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Let's go fishing!!!!
Math
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- #62
Thread Owner
I take it you find math very boring.
You know if you don't like math doesn't mean you have to post at all anything far from math. Nobody is forcing you to. Those that haven't posted at all probably aren't interested in math either. I don't post a lot in Grosser is Gross because I don't personally like everything discussed there, doesn't mean I have to post something different there and change the subjective integrity.....
You know if you don't like math doesn't mean you have to post at all anything far from math. Nobody is forcing you to. Those that haven't posted at all probably aren't interested in math either. I don't post a lot in Grosser is Gross because I don't personally like everything discussed there, doesn't mean I have to post something different there and change the subjective integrity.....
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danceswithfish: please try to keep your posts on topic, and avoid double posting. as psi said, if a thread bores you there's no reason to inform the rest of us.
psi: is it true that fractals appear often in nature, like in rock formations and tree growth? i was told that once a long time ago and i seem to have observed it a few times, but i'd like some verification
psi: is it true that fractals appear often in nature, like in rock formations and tree growth? i was told that once a long time ago and i seem to have observed it a few times, but i'd like some verification
- #64
Thread Owner
monsieurjohn said:
Yes a few scientists and mathematicians are quoted with such. I will write about it; but I have to look it up because history of math is not my forte, I just remembered it being a big deal to some. I know for sure that Mandelbrot was one. He cut up a cauliflower, and noticing that when you get one of the branches it was another flowering cauliflower. He kept cutting it and as it got smaller it was still a flower/tree shape. But yes I will show you who (there are multiple people who studied such) soon.
By the way still working on notation list, but working on some business (antique) deals in California (which sadly take precedence); will post it as soon as I can.
- #65
Thread Owner
Fractal History
Well first off let's say that Euclidean shapes are objects such as the sphere, circle, cube, square, etc. They are ideal shapes that do occur (but mostly of man-made objects). Now the universe is rough, broken, wrinkled, uneven shapes for the geometry of nature, but hold patterns. This is fractal geometry: fractal comes from the word fractus meaning fragmented, broken and discontinuous. Benoit Mandelbrot coined the term "fractal" in 1975 to describe shapes that were detailed at all scales; characterized by infinite detail, infinite length and the absence of smoothness. Fractal geometry is the geometry of the irregular shapes we find in nature. It takes Classical geometry one step further. Models can be made to describe all things of nature: clouds, shells, webs, galaxies, etc.
Now most of early mathematics when calculus was developed it could describe the smoothness of objects. Even when there were rough turns it could be split up to describe parts through calculus. This brought in the motions of orbits, electromagnetic fields, and rate of changes to the forefront described through calculus Kepler, Newton, Leibniz, Maxwell, Laplace were the key figures in this material.
However this left out objects that were rough with corners but still could be continuous but couldn't be fully described by calculus. It then had to be described by numbers which is the birth of "analysis" by Karl Weierstrass. He made a curve that consisted of nothing but corners with no smoothness anywhere (so it couldn't be taken apart). So analysis explained numbers through logic into "sets". Not going in to describe sets, but simply they are group of numbers, a collection let's say.
