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jehovajah
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« Reply #105 on: February 09, 2014, 07:43:33 PM » |
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Thanks both of you.
I have looked again at the text and Hermann wrote Berührungspunkte. Your suggestion of tangential points is interesting and gives me something to review the translation with.
While I welcome your comments and corrections, please do not feel you have to keep up with me! I have tried to clearly mark my translations and my meditative comments. This format I hope allows any one to translate any piece in the Vorrede, just giving the page reference. While commentaries may become a bit intertwined if we agree to identify commentary on the text as such , I think it will be clear by author who is commenting on what. Then general discussion like this need not be distinguished unnecessarily.
I am the first to admit my German is bad. But I find it more than rewarding to engage with he text rather than a translator!
However if anyone wants to use Kannenberg I am happy with that, and it is a fine translation.
Finally there are many versions of Die Ausdehnungslehre authored by Robert, not Hermann, because Robert saw it as a duty to expound his Fathers ideas as did Hermann. However it is Hetmann who actually corrected a supposed flaw in his Father's theory. And it is Hetmann who laid the philosophical groundwork for the later 1862 redaction mostly edited by Robert, in collaboration with Hermann.
And of course it is Hetmann ho singlehandedly set out the "vector" or Strecken algebra as an example of the power of his analytical and synthetical method.
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« Last Edit: February 10, 2014, 01:03:50 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #106 on: February 09, 2014, 09:38:30 PM » |
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Hallo Jehovajah, I hope this post will give some help on the use of german expressions. For translations between english and german expression I use the internet page leo which, for my purposes does an excellent job: http://www.leo.org/index_de.htmlDoes this translations meet your meaning or would you like to express something different? Hermann Thankyou so much Hermann and kram1032! First for contributing to the thread and secondly for empowering me or anyone to engage with the text directly themselves! The Berührungspunkt is a case in point. The tangential contact points or points determined by tangential contact now refer me to incircle and exterior circle constructs! Schwerpunkt as weightpoint ( a point where weight acts, or a weighted point) seems to suggest itself as appropriate. Even point mass( Newton) or point weight carry the notion succinctly. It is a physical point even in " empty" space. I am reviewing the translation in this light, and will amend some commentary accordingly, but without obscuring my tracks I hope! |
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« Last Edit: February 10, 2014, 07:50:16 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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kram1032
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« Reply #107 on: February 10, 2014, 12:09:32 AM » |
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A weighting point with its associated weight would typically be a Gewichtspunkt with its Gewicht. That being said, some people may refer to it as Schwerpunkt. Still, normally, Schwerpunkt should refer to center of mass. - And barycentric coordinates typically indeed are based on a center of mass. In the simplest form, you take the three corner-vertices of a triangle and replace them with just two coordinates that describe the distance from the center of mass - the average location of the three points. This coordinate system is great for procedural texturing the surface of an object. Though with more vertices (e.g. if you are dealing with quadrangles or other n-gons with n>3), there is no longer a single, simple, natural, well-behaved barycentric coordinate system along the surface. There are two reasons for this: First of all, in case your n-gon is actually flat, e.g. all the vertices lie in the same plane, and convex, e.g. there is no "dent" in the n-gon, then only 3 points are needed to determine a fully valid barycentric coordinate system. So for n>3, anything more complex than a triangle (which notably always is planar and convex in euclidean geometry), the problem is over-determined and you get a valid instance of barycentric coordinates for each combination of three of the n-gon's vertices. None of them, however, will have the nice property of being constrained to a nice, compact interval anymore. Usually, you define your coordinates such that to go from one point to another, you need to go from 0 to 1, e.g. one of the vertices will have the coordinates (0,0), one will be at (0,1) and the last one will be at (1,0). For an n-gon with n>3 you can't easily have all the points lie on such notable, "nice" coordinates. And in case you look at a bent or concave shape, things get even worse. Instead of being over-determined, the problem suddenly naively is unsolvable. You'll need to think of more complex ways to define yourself a nice, "well behaved" barycentric coordinate system. Ideally you'd want such a system to: - describe all corner points, e.g. all vertices with simple coordinates like [0,0], [0,1] or [-1,1]
- be smooth (e.g. arbitrary derivatives at either the center point (the center of mass of the vertices) or along the edges and vertices should always remain continuous)
- deal with bent or concave shapes
- be reasonably fast to calculate, ideally in a closed-form expression
- be smooth in another sense (I don't know how this property is actually called but basically you don't want your coordinates to wildly oscillate - in a perfect world, the second derivative would already be 0 everywhere)
- be unique - e.g. no point inside your n-gon should have more than a single coordinate associated with it (this is another problem for n-gons that doesn't happen for triangles)
I believe some of those are mutually exclusive and I'm not sure if all of them even are possible for all n-gons, so which coordinate system you end up using partially depends on the things you want to accomplish. Finding "the perfect barycentric coordinate system for arbitrary n-gons" is still an open problem and a lot of research goes into this. And all I said above also applies to volumes, in which case the natural shape that has no problems is the tetrahedron, while any volume with more vertices necessarily has some problems. This is especially important for finite elements methods which have become one of the most important tools for simulating complex physical systems like car crashes or other strongly deforming problems. |
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jehovajah
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« Reply #108 on: February 10, 2014, 12:58:58 AM » |
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Thanks Kram1032.
