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 Author Topic: V9  (Read 7868 times) Description: The general group structure on a circle with a specific example 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « Reply #120 on: May 17, 2017, 08:00:47 AM »

In this seminar Norman illustrates the combinatorics of line segments .
Of particular interest is the Cyclus groups, for which the permutation count formulae is degenerative. It is not n! , but nx1x....x1

I think that complexions is the word used to describe each instance of a permutation .
Everything set out for straight lines of course can be set out for circular arcs. The additional requirement of construction radii to determine arc centres introduces more intersecting lines whic then  may become the subjects of study . It is useful therefore to distinguish between construction lines or radii and the lines of interest which they carry( vectors / Träger)
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jehovajah
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 « Reply #121 on: September 04, 2017, 09:17:12 PM »

In seeking to understand the notation Justus introduced and the term geometric product I have concretely identified it with standard area formulae.
But Justus inspiration was from the geometry of his time which redacted heavily the Greek geometry, rincipally that  of Aristotle as passed down by Islamic scholars. Later Greek sources of the Stoikeia were found but they had little impact except on a few like Newton and Barrows fir example.

The Stoikeia takes the line segment as its fundamental magnitude , but not as it's fundamental measure. The fundamental measure is the arallelogram and the rectangle.
The parallelogram is defined by two contiguous line segments. They form a corner that contains the Rectilineal forms.

This is the product notated by ab where a and b are notation for general line segments.

The triangle is formed by using a third line segment that is a diagonal of the parallelogram , or a diameter of a Rectilineal form. This is usually a factor of a half.

The next product is that formed by drawing a right triangle in a figure that isbrectilineal. This is called dropping a perpendicular from a given point,vertex onto a Rectilineal line. . This product is a right triangle and is denoted by a|b in Hermann Grassmans notation.  While this is associated with the altitude or height of a general triangle, in fact it can be a line dropped any where on another segment , by extending the segment from which it is dropped. . The various products formed in this way are similar or in a proportion so the factor is not restricted to a 1/2. The factor for this product is determined fro trigonometric tables.

There are other products like for examp,e trapezoids  and regular polygons. These products have distinctive firms but are made up of some combination of the basic 2, some combination of parallelograms and right triangles.

The Middler product is thus some combination of right triangles and parallelograms, the factors of each are determined by the form .

In 3d does the general form become a parallelepiped?  Yes, but the form is still dealt with by appropriate right triangles nd parallelograms in the appropriate planes.

The thing to remember from the Greek Stoikeia is that all these products are dynamic. And so a rotational dynamic is inherent in these geometrical products.

That rotational dynamic s form
Malised in the spheres which in general construct the segments and forms by intersection of their surfaces.
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jehovajah
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 « Reply #122 on: September 05, 2017, 06:53:34 AM »

The importance if parallelograms as the general form for products was missed by Justus. He after all was trying to logically found arithmetic, and that is based on the rectangular parallelogram, and particularly the square.
However this is a misconception carried over by Islamic cultures from Aristotelian analysis.

The Sumerians had a greater respect for factors, factoring , the rectangle and the parallelogram inscribed in a circle.
Recent work by Norman Wildberger and his colleague is uncovering the richness of their sexagesimal system.

Factors become the bedrock for apprehending scale, proportions( Analogos) squares( duoplassoi) cube ( triplassioi) and other logoi( dynamic ratios) relationships found in book 5 of the Stoikeia. These ratios are all based on the intersection of chords and secants associated with the circle, in the most general propositions of books 1 to 4

Book 2 is the clue that is overlooked its propositions are based on the general parallelogram with a diameter, the general half parallelogram. And the special rectangular parallelogram, and it's half.

Tables have since Sumer been associated with quick and handy calculation. The process using the tables was therefore key to being competent. In a real sense algebra( symbolic arithmetic) is really an expression of the general process for using a set of tables!

Justus understood that his ring theoretical approach had to be based on this dynamic geometry of the Greeks, but he was not clear how multiplication old be logically founded. There was no proposition for multiplication, only the rectangular form.

Un fact Hermann his son realised that the proposition for multiplication is based on the dynamic parallelogram. Not the static rectangular form. Like all propositions every type of multiplication had to be constructed to demonstrate its validity.

Several products can be constructed geometrically. Books 1 and 2 deal with those as foundational concepts. Pythagoras theorem is a product, intersecting chords is a product, gnomon, parallelograms and rectangles are products.

Where multiplication enters the discussion is in the notion of factoring. Factoring is breaking into parts, and it is this notion of breaking into regular or irregular parts that is emphasised over and over. Those parts that are equal are called artios or perfect, those that are not equal are called perisos or approximate( near to some perfect part) .

The process of multiplication is founded on duplication. The number of times a form is duplicated( isaskis)  is punted and this product is called a polyplassios or multiple form .

The discourse in book 6 is about dynamic ratios . Skesis is a held stage in that dynamic process( kinesis) . The ratios are thus studied in proportions, that is in some relationship of 2 or more ratios. . The first relationship is called Analogos, and involvesc2 ratios . Book 6 goes on to define squared, cubed, inverse and other relationships rarely mentioned.
All of these have a foundation in a geometrical form inscribed in or exterior to a citcle.

Measuring the circular arc was done by using the chord bow.. Remarkably the radius of a citcle is a chord that cuts the circle precisely 6 times by definition! From a proportion perspective one might think  twice the radius( diameter) should cut it 3 times, but in fact it cuts the cicle in equal semicircles. What cuts it into 3 is the diagonal of a parallelogram called a rhombus .
As you study chords more you fing these rhomboids appealing regularly in the dissection of chords. So it is most diagonal length that is important, but what rhomboid form is it that bisects, trisecting quadrasects and quintasects a chord bow.
Proportions in a circle are based on area not length .
 « Last Edit: September 05, 2017, 07:00:09 AM by jehovajah » Logged

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #123 on: September 05, 2017, 07:08:56 AM »

Benoit Mandelbrot RIP
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #124 on: September 05, 2017, 07:21:17 AM »

The logoi that are Analogos ø.
Here they are presented as segment , but in fact they should be presented as rectangular parallelograms.

 « Last Edit: October 15, 2017, 08:16:51 AM by jehovajah » Logged

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #125 on: November 03, 2017, 09:46:16 AM »