Now "topology" is the study of geometrical properties and spatial relations that remain unaffected by continuous stretching, bending, twisting....Dimensional studies of objects like Giuseppe Peano's Curve that filled up space on a plane. It was a line and took up all the points in a plane. This was a type of "mapping" of a line to a plane, complicated shape it may be. Felix Hausdorff described dimension through complicated shapes by fractional dimension, like one and half dimensions. Shapes that lie between dimensions are "fractals"
If any of you tried the applet for the Mandelbrot Set above you probably saw the set over and over again as you went smaller and smaller into it. That is called "self-similarity" A simple set to describe this is like below which is the Cantor Set below. You can draw this out on paper. Take a line and then take the middle third out. Then you are left with two third parts that are lines again. Do that again and you split it up taken out the middle third. Here's how to make it and see it -> http://www.mathacademy.com/pr/prime/articles/cantset/
If you did this infinitely you would have the Cantor Set. Well it has "zero measure", not literal zero measure but it's so far from dense you couldn't find points in being connected. Now this set has "self-similarity" too. This important word to describe Mandelbrot Set, Cantor Set, etc. Even the cauliflower idea from above post. You slice up a circle you have an arc. An arc is not a circle again. A side of a triangle is not triangular, etc. But if you took something from nature, a fractal, a break a piece off you still have the something. They are endless motifs "repeated at all scales". A cloud, tree, rocks, etc. all resemble smaller parts of themselves. Now these natural shapes are very hard to discuss through Euclidean Geometry and are called "pathological" shapes. A snowflake is a pathological shape, it succumbs to a bigger dimension then Cantor Set butit consists of the Koch Curve from Helge von Koch. Everyone has made this curve in school. It's a infinite representation of the Cantor Set through angles. Here is easy link to how to make it -> http://math.rice.edu/~lanius/frac/koch.html
Now mix this self similarity and "fractal" dimension and we are nearing the fascination of fractals. With a Euclidean shape like cube you can cut up cube divisible by 2. You'll have eight smaller cubes. The square will have four squares, the line have two lines. Cut it up like a Rubik's Cube and have fractional pieces of smaller cubes, division by 3 and you have 27 cubes, nine squares, 3 lines moving down the dimensions. Basically any n-dimensional object will have m to the power of n 1/m sized copies of itself.
This is where we get technical: the Cantor Set has dimension 0.63, the Koch Curve is 1.26, clouds 1.35. Coastlines? 1.26. In fact Koch Curves when infinitely produced look like coastlines. Again Felix Hausdorff's similarity dimension as the cutting up cube can be extended to objects NOT self-similar. Fractal shapes that lie somewhere between dimensions have dimensions known as "fractional" Hausdorff dimensions. And because we are talking about fractals it's sometimes called the fractal dimension. This was a theory of Hausdorff's until Mantelbrot revived it via the Julia Set.
Lewis Richardson, using idea of Jonathan Swift's quote:
So, Nat'ralists observe, a Flea Hath smaller Fleas that on him prey. And these have smaller Fleas to bite 'em. And so proceed ad infinitum.
took in natural occurrences like currents and turbulence as collections of smaller eddies and so on that he wrote like Swift as:
Big whorls have little whorls,
That feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.
Lewis Richardson asked many questions about nature, like how long is the coastline of Britain. With a map it's measured by the road near the coastline. But if you walked along the coast and measured it would be more accurate. If you held out a ruler to map it would be too straight. Make the ruler smaller and then you could accomplish measuring at angles. He did this for many coastlines around world comparing data from maps to the logarithm of the size of measuring stick. When Mandelbrot reviewed Richardson's findings he found them as Hausdorff dimensions of the coastline 1.26
Now cellular evolution spans from much of chaos theory of Michael Barnsley in the 1980's. He worked from Sierpinski's triangle (look this up here--> http://math.rice.edu/~lanius/fractals/ )
Cellular evolutions follow patterns to become fractals because it becames layered like sea shells or fern leaves. His chaos game of rolling a dice to chance from points scribed would sooner or later produce Sierpinski's triangle if one initial point was taken. But in some instances it might produce a fern leaf if a different point was chosen as initial point. Once an initial point was chosen the fate of the game was decided what shape would be. This is the idea behind the "strange attractor".