The more I research the more "applied" the Barycentric method seems to be! Still the tangential contact points are only vaguely associated in my searches, but I placed a link to the most direct I could find.
My interest in such a coordinat system is not just for smooth curves or surfaces. I am more inclined toward contiguous surfaces or curves.
However, in this thread I am seeking a global understanding of Grassmanns thinking. As I translate I refine and review my wild conjectures closer to what the text probably means. I do it this way to avoid being too narrowly constrained, but as I go along backward constraints necessarily apply.
I do not think at this stage Grassmann got it all right. He certainly freely recounts modifying his concepts.. I am ok with that because it is heuristic and natural. Yes I agree it is confusing, but it provides me with insights and sensitivities that keep me far from any dogmatism I hope.
Nice survey, by the way! Very informative and clearly presented.
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #109 on: February 10, 2014, 09:29:38 AM » |
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Commentary The Clifford algebra even the Geometric algebra derive from this analysis and synthesis. But I have to say it is more accessible to the student in Hemanns format than it appears to be in modern presentations. Hermann reveals his basic product rules in geometric descriptions. He also highlights the role of the product of points, and finally he relates the summation format with the inclusive point products as the principal and simplest law or rule. In every discussion of the Barycentric method I have seen the ' vector " approach has been the fundamental superstructure of the presentation. In that regard it has been more like Grassmann than late and lazier Möbius notation. The product rules laid out in this section, plus the summation format of those products is what begins the meaningful relationship of this method to geometry. Starting with a product of points gives me a line segment or Strecke. But the Line segment is bounded by those 2 points. A sum of 2 points gives me a weightpoint or a Schwerpunkt. Now I know it is almost exactly early Möbius Barycentric calculus. That is the 2 points are extended to 2 other points by parallel lines to form a parallelogram with the product of the 2 points being one diagonal and the product of the 2 new points being the other diagonal for equally weighted or equal weightpoints! As the weighted points are independently varied the parallelogram becomes a quadrilateral with only one pair of parallel sides and the Barycentre of the system of weighted points moves along the originating diagonal. So contrary to Möbius instinct, dropping the construction lines for the proof of his Barycentric theorem actually obscures how the system works, not clarifies or simplifies it! I can now see how in Hermanns system, a point added to a line segment ( Strecke) actually gives a weightpoint or Schwerpunkt dependent on the weighting of the added point, and the direction associated with that "construction line" weight. Ideally to work it has to be parallel to the line segment in the sum. For any 3 arbitrary points we can work out the Schwerpunkt or weight point, the sum of a point and a product of the other 2 or the sum of any pairs of produced points given the fundamental rule for the sum of 2 Strecken in a 3 point system. This generalises to 4 points in the combinatorial way , based on the products of points. A point in this system is clearly a Schwerpunkt with a weighting of 1. For me the confusion starts with the imposition of the notion of a vector space. This abstract entity is written in set notatin so RxRx...represented by R n is the n tuple vector notation for an axes system of n orthogonal axes generally( but this is not clear). The direct sum of n R sets R + R +... Would seem more in keeping, but the internal products in Grassmans method preclude this set structure. Thus we do not naturally reflect the lineal combination of basis " vectors" in this set structure. In fact , due to the heavy number line use of R the lineal basis of a Dedekind cut for example is lost and we look for abstract numbers.
