: An UDE - Or Not An UDE: That Is A Question Jan 03: The Privatization Of (Logical) Trees Jan 04: The Highway System Jan 05: A Happy New Year: A Logical-Haiku Jan 06: Law Enforcement And Racial Profiling Jan 07: My "Feynman-Test": Do Something. Jan 08: Mary Poppins And System's Improvement Jan 09: The Shortcut Dilemma Jan 10: The Visual Display Of Information Jan 11: Arthur C. Clark: Envisioning A Geosynchronous Orbiting System Jan 12: In Search Of A Rational Penal System Jan 13: The Aurora Jan 14: In Search Of Simplicity Jan 15: Anomalies Jan 16: The Ant And The Grasshopper Jan 17: Black Holes, Strange Attractors, And Basketball Jan 18: Music And Meaning Jan 19: The Moral Meaning Of Chess Jan 20: The Poetry Of The Automatic Garage Jan 21: To Hit Or Not To Hit? That Is My Question Jan 22: The Logic Of Quadrilaterals Jan 23: Is A Puzzlement! Jan 24: Julia Sets, Fractals, And Thinking Processes Jan 25: Architects Of Their Own Future Jan 26: The Goal Of Education Jan 27: Why Can't We Be Friends Jan 28: The Dustbowl, The Caveman, And Galloping Gertie Jan 29: Presidents Through The Ages Jan 30: The Caveman And Chaos Theory Jan 31: Math Formulas And Frigid Weather
Feb 01: Architects of Their Own Future: Chapters 3 & 4 Feb 02: Caught in a Timeless Dilemma Feb 03: The Super Bowl, Roman Numerals, and Cognitive Development Feb 04: A Preference to Apathy over Ignorance Feb 05: Reality, to be Conquered, Must be Obeyed! Feb 06: Economic Fallacies and Uneaten Cakes Feb 07: Flavors of the Month all Taste the Same (when they're still in their wrappers) Feb 08: Architects of Their Own Future: Chapters 5 & 6 Feb 09: The Quadrilateral Jamboree Feb 10: Iterative Effect-Cause-Effect Logic and the Context Syllogism Feb 11: The Dénouement-Detective (untying the knot) Feb 12: First-Hand-Accounting Feb 13: Visual Vocabulary Feb 14: The Ultimate Occupational Dilemma Feb 15: Chronologic Basketball Feb 16: Architects of Their Own Future: Chapters 7 & 8 Feb 17: The Simultaneous Nature of the Predicted Yet Unintended Feb 18: Systems Theory, the Federal Reserve, and the Fuzzy Cognitive Map Feb 19: Coal and Electricity Feb 20: The Highest Office Feb 21: A School Bus Tragedy Feb 22: The Arete of Line Designs Feb 23: Architects of Their Own Future: Chapters 9 & 10 Feb 24: A Lunar Light Show Feb 25: A Foreign-Language Wish-List Feb 26: How NOT To Argue Regarding HealthCare Feb 27: Iatosthenes' Revenge Feb 28: Architects of Their Own Future: Chapter 11 Feb 29: In Honor of Leap Year
Mar 01: Robert Frost, Mending Wall, and Poetry Mar 02: The Supreme Count - Visually - Through the Year Mar 03: The Sub-Prime Fiasco Mar 04: An American Grammar Mar 05: Architects of Their Own Future: Chapter 12 Mar 06: Our Own Writing System Mar 07: The Origins of Logical Haiku Mar 08: A Letter to the Philamath Society Regarding Morris Kline Mar 09: Challenging Intuition Directly Mar 10: Rivers and Dams Mar 11: In Search of Simplicity Mar 12: The Simplest Equation in the World: The Mandelbrot Set Mar 13: Understanding Shakespeare and Hamlet Mar 14: Integrity Mar 15: The Incredible Bread Machine Mar 16: Three Layers of Causality Mar 17: A Good Man: Lew Anderson: Part 1 Mar 18: Good Intentions Gone Bad Mar 19: Cars, Trains, and Systems Mar 20: The Federal Reserve, the DJIA, and the Visual Display of Quantitative Information Mar 21: Rethinking Pi-Day Mar 22: POLITE DISCUSSION Mar 23: Candy Cane Pipes "about town" Mar 24: Architects of Their Own Future: Chapter 13 Mar 25: Counter-Intuition: Revisiting the TOC P-Q Game Mar 26: O Captain! My Captain! and the Metaphor Mar 27: Strangers in the Night Mar 28: No Season Better: Sports Forensics Mar 29: Sediment Revisited Mar 30: Prime Numbers from a "Manipulative" Perspective Mar 31: Why I Could / Would Never Go Back to School