It was known earlier but not overly focused on since chaos theory is fairly new. However because I'm getting tired of typing here are some names major to focusing on bifurcations (in population growths, yeah Differential Equations!), chaos theory, iterations and mapping, fractals! I am tired of typing seriously....here are important names to look up to learn more along the way. Most of these names are scientists noticing fractal and chaotic aspects to objects or systems. Everything from magnetic material to astronomical objects to cancer cells. I'll save to discuss more later-- Robert May, Thomas Malthus, Pierre Francois Verhulst, Henri Poincare, Edward Lorenz, Tien Yien Li, James Yorke, Kurt Godel, Mitchell Feigenbaum, Gaston Julia (Julia Set!), Pierre Fatou, Heinz-Otto Peitgen, Peter H. Richter, Tan Lei, Roger Penrose (will post Penrose Triangles in here sometimes), Arthur Cayley, Thomas Witten, Leonard Sander, Stephen Hawking, Alan Penn, Wilheim Olbers, Hannes Alfven, Ian Stewart, etc.....going back to bed.....lol
me --> :blah:
Well first off let's say that Euclidean shapes are objects such as the sphere, circle, cube, square, etc. They are ideal shapes that do occur (but mostly of man-made objects). Now the universe is rough, broken, wrinkled, uneven shapes for the geometry of nature, but hold patterns. This is fractal geometry: fractal comes from the word fractus meaning fragmented, broken and discontinuous. Benoit Mandelbrot coined the term "fractal" in 1975 to describe shapes that were detailed at all scales; characterized by infinite detail, infinite length and the absence of smoothness. Fractal geometry is the geometry of the irregular shapes we find in nature. It takes Classical geometry one step further. Models can be made to describe all things of nature: clouds, shells, webs, galaxies, etc.
Now most of early mathematics when calculus was developed it could describe the smoothness of objects. Even when there were rough turns it could be split up to describe parts through calculus. This brought in the motions of orbits, electromagnetic fields, and rate of changes to the forefront described through calculus Kepler, Newton, Leibniz, Maxwell, Laplace were the key figures in this material.
However this left out objects that were rough with corners but still could be continuous but couldn't be fully described by calculus. It then had to be described by numbers which is the birth of "analysis" by Karl Weierstrass. He made a curve that consisted of nothing but corners with no smoothness anywhere (so it couldn't be taken apart). So analysis explained numbers through logic into "sets". Not going in to describe sets, but simply they are group of numbers, a collection let's say.
Now "topology" is the study of geometrical properties and spatial relations that remain unaffected by continuous stretching, bending, twisting....Dimensional studies of objects like Giuseppe Peano's Curve that filled up space on a plane. It was a line and took up all the points in a plane. This was a type of "mapping" of a line to a plane, complicated shape it may be. Felix Hausdorff described dimension through complicated shapes by fractional dimension, like one and half dimensions. Shapes that lie between dimensions are "fractals"
If any of you tried the applet for the Mandelbrot Set above you probably saw the set over and over again as you went smaller and smaller into it. That is called "self-similarity" A simple set to describe this is like below which is the Cantor Set below. You can draw this out on paper. Take a line and then take the middle third out. Then you are left with two third parts that are lines again. Do that again and you split it up taken out the middle third. Here's how to make it and see it -> http://www.mathacademy.com/pr/prime/articles/cantset/
If you did this infinitely you would have the Cantor Set. Well it has "zero measure", not literal zero measure but it's so far from dense you couldn't find points in being connected. Now this set has "self-similarity" too. This important word to describe Mandelbrot Set, Cantor Set, etc. Even the cauliflower idea from above post. You slice up a circle you have an arc. An arc is not a circle again. A side of a triangle is not triangular, etc. But if you took something from nature, a fractal, a break a piece off you still have the something. They are endless motifs "repeated at all scales". A cloud, tree, rocks, etc. all resemble smaller parts of themselves. Now these natural shapes are very hard to discuss through Euclidean Geometry and are called "pathological" shapes. A snowflake is a pathological shape, it succumbs to a bigger dimension then Cantor Set butit consists of the Koch Curve from Helge von Koch. Everyone has made this curve in school. It's a infinite representation of the Cantor Set through angles. Here is easy link to how to make it -> http://math.rice.edu/~lanius/frac/koch.html
Now mix this self similarity and "fractal" dimension and we are nearing the fascination of fractals. With a Euclidean shape like cube you can cut up cube divisible by 2. You'll have eight smaller cubes. The square will have four squares, the line have two lines. Cut it up like a Rubik's Cube and have fractional pieces of smaller cubes, division by 3 and you have 27 cubes, nine squares, 3 lines moving down the dimensions. Basically any n-dimensional object will have m to the power of n 1/m sized copies of itself.