<![CDATA[<a href="http://www.youtube.com/v/wwmdf5m9khg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/wwmdf5m9khg&rel=1&fs=1&hd=1</a>]]>Applies, I think! Normans simple device of square and round brackets emphasises this subtle distinction . But in addition his primitives are natural numbers or extended to rational numbers, which are coordinated by the underlying axes geometry , and then distinguished as numbers or vectors by bracket notation. The sum of vectors then becomes clear as a geometrical type constructed from differentials in the R n space. To convey all this set up it is usual to say let R n be a vector space. Then we define an element of it as a lineal combination of basis vector primitives. All very neat, but somehow it cuts me off from the actual mechanical space that gives rise to geometrical intuition! Also points are not clear in this set up. In fact R n is used to represent both! The dot product and the wedge product and the general product also create a mental strain, especially if one is not really comfortable with them as a product or multiplication! The generalising of product into a complex aggregation process is best compared to long multiplication! At least then a student might grasp that a product is essentially a sum of sub products! Further, the arithmetical nature of these set ups is crucial to communicate. In that regard Algebra confuses the issue by purporting to be something other than symbolic arithmetic! Looking at the reference Hermann gave to a Kannenberg translation of the 1862 redaction, it is clear from the get go that this is all about arithmetical summation of multiples as sub products. By this stage in his thinking, the lineal aspect is lost in the number theoretical approach. This I guess is down to Roberts influence. While not called such the system is reminiscent of the p-adic number systems in which relatively prime numbers replace the prime powers. In Euclid these are called protoi Arithmoi. That being said the application of Grassmans insights derived from his own Schwerpunkt calculus, which coincided with Möbius Barycentric calculus have still to be explored in his own treatment. |
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« Last Edit: February 10, 2014, 10:30:53 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #110 on: February 11, 2014, 01:19:55 AM » |
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Commentary
The Schwerpunkt Kalkül that Grassmann developed applies equally to arc segments as it does to line segments . In fact a mixture of line and arc segments gives many naturally trochoidal curve locii and surfaces. The flexibility of this method and it's power has much to recommend it. But it is only the technologists and the mechanical engineers who exploit it to the fullest potential in their manufacturing processes..
Again, Laz Plath at qqazxxsw on youTube showcases the extraordinary potential of trochoids, which are a Barycentric calculus of incredible potential for curved surfaces as well as lineal ones.
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« Last Edit: February 14, 2014, 04:29:11 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #111 on: February 11, 2014, 09:24:38 PM » |
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Commentary http://www.cut-the-knot.org/Generalization/ceva.shtmlChevas theorem may relate to Hermann's comment on the tangential points |
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« Last Edit: February 14, 2014, 04:27:40 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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kram1032
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« Reply #112 on: February 11, 2014, 11:13:38 PM » |
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Kalkül = Calculus
So Schwerpunkt Kalkül probably is the calculus of the center of mass or, perhaps, something like the barycentric calculus you mentioned before
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jehovajah
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« Reply #113 on: February 14, 2014, 05:23:11 AM » |
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Commentary The Schwerpunkt in Grassmann's method naturally associates to Möbius' Barycentric calculus Hermann was overjoyed to see. This was because his Schwerpunkt concept used the same notation as Möbius. But, something changed! Hermann seems to say that the roles switched! He was no longer the student of Möbius but the teacher of Möbius in the succeeding episodes! As far as I can tell Möbius published his method in 1827. But continued to develop and refine it over his lifetime. He published in the Crelle, a journal that Hermann published in and even for a time was on the editorial board, until he fell out with the political line the journal started to take. The only real change Möbius made was in notation. It seemed minor and insignificant to him. He replaced the constructing Strecke with the the initial point label. What he had done, in German, was replace a Strecke with a Punkt! In English he replaced a line segment with a point! Now I called this laziness in notation, but there is a powerful precedent