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we need the Derivative as a Vector, not as slope or tangent Re: definition of "cell" #16.3 Unitextbook 6th ed.:TRUE CALCULUS
Posted:
Jul 5, 2013 1:46 PM


#15.3 fine structure and hyperfine: proof from the Maxwell Equations, that physics has no curves; #15.3 Unitextbook 6th ed.:TRUE CALCULUS
Sorry, previously I said that radioactivity should be a good place to look in physics to prove that there are no curves in physics, but rather instead, a collection of tiny straightline segments that takes on the appearance, deceptive appearance of being a smooth curve when in reality it is tiny straightline segments compiled together. Radioactivity may render another proof, but the better proofs are light rays in Newton's prism and the fine structure in physics.
So we have angular momentum in physics and spin in physics and we have electrons going around the nucleus in atoms. Now if there are no curves in physics, but rather all of these circuits are a bunch of tiny straightline segments collectively put together, then we should have spectral lines with forever finer structure. If no curves exist, but only tiny straightline segments, then there is always a finer structure of splitting as we get smaller and smaller in physical measurement of circles.
So, the meaning of pi is that in Euclidean geometry, no matter the size of the circle, that its circumference is about 22/7 times its diameter. The number "e" is the same as pi, only the circle is a logspiral and is now open and 19/7 relates the circumference with a diameter that is a whirling square inside the spiral.
Usually mathematicians do not include the finestructure constant along with pi and "e". In fact, most mathematicians do not attempt to include pi and "e" together. But I relate all three. And the fine structure constant (its reciprocal) of 137 relates the circumference of a circle (a circuit for the electron in a hydrogen atom), by telling us it must have a minimum of 137 straightline segments that compose into making that circle, in other words, the electron circle is at minimum a 137 sided polygon.
I spent most of the day pondering the proofs of physics that no curves exist in physics, and the places to look are where physics thinks of a continuum. A continuum of colors in the spectrum of light in Newton's prism. A continuum of points in a circle, whereas the real truth is that the circle is composed of a finite number of points all connected by tiny straight line segments.
Now in this textbook we can look at pi as the number of subshells in plutonium divided by shells, which is 22/7 and in plutonium only 19 subshells are filled at any one instant of time, so "e" is 19/7. And now the fine structure constant of physics is ((22/7)^7)/22.
Now I wish in this page I could report to you how those three numbers relate to the infinity borderline of 1*10^603. Pi relates of course in that we know infinity border has to be where every important theorem of geometry is true and that is Floor pi*10^603. And the number "e" simply follows along and cuts at the exponent 10^603. But how does fine structure constant fit into this 10^603 picture? Well, it is pi to the 7th power, divided by 22. It is a tangled mess of pi coefficient numbers of 7 and 22. I do not know what to make of it tonight, with respect to a minimum number of straightline segments that goes to make up the electron as it circuits around the proton nucleus.
AP
END
#15.4 fine structure and hyperfine: proof from the Maxwell Equations, that physics has no curves; #15.4 Unitextbook 6th ed.:TRUE CALCULUS
 show quoted text  Perhaps I do know. If you take a graph paper and make a square of 11 by 11 points and another square of 12 by 12, that 137 points falls between 11 by 11 and 12 by 12. Now draw a circle that includes as many of those 121 points and those 144 points. And now draw straightline segments connecting points to make a circle appear.
If you had a 5 by 5 square, your best circle would look too much like a square rather than a circle of the total possible 25 points.
The fine structure constant (reciprocal) of 137 starts to capture enough points that it actually begins to look like a circle.
So the fine structure constant is a mathematical constant where we have the minimum points of a grid to form a dodecahedron.
AP
inverse fine structure constant is the minimum number finite points to construct polygons #15.5 Unitextbook 6th ed.:TRUE CALCULUS
Alright, I got a piece of graph paper and with a black magic marker so the ink goes through I marked off a 11 by 11 grid, and then my task was to see if I can retrieve a 7 sided polygon called a heptagon and a 12 sided polygon called a dodecagon from this grid that has 121 points. And then check out if the 12 x 12 = 144 allows those two polygons for an easy construction.