This is where we get technical: the Cantor Set has dimension 0.63, the Koch Curve is 1.26, clouds 1.35. Coastlines? 1.26. In fact Koch Curves when infinitely produced look like coastlines. Again Felix Hausdorff's similarity dimension as the cutting up cube can be extended to objects NOT self-similar. Fractal shapes that lie somewhere between dimensions have dimensions known as "fractional" Hausdorff dimensions. And because we are talking about fractals it's sometimes called the fractal dimension. This was a theory of Hausdorff's until Mantelbrot revived it via the Julia Set.
Lewis Richardson, using idea of Jonathan Swift's quote:
So, Nat'ralists observe, a Flea Hath smaller Fleas that on him prey. And these have smaller Fleas to bite 'em. And so proceed ad infinitum.
took in natural occurrences like currents and turbulence as collections of smaller eddies and so on that he wrote like Swift as:
Big whorls have little whorls,
That feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.
Lewis Richardson asked many questions about nature, like how long is the coastline of Britain. With a map it's measured by the road near the coastline. But if you walked along the coast and measured it would be more accurate. If you held out a ruler to map it would be too straight. Make the ruler smaller and then you could accomplish measuring at angles. He did this for many coastlines around world comparing data from maps to the logarithm of the size of measuring stick. When Mandelbrot reviewed Richardson's findings he found them as Hausdorff dimensions of the coastline 1.26
Now cellular evolution spans from much of chaos theory of Michael Barnsley in the 1980's. He worked from Sierpinski's triangle (look this up here--> http://math.rice.edu/~lanius/fractals/ )
Cellular evolutions follow patterns to become fractals because it becames layered like sea shells or fern leaves. His chaos game of rolling a dice to chance from points scribed would sooner or later produce Sierpinski's triangle if one initial point was taken. But in some instances it might produce a fern leaf if a different point was chosen as initial point. Once an initial point was chosen the fate of the game was decided what shape would be. This is the idea behind the "strange attractor".
It was known earlier but not overly focused on since chaos theory is fairly new. However because I'm getting tired of typing here are some names major to focusing on bifurcations (in population growths, yeah Differential Equations!), chaos theory, iterations and mapping, fractals! I am tired of typing seriously....here are important names to look up to learn more along the way. Most of these names are scientists noticing fractal and chaotic aspects to objects or systems. Everything from magnetic material to astronomical objects to cancer cells. I'll save to discuss more later-- Robert May, Thomas Malthus, Pierre Francois Verhulst, Henri Poincare, Edward Lorenz, Tien Yien Li, James Yorke, Kurt Godel, Mitchell Feigenbaum, Gaston Julia (Julia Set!), Pierre Fatou, Heinz-Otto Peitgen, Peter H. Richter, Tan Lei, Roger Penrose (will post Penrose Triangles in here sometimes), Arthur Cayley, Thomas Witten, Leonard Sander, Stephen Hawking, Alan Penn, Wilheim Olbers, Hannes Alfven, Ian Stewart, etc.....going back to bed.....lol
me --> :blah:
- #66
Thread Owner
History of Mathematical Notation
For whetting the appetite before I finish my rendition of some special mathematical notation here are a couple of links in regards to the history to some that might be familiar to you from arithmetic to calculus:
http://members.aol.com/jeff570/mathsym.html
http://www.stephenwolfram.com/publications/talks/mathml/
For whetting the appetite before I finish my rendition of some special mathematical notation here are a couple of links in regards to the history to some that might be familiar to you from arithmetic to calculus:
http://members.aol.com/jeff570/mathsym.html
http://www.stephenwolfram.com/publications/talks/mathml/
- #67
Thread Owner
Mom and Dad did say math would be hard.....thanks mom and dad!
Ok within my busy schedule I managed to finish half of the list. Once completed I will post it as pdf file since it will be larger than some you see elsewhere. About one third will be recognizable to most familiar with math. The other two thirds might look new or you've seen before but didn't know what it was for....well that's what this list will be about. Probably at end of month it will be up! I bet you're having restless nights just waiting for it to show up..... :tongue: Math can be enjoyable and difficult at same time so if you're that way let the toughness sink in and relax. The pressure is ON!