in Newton, who replaced a corporial mass(Körper) with a point mass. However the significance is entirely different. In his establishment of the Barycentric principle, möbius does not use a mass at all, he uses 2 parallel Strecken passing through the points A and B. Thus , note bene!, these are point Strecken! They were labelled AA' and BB'. The length of these 2 Parallel Strecken corresponded to the mass or more accurately the mass weight hung on that point. By a principle of Archimedes, the Baryos, or mass of the system was balanced at the fulcrum, the barycentre. This fulcrum was represented by a point Hermann therefore understood the Schwerpunkt as a point Strecke! A point line segment does not make it any clearer, because the intuitive gut idea of Hermann's Förderung was that line segments could represent quantities! Thus the weightpoint is the point at which the mass is attached and through which the associated weight force acts. But geometrically this was represented by a line segment. Being a line segment it has a direction. While it is tempting to draw both line segments as acting in the same direction, ie as a force vector , Möbius equation specifies the line segment must have contra direction! This is geometrically justifiable, because we wish to construct a parallelogram in which the second diagonal cut the focus diagonal at the point of balance. However, the parallelogram is not general enough for differing masses. The constructed figure is actually a trapezium in general. The Barycentric calculus is actually based on trapezoidal geometry. Now the point is that Hermann's Schwerpunkt is his own conception, and his own meaning.mwhatever it translates as, hermanns concept was lineal and thus a directed line segment. Returning to the concept of a point vector, as it is commonly called, Norman, using point vectors describes how the Barycentric coordinates may be found, by given proportions. However Hesse point vectors come from an arbitrary point origin. Using a concept of a point at infinity, common now in geometry, we can see that we recover the parallel lines that Möbius and Hemann based their calculii on. The contra Line Segments can be marked off on these parallel lines and the trapezium constructed, which determines the Barycentre. The new idea, developed it seems with this point vector notion is that these line segments do not need to be parallel! It is the proportions of similar triangles that was key. The Logos Analogos of Eudoxus underpins this method , and a Schwerpunkt can also be identified as a point line segment, or a point Strecke. The Trapezoid (UK: Trapezium)
Trapezoid
Isosceles Trapezoid
A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.
It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown.
And a trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides:
Trapezoid Trapezium
US: a pair of parallel sides NO parallel sides
UK: NO parallel sides a pair of parallel sides
OMG! http://mathworld.wolfram.com/Trapezium.htmlI find from this quote that Möbius was a student of Archimedes? http://books.google.co.uk/books?id=AXl9j4n2iP4C&pg=PA29&lpg=PA29&dq=trapezium+barycentre&source=bl&ots=WfSfujRmmn&sig=FsxOwJf-5UGijS5jeqhjtd4rte8&hl=en&sa=X&ei=A6X9Uu2hItGRhQfW3IFA&ved=0CDYQ6AEwAw#v=onepage&q=trapezium%20barycentre&f=falsehttp://uk.answers.yahoo.com/question/index?qid=20070422093416AAVaU4sNote in these answers no mention of Möbius method!i There is also one subtle point usually finessed by referring to Archimedes fulcrum: the coordinates switch the line segment relationship to the point. Using the point vector approach the correct relation is maintained. The finesse overlooks the important anti commutativity relations in space. |
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« Last Edit: February 16, 2014, 10:08:53 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #114 on: February 14, 2014, 07:11:26 AM » |
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« Last Edit: February 14, 2014, 03:26:39 PM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #115 on: February 14, 2014, 07:23:19 AM » |
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This discussion of the centroid is the nearest I have found to the allusion to the weightpoint and the tangential contact points. http://math.tutorvista.com/geometry/centroid.html?view=simple |
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« Last Edit: February 14, 2014, 03:25:56 PM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #117 on: February 17, 2014, 11:36:05 AM » |
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Ausdehnungslehre Vorrede p vi
Footnote : in practice it usually shows itself quickly how through this method of analysis the difference between the analytical and synthetic treatments of Geometry are completely removed.
Back to text: p vi and vii.
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« Last Edit: February 17, 2014, 11:55:08 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #118 on: February 18, 2014, 04:42:01 AM » |
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Commentary
An earlier passage in the light of the Barycentric calculus now makes more sense!
