jehovajah


« Reply #120 on: May 17, 2017, 08:00:47 AM » 

http://m.youtube.com/watch?v=TsMWT7U_AIcIn 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


« 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 ab 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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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!



jehovajah


« 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.


« Last Edit: September 05, 2017, 07:00:09 AM by jehovajah »

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jehovajah


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




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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!



jehovajah


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



« Last Edit: October 15, 2017, 08:16:51 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!



jehovajah


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




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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!