Apr 01: The Incomparable Chuck Jones Apr 02: The Interests of the One - The Interests of the Many Apr 03: The ACT: A Conversation with Myself Apr 04: The Quadratic Stream Apr 05: The Aluminum Bat Apr 06: Deal or No Deal Apr 07: LOGICIONARY-1 Apr 08: Architects of Their Own Future: Chapter 14 Apr 09: Questions I Get Apr 10: Turkey Dinners and Steering Wheels Apr 11: In Search of The Missing Yard Apr 12: Logical Thinking Apr 13: To Have a Neighbor, You've Got to be a Neighbor Apr 14: Visualizing Prime Numbers Apr 15: The Main Way in Education Apr 16: Hypocrisy in Action Apr 17: The Olathe School System - Visually Apr 18: The Case of the Submerged Theory Apr 19: Fall, Water: Just Don't Fall on Me Apr 20: The Time for Proper Action Apr 21: I Declare April "System A" Month Apr 22: Communication Breakdown: Part I Apr 23: The Politics of "Interest" Apr 24: If I Were a Blossom Apr 25: Center Pivot Irrigation Apr 26: These are a Few of My Favorite Things Apr 27: Digital Turkish Rugs Apr 28: Aunt Polly and the Get-Back Plan Apr 29: Communication Breakdown: Part II Apr 30: A Few Degrees of Separation
May 01: Z←Z2 + c May 02: Plus or Minus May 03: Energy for the 21st Century + May 04: Rabbit Seasoning, Eclipses, and Thales May 05: Celebrating Cinco de Mayo properly May 06: 2-Dimensional Cellular Automata May 07: CHOICE May 08: Architects of Their Own Future: Chapter 15 May 09: Recommended Changes to NCAA Basketball May 10: Let's Begin with Level Flight May 11: Architects of Their Own Future: Chapter 16 May 12: Spherical Graphics in Excel May 13: Oil and Gas May 14: Architects of Their Own Future: Chapter 17 May 15: Iatosthenes' Revenge (part 2) May 16: Architects of Their Own Future: Chapters 18 and 19 May 17: Visual Remainders: Pascal and the Mod Squad May 18: Three Haikus May 19: To Kalon! May 20: Architects of Their Own Future: Chapter 20 May 21: Euclid and His Prime Numbers May 22: America's "Forgotten War" May 23: Architects of Their Own Future: Chapter 22 May 24: The Viability of the Computational Universe? May 25: The "Class 3" Aberration May 26: An "Alfred Nobel" Siting - in Nebraska / 2008 May 27: Function Maximums and Minimums May 28: EUCLID ALONE? May 29: Spherical Cellular Automata May 30: The Infield Fly Rule May 31: Architects of Their Own Future: Chapter 20
JUNE REORGANIZATION / REVIEW MONTH
July 01: Beyond the Second Amendment July 02: Beyond Our Borders July 03: An Inevitable Crisis July 04: The Simultaneous Nature of the Predicted Yet Unintended July 05: Battle Fatigue July 06: Social Engineering July 07: The International Community and Africa July 08: On Open Letter to Dennis Moore July 09: A Clarion Call For Even-Ness July 10: The Logical Educational Extension July 11: Law Enforcement and Racial Profiling July 12: Honesty: Consistency Between Words and Actions July 13: The Power of the Mind July 14: Plus Or Minus July 15: Be Careful What You Wish For July 16: The Nature of "Bias" In Reporting July 17: The Incredible Bread Machine July 18: Airbag Deployment July 19: Theological Cake July 20: The Logic of Light Rail July 21: Youth and Athletics July 22: A Protection Racket July 23: The Visual Display of Information July 24: Kelo Revisited July 25: If You Call A Tail A Leg ... July 26: Advanced Placement - Reconsidered July 27: The Educational Musical July 28: The Danger of Awards July 29: Why Don't We Vote ... July 30: The Real Source of Inflation July 31: The World As It Can Be - As It Ought to Be
August 01: The Proximate Event (Chapters 1 and 2) August 02: NUMB3RS August 03: The Proximate Event (Chapter 3) August 04: Career Wins August 05: Rethinking Baseball Strategy August 06: The History of Paved Roads August 07: What to do - oh, What to do! August 08: Algorithmic Botany August 09: Houston, We Have A Problem August 10: NBA Basketball and The Matrix August 11: The Shadow I Cast August 12: Architects of Their Own Future: An Introduction August 13: The Geometry of the Game August 14: The World's Greatest Athlete August 15: The Proximate Event (Chapter 4) August 16: A Random Road Normally Traveled August 17: Midwest NKS Conference Abstract August 18: From Leonidas To Phelps August 19: Eccentricity August 20: Flying High August 21: GIVE 'EM HELL! August 22: The Montessori Method August 23: The Olympics Come To A Close August 24: My Random Walk August 25: The Proximate Event (Chapter 4: Part 2) August 26: Brother, Can You Spare a Token August 27: The Proximate Event (Chapter 5: Part 1) August 28: The Santa Fe Trail August 29: The Santa Fe Trail August 30: Low Hanging Fruit August 31: The Proximate Event (Chapter 5: Part 2)