" I did not realise what a rich and fruitful field of study I had mapped out, and thought it not worthy of consideration until I combined it with a related idea, found in the notation of geometrical products as presented by my father!
In following this notation, as my father would apprehend it , it dawned on me that not only the rectangle, but more importantly the parallelogram, should be considered the same as the product of 2 adjacent sides, jostling the one against the other, even if One specifically Once again apprehended the product not as of lengths but as of both line segments with their length and direction firmly combined !
In which thought I now brought into combination the aforementioned notation of the sum and this notation of the product, and thus it was revealing itself the most stumbled upon Harmony, specifically, even if I
• replaced any perceived sum of pairs of line segments (in the sense already given to you) by a third, lying in the Same plane line segment, in the planar constructed sense to multiplying;
• multiply the individual pieces with this same line segment;
•and add the products with the appropriate careful observation of their positive or negative values;
Then it shows itself that in both cases every time the same result has to be coming about and has to proceed onward"
Thus for 3 points A,B,C 3 Strecken / line segments have direction and length AB length c, BC length a and AC length b.
In combination
cAB + aBC=bAC.
Combining the length and the direction.
Thus replacing
cAB + aBC by bAC and multiplying it by itself gives
(bAC)2 = bAC(cAB + aBC)
—>bcACAB + baACBC
This is in the planar or 2 dimensional sense attributed to multiplying lengths in the plane to calculate areas. However, the usual convention was to use rectangles and Pythagoras theorem for line segments not rectangular. Hermann innovates on convention by allowing any adjacent line segments to multiply(ie no overt use of Pythagoras theorem).
But now look what happens:
c(cAB + aBC )AB + a(cAB + aBC )BC –> (cAB)2 +caBCAB + acABBC+ (aBC)2
Pythagoras appears !
It can be set to 0 revealing again the anti commutativity.
Although Hermann does not state this , extrapolating on the concept of lineal summation, we can consider the product of a line segment with itself as lineal " multiplication". Thus planar multiplication can then be thought of as the product of any two general Strecken ( in parallelogram formation and adjacent) . These necessarily lie in the same plane.
Thus using the perceived sum in this way reveals Pythagoras as lineal multiplications and 2 planar multiplications or planar products. , by zeroing Pythagoras we get lineal products are zero planar multiplications, and planar multiplications are anti commutative!
This planar construction to multiply quantities is not properly understood. Justus got stuck here in his analysis of Mathematics and it's foundation in arithmetic. He got stuck because arithmetic is founded in geometry, which in turn is founded in and refined by mechanics,
I was Newton's opinion that the two worked hand in glove to perfect each other, but at the end of the day for me Mechanics and interaction with spatial objects or regions founds and refines our Metrons, which , providing we use the concept of Monad, establishes our arithmetic.
However we do not need to use Monad, Monas is just the concept of 1( one). Many of us do not even realise it is a concept!
If we do not use this concept, we may still use our concepts of comparison: greater, lesser, dual. This leads to Logos Analogos thinking, often called proportioning, but less well recognised as reason, or rational thought. Within this praxis systems of reasonings often called logic developed, but that is another story.
Thus planar multiplication is not well understood, nor is multiplication in general.
The Pythagorean school started not with multiplication, but division into factors and or fragments or parts. From this the concept of a whole as a sum of parts is crucial. This gives a singular purpose to summation or aggregation.
Within this summation context a Metron as an entity representing Monas becomes psychologically important. With a Metron we can count by measuring the Metron against the focus object. The focus object or any entity can be determined as the whole , the parts of this whole may then be counted as varying parts, or a standard part can be used as a Metron to measure and count all other parts.
These parts are factors of the whole. Nevertheless, with a standard Metron, various factors can be defined as counts of this Metron.
Recognising the whole and a lesser entity as a Metron, allows the conception of the whole as a multiple form. This simply means the whole is factored from many parts or many repetitions of the Metron.
The form of the whole may be geometrical, or spaciometrical. That is to say it may be set in the ground plane or freestanding in space.