I love you mom and dad! :inlove:
Ok within my busy schedule I managed to finish half of the list. Once completed I will post it as pdf file since it will be larger than some you see elsewhere. About one third will be recognizable to most familiar with math. The other two thirds might look new or you've seen before but didn't know what it was for....well that's what this list will be about. Probably at end of month it will be up! I bet you're having restless nights just waiting for it to show up..... :tongue: Math can be enjoyable and difficult at same time so if you're that way let the toughness sink in and relax. The pressure is ON!
I love you mom and dad! :inlove:
- #69
Thread Owner
Notation: Relational Symbols and Order, and Logic
okay so referencing all my books without getting too elaborate I made a few lists. I will be posting pictures in gif form of each section this month. At end I will post a complete full size list in pdf form. Feel free to print out the pictures or pdf when posted.
To start list 1/10 is Relational Symbols and Order, and Logic
UPDATE had to delete Logic.gif and insert a new one since it cropped out bottom "Because" line.
okay so referencing all my books without getting too elaborate I made a few lists. I will be posting pictures in gif form of each section this month. At end I will post a complete full size list in pdf form. Feel free to print out the pictures or pdf when posted.
To start list 1/10 is Relational Symbols and Order, and Logic
UPDATE had to delete Logic.gif and insert a new one since it cropped out bottom "Because" line.
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i took a whole logic class that dealt with the symbols in the first pic there. drove me nuts, i couldn't see why we didn't just use conventional programming symbols for and, or, xor, xand, etc. plus it hurt my brain to think that hard.
- #71
Thread Owner
monsieurjohn said:
it's gone through a lot of changes you know in history. They used to use accents above letters and punctuation you would find on typewriter. GAHHHH that my friend would drive you nuttier indeed. :nuts:
I'm saving advanced logic courses in my last terms in school. That and statistics...... up next Sets......joy....
- #72
Thread Owner
Notation: Sets
Okay here is second grouping: Sets!! In two parts since I didn't want to make the font smaller for view.
By the way if any of you see a mistake or know of a different form (or if I left something out of a category) let me know and I'll add it or correct it!
I want this list to be usuable and trustworthy for you all!
Okay here is second grouping: Sets!! In two parts since I didn't want to make the font smaller for view.
By the way if any of you see a mistake or know of a different form (or if I left something out of a category) let me know and I'll add it or correct it!
I want this list to be usuable and trustworthy for you all!
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psi said:
I love maths cos its the one thing I can actually do, I'm doing AS maths at the mo, 2 exams on tuesday afternoon eeek !
- #74
Thread Owner
Which ones exactly are you taking eh?
I haven't heard the term 'maths' in a long time by the way. Last I heard it was from a European friend years ago. In the U.S. it's usually called 'math' or 'mathematics' but 'maths' is fun since it sounds plural and is short.
Oh, and I'll have the next list up this week!
You all are thrilled, I know....you can thank me later....lol....
I haven't heard the term 'maths' in a long time by the way. Last I heard it was from a European friend years ago. In the U.S. it's usually called 'math' or 'mathematics' but 'maths' is fun since it sounds plural and is short.
Oh, and I'll have the next list up this week!
You all are thrilled, I know....you can thank me later....lol....
- #75
Thread Owner
just an update.....
I haven't forgotten about this; just been extremely busy to finish writing it up, the only breaks I get is when I'm asleep....I need a vacation....someone injure me to give me sick leave from business (and school maybe)!!!
I haven't forgotten about this; just been extremely busy to finish writing it up, the only breaks I get is when I'm asleep....I need a vacation....someone injure me to give me sick leave from business (and school maybe)!!!
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i like math very much even though i got a A+ in sports math
- #79
Thread Owner
Notation: Geometry and Algebraic Topology
Well here are two parts for this one list (split it in half so as not to minimize size of font to read if I kept it in one picture)
side note: some notation you may notice are missing and may be in a different list or I just plumb forgot it {send me message or post the name of the notation here if I left it out of previous list(s)}
coming up next (HUGE list): Algebra
Well here are two parts for this one list (split it in half so as not to minimize size of font to read if I kept it in one picture)
side note: some notation you may notice are missing and may be in a different list or I just plumb forgot it {send me message or post the name of the notation here if I left it out of previous list(s)}
coming up next (HUGE list): Algebra