What I am suspicious about, is that the inverse fine structure constant of 137, is the Minimum number of points to allow for these two regular polygons to come into existence.
So far, it is looking promising in that of constructing right triangles of 1 by 2 sides.
So, I have the instinct that 137 is the minimum number of evenly spaced points to form these polygons.
AP
definition of "cell" in New Math is crucial to calculus and relates to vector #16 Unitextbook 6th ed.:TRUE CALCULUS
definition of "cell" in New Math is crucial to calculus and relates to vector #16 Unitextbook 6th ed.:TRUE CALCULUS
Alright, a question arose about my definition of "CELL" in mathematics of calculus. I thought I made it quite clear and plain to understand.
When you have an infinity borderline of 1*10^603 then the inverse is the microinfinity borderline of 1*10^603. So the borderline marks out the entire Cartesian Coordinate System of the xaxis smallest finite interval is 1*10^603
. . . . 0 a b c
So that a = 1*10^603 and b= 2*10^603 and c= 3*10^603
So the CELL is defined as the smallest finite width of a rectangle that is possible and the length or height of the cell can go from 0 to 1*10^603. And the CELL can accommodate a picketfence or a rectangle or triangle.
We use only the 1st quadrant since other quadrants are redundant in Calculus, for we can easily translate a function curve to be placed completely inside quadrant 1.
So in New Math, in the 10 Grid system where we pretend 10 is the borderline of finite to infinite, the Cell width is 0.1 and be any rectangle or picketfence or triangle inside that xaxis width of 0.1 and rise to be of length 10. Now on a special occasion, such as division by 0, the cell can rise a tiny bit further to be 10.1 in the 10 Grid, since 10.1 is the first macroinfinity infinite number.
Now the cell is crucial to New Math because the integral is the summation of the picketfences or the rectangles or the triangles in all of the 100 cells of the 10 Grid or all the 10^1206 cells of the 10^603 Grid.
So in New Math, for calculus, the integral is the summation of the areas of the picketfence or rectangle or triangle in each of the cells.
The CELL is why New Math is transparent and clear, because we can zoom in on any particular cell and see what the derivative or integral for that cell is. In Old Math, when you zoom in on a point of the graph, you are fiddling around with limits, limits everywhere, and not because you want to make clear or transparent, but rather to hide and obfuscate the function and graph at that point.
Now the CELL is related to the Vector in New Math. The Vector in New Math is the connecting of two finite points and with a arrow at both ends indicating the vector may have more finite points if the vector is extended. The vector has direction and has quantity of length. The vector does not have area, for the cell has area.
Now Vectors have unit vectors of length 1 unit distance. Cells do not have unit cells because their widths are all the same width and their lengths all have the same potential to be macroinfinity long. But there is a concept of a Unit area in New Math. For the 10 Grid, the unit area is the smallest nonzero area inside a cell. For the 10 Grid, that UnitArea would be 0.1 width and a triangle of 0.1 height for an area of 1/2(.1*.1). But there is a minor problem issue with unit area in New Math, in that the number 1/2(.1*.1) = 0.005 is not a finite number in 10 Grid. And the same issue crops up in the 10^603 Grid, but what happens is that in science or mathematics, we do not worry ourselves with widths of 10^603 for most of physics is done with numbers no smaller than 10^22. So in New Math, we do have unitareas as we have unit vectors.
Now in New Math, vector theory is altered also, but this book is focused on Calculus and I would stray too far if I went into vector theory.
AP
Derivative is a vector related to cell Re: definition of "cell" #16.1 Unitextbook 6th ed.:TRUE CALCULUS
Sorry, I should have said that the vector is related to the cell, for the derivative is a vector. So in New Math, the Calculus without the phony limit concept, relates the derivative as a vector to the integral as the area inside a cell. For example, in the function y = 1/x, and its derivative as y' = (1)(x^2) that we see the derivative is a vector of direction (1) and whose vector quantity is (x^2).