The implications of multiple forms is down to each individual to determine, but most learn factor tables for standard units. These are misleadingly called multiplication tables. They are factorisation tables or factor product tables.
The geometrical representation of these multiple forms, usually set in a mosaic, led to various geometrical standard shapes being formed or created and measured by standard Metrons. These measurements are the basis of the planar sense of multiplication.
The various formulas for the areas of the standard geometrical forms further define the planar sense of multiplying. Their derivation however, were strictly by geometric proofs. Thus any geometrical manipulation in a sense is part of the process of establishing a formula to " multiply".
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« Last Edit: February 18, 2014, 01:34:29 PM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #119 on: February 18, 2014, 06:16:41 PM » |
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Commentary The lineal product is not the same as scalar multiplication. If we enter Grassmanns promoted geometrical view, his Förderung, we consider length and direction as bound together. Notation ally or rather how we get a label or Handel on this is by using lowercase letters for lengths and uppercase letters for points. Yen a line segment " opposite" a point is labelled in its length with the lowercase letter to the letter labelling the point. This was common practice when I learned trigonometry, and the advantage was cyclic or rotational symmetry! This frequently meant, tht if you had proved something to hold for the letters upper and lowercase in one interpretation, you had proved it for all interpretations in the cycle of the letters.. In addition to this labelling system the labels were directly combined! Thus cAB represents a line segment between points A and B whose length ( in some arbitrary metric) is c. This length and this line progresses in the direction A to B! This notation is none other than the notation of a basic vector concept! The length c is a quantified magnitude, and AB is the designated direction. This product represented the notion of a magnitude with a direction. In the early lessons of physics I remember being introduced to a vector in these terms, bu without these labels or notation! Now, this being the fundamental idea Hermann had, expressed notationlly by letter labels in this way what are the implications? First of all the length of a line segment: the dimensional units are arbitrary the fundamental thing is the numeral names the count of some Metron . What is this Metron? It turns out to be a standard piece of matter , usually rigid, used to compare. But we can and do represent this standard unit by a straight line segment!, we do not move away from the line segment, but rather gravitate to it as if drawn to it psychologically.. Thus 3 cm means the unit line segment is 1 cm long . What is nowhere to be seen is its direction! This 3cm AB specifies it completely. Nw conventionally we emphasise the length unit but not it's direction points , but when that is done I hope you can see that 3 AB cm specifies the unit direction leaving the numeral free and unrncumbered.. We can now apply this to the number line concept as shown in this video.
<![CDATA[<a href="http://www.youtube.com/v/t0aHtXud9r4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/t0aHtXud9r4&rel=1&fs=1&hd=1</a>]]>The thing to note, is that when we do linel multiplication we do not mention the units! We just use the numerals nd add in appropriate units at the end! Well, now Hermanns notation allows us to define precisely what we are doing, and how precisely units fundamentally work in the lineal space! Leaving Sade the sums and products of signed rational numbers I focus just on the product of the line segments Thus AB x AB is AB 2. Now as a product it is a line segment considered as 2 parallel and point wise collinear line segments. It would seem consistent to define it as just AB , in this special case of lineal multiplication. However it can also be seen as a special product of planar multiplication, in which case it would be more consistent to define it as 2AB. But again in the planar sense it is a collapsed parallelogram and therefore a null or 0 planar form. It is completely lineal and lineal projections within that space are defineable which again is consistent with a line segment defined as 2AB. We would then define lineal products as 2 times the coefficient product in the direction AB. By a similar projective pricess AB n would be defined consistently as nAB. When we use a number line concept to define multiplication in a lineal sense we ignore this analysis and just use AB n = AB. As a notation it leads to minor inconsistencies which we plainly just ignore. If we do not ignore it we can actually use the collapse of space to model an instantaneous impulse in the line of collapse, even beyond the square of the coefficients for the planar case. There are other products that Hermann explored in the planar setting, so the lineal case has not received much attention. The other thing we do is we use a concept of scalars! They are not numbers on the number line they are counts in our heads. Using these we do not get the expansion of space on multiplication, instead we get multiple copies of a Metron. |
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« Last Edit: February 19, 2014, 12:19:48 AM by jehovajah »
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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August 18, 2014, 10:59:39 AM
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