September 01: Right Around the Corner September 02: The Santa Fe Trail (Part 3) September 03: The Proximate Event (Chapter 6) September 04: Proper Context September 05: The Proximate Event (Chapter 11) September 06: In Search of the Greatest Baseball Achievement September 07: The Louisiana Purchase September 08: Polygons, Circles, and Playing Around September 09: The Star Spangled Banner (revisited) September 10: The Proximate Event (Chapter 7) September 11: The Santa Fe Trail (Part 4) September 12: The Proximate Event (Chapter 10) September 13: Morris Kline: A Commemorative Stamp Campaign September 14: The Proximate Event (Chapter 8) September 15: Conduction: A Natural Integration of Induction & Deduction September 16: Weather War September 17: The Santa Fe Trail (Part 5) September 18: The Moral Meaning of Money September 19: A Letter to Olathe Police Chief Janet Thiessen September 20: Lissajous Figures September 21: The Magnificent Tower Crane September 22: Fall Season - Just Don't Fall on Me! September 23: A Drunken Random Walk September 24: The Proximate Event: Writing the Book September 25: Common Language September 26: The Spectacular Stradivirius September 27: Paul Newman: RIP September 28: The Practice of Polygony September 29: Warning the Rulers from Time to Time September 30: Connecting the Dots
October 01: The Fractal Phenomenon October 02: My Line Design Black Hole October 03: The Louisiana Purchase: A Brief History October 04: A Logical Haiku Invitation October 05: Ideas Have Consequences October 06: A Moratorium on Prices October 07: Intellectual Flotsam and Jetsam October 08: Of Value to Whom - and For What? October 09: An Early Halloween October 10: A First Hand Look at the Santa Fe Trail October 11: Bankruptcy - of a Moral Kind October 12: A Growing World October 13: Columbus in Context October 14: The Story of An Athletic Asterisk October 15: A Bilge Bay Reservation October 16: Robin Hood - as he Was - and As He's Remembered October 17: The Geometric Mind October 18: Somebody Do Somethin'! October 19: I'll Meet'Cha Halfway October 20: Public Seating October 21: No Taxation - EVEN WITH Representation! October 22: A General State of Dissatisfaction October 23: Chloro-phyll-osophic October 24: It May be Mere Face Paint to You October 25: Zooming in to See a Picture of Yourself October 26: Revisiting "The Fairness Doctrine" October 27: 53 - 47 October 28: The Lattice Method of Multiplication October 29: Off To Indiana October 30: Sampling the Computational World of "Turkish-Rug" Skylights. October 31: The Presidential Vote Over Time
NOVEMBER REORGANIZATION / REVIEW MONTH
December 01: Let it Snow! December 02: When the Bough Breaks: Part I December 03: The Geometric Mind: II (Chapter 1) December 04: An Introduction to the Syllogistic Dictionary December 05: When the Bough Breaks: Part II December 06: The Geometric Mind: II (Chapter 2) December 07: Bingo December 08: When the Bough Breaks: Part III December 09: The Geometric Mind: II (Tentative Chapter 3) December 10: Ominous Parallels December 11: The NBA Draft December 12: The Geometric Mind: II (Tentative Chapter 4) December 13: The Greatest December 14: The Geometric Mind: II (Tentative Chapter 5) December 15: Bending the Twig December 16: Second Wind December 17: Second Wind: Finishing Up Part I December 18: Second Wind: Part II December 19: Second Wind: The Denouement December 20: The Geometric Mind: Part II December 21: Goals for 2009 December 22: The Geometric Mind: Part II (next chapter) December 23: The Geometric Mind: Part II (next chapter) December 24: Global Haiku December 25: The Fractal Nature of Reality December 26: On The Brink of ... December 27: The Geometric Mind: Part III December 28: A History of the Thermometer December 29: Enough is Enough December 30: The Geometric Mind: Part III December 31: The Year in Review
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