So in New Math, we relate vectors to cells.
AP
more clarity on Derivative as a vector related to Cell Re: definition of "cell" #16.2 Unitextbook 6th ed.:TRUE CALCULUS
No, I have not made myself as clear as what I need to make. Thanks to a response, I still see I have obfuscation to make clear.
A CELL in mathematics of Calculus is the width of the smallest nonzero finite distance times the length of the largest finite distance. These two I have often called microinfinity and macroinfinity.
Now all of Calculus takes place in the 1st quadrant only, since all other functions can be translated and transformed to be in the 1st quadrant.
So, macroinfinity is the borderline between the last and largest finite number where the next number is an infinity number, and microinfinity is the smallest nonzero finite number (the inverse of macroinfinity).
Examples make more clear than words of explanation:
So in 10 Grid where we pretend that 10 is the last and largest finite number, and thus macroinfinity is 10 and where 0.1 is microinfinity.
So a CELL in 10 Grid is of width on the xaxis as 0.1 and whose length is 10, so the full cell is 0.1 *10 = 1 square area. In the 10 Grid there are exactly 100 cells, no more and no less.
In the 100 Grid the Unit Cell is 0.01 * 100 = 1 square area and there are exactly 100*100 = 10000 unit cells.
In the 1000 Grid the unit cell is 0.001 * 1000 = 1 square area and there are exactly 1000*1000 = 10^6 unit cells in 1000 Grid.
In the 10^603 Grid the Unit Cell is 1*10^603 * (1*10^603) for 1 square area, and there are exactly 10^1203 unit cells in the 10^603 Grid.
In previous posts I was mistaken or unclear as to Unit Cells, for there are unit cells and they have 1 square area as the unit cell.
Now also in Cell theory is related to Vector theory for the two are part of one larger theory Calculus.
The unit vector in Cell theory is not that of Old Math where it has length of 1 but is rather instead of length microinfinity. So in 10 Grid the unit vector is 0.1 and not 1. In the 100 Grid the unit vector is 0.01 and not 1.
Now in Old Math, they never had a borderline between finite and infinite and so they would never have a Cell theory to relate to a Vector theory. In New Math, the derivative is a vector and the cell is where the integral counts up the areas inside of each of the cells.
The vector in New Math is a line segment that has at least two finite points and has one arrow of direction and has quantity. In New Math, vectors are composed of mostly empty space in between two finite points of the vector. But the major difference is that the unit vector is not of length 1 but is rather the length of microinfinity.
Now I do not know if mathematics ever had a Algebra of Vectors, much like Group theory is to numbers. But I doubt Old Math could have a Algebra of Vectors because they never had a Cell theory to be inverse to the Vector theory. And to have a Group theory Algebra you need a units. So the units have to be the same units for the cell as for the vector. For the cell, the width is microinfinity, and for the vector, the unit vector length is microinfinity.
AP
Elliptic and Hyperbolic are subsets of Euclidean geometry #14.1 Unitextbook 6th ed.:TRUE CALCULUS without the phony limit concept
Sometimes a takes awhile to realize the importance of a discovery. In this case, of finally finding a proof that Euclidean Geometry = Elliptic Geometry unioned with Hyperbolic Geometry.
Finding the proof in Calculus is not surprising.
And the proof is as simple as a diagram. We draw a circle as best we can and along side it we draw a logarithmic spiral, the equiangular spiral so that it is tangent to the circle. And the proof consists of showing that a segment of the circle arc equals a segment arc of the log spiral. So that as we put the circle segment of ) next to the log spiral segment of ( that the two are equal but curved in opposite direction and that the two together cancel and forms a straight line segment of )(
That is the proof. And it is also supported with the idea that the number e is the pi of Hyperbolic geometry for which I want the student or reader to do some exercise. For this is the very best way of learning the meaning of "e" and of pi together. I am of the school of education thought that learning is best done by "doing" the pragmatist viewpoint of life. Learning is doing. If you only read and not do, then you only learn at best, half as much. Our very best learning is doing, mixed with some reading.
Now there is a math textbook that all math textbooks should be written in the manner of. It is "Mathematics: A Human Endeavor" by Harold Jacobs, 1970. If all math textbooks were written in this style, we would have a lot more people doing mathematics and doing math without the need of teachers. So much of our education system and textbooks in college are so poorly written, that you need a teacher to navigate through that trashy textbook. When a textbook is well written, then the curious person who wants to learn the subject, needs no teacher. One of the reasons math classes in college are low enrollment is because, the text is piss poor and the teacher is poor at teaching. If all math texts were as well written as Jacobs, the USA would not be deficient in people learning math and liking math. What turns people off of math is when they encounter an incomprehensible text and a poor teacher using that text.
This is the 6th edition of my Calculus text, and I have not yet reached that teaching excellence, for I am learning new things myself. But in the 7th edition of this text, I should begin approaching the teaching excellence.
Now in that Jacobs text on page 291 is a picture of a logarithmic spiral inside a rectangle of whirling squares. It is probably on the Internet somewhere, but I want the student or reader to photocopy that page of Jacobs and then get a piece of flexible wire, and cut the wire of a length that matches the radius as shown on that page of length 55. Now, three and a tiny bit more of those 55 lengths should be as long as two of those arcs in the 55 square shown, (that is a semicircle). Now, however, using that same wire track down the length of the wire that it takes to cover the 55 square and the 34 square and finally the 8 square, note that the 8 square has to be extended over to the right inside the 21 square.
What the student or reader should find is that it takes roughly 2.71 of the wire to cover that arc.
So here we learn best meaning of the number "e". The number "e" is pi in hyperbolic geometry where circles are not closed but are open and spiraling outwards. In Elliptic geometry, a circle is closed and its circumference is pi times diameter. In Hyperbolic geometry, circles are log spirals that are open, and it is "e" that relates the diameter of the log spiral with the circumference.
Now that is a technical understanding, but the deeper meaning, that takes time to settle in, is that Geometry is just Euclidean Geometry and that Elliptic and Hyperbolic geometry are not independent, stand alone geometries. Much like in physics, a particle is a wave and you cannot separate the two.
In astronomy and physics, there is an open question of whether the Universe is Euclidean or Elliptic or Hyperbolic, well that is a dunce question, because there is only one geometry of Euclidean and you can rescue Elliptic and Hyperbolic out of Euclidean.
In physics, whereever there is electricity, there is magnetism, for the two are not independent but always dependent on one another. You can never have one without the other, for they are duals of one another. We can write a formula in physics of this:
Electromagnetism = electric unioned magnetism
Likewise we can write:
Euclidean geometry = Elliptic unioned Hyperbolic geometries
In Old Math, they had the notion that these three geometries were independent, and in New Math we learn there is but one geometry and that Elliptic and Hyperbolic are duals and are Euclidean. I think the best way of saying this is that Euclidean geometry is the only geometry and that Elliptic and Hyperbolic are subsets thereof.
AP
we need the Derivative as a Vector, not as slope or tangent Re: definition of "cell" #16.3 Unitextbook 6th ed.:TRUE CALCULUS
On second thought, perhaps we need the Vector as essential to Calculus, much as we need the Cell as essential to the integral in Calculus.
What does the vector have that the slope or tangent line do not have? Well, for one item, the vector has direction and as we see in y = 1/x, the derivative has the 1 direction. The concept of slope or tangent does not capture this notion of a direction for the derivative, but a vector does capture that feature.
And, in Old Math, they never really sorted out this ambiguity of whether the derivative is a slope or is a tangent, they sort of mix and mash the two together whenever it suits their discussion, and if one professor likes tangent more than slope we have a tangent favoritism and bias versus another professor who adheres to slope. But probably both are wrong and incorrect because neither has a direction or orientation for the derivative which the vector as derivative would have that orientation.
AP



