Vesica Piscis--Real Sacred Geometry

[Turtle]

___Here is a very good link which describes the geometry of the vesica piscis as we have developed it & goes yet even further to develop geometrically, spirals on the form. Curiouser & curiouser.
http://www.nexusjournal.com/GA-v6n2.html
___I still haven't found the calculus solutions I seek in any of the refeerences so far...hint, hint.

[Turtle]

___Just a note to say to C1ay that the demonstration of the pentagram using circles you attached in post #18 is rather elegant. As you said, arcs would suffice, but the circles leave no doubt.
___One further note, some of the references I linked to begin their constructions on a vesica piscis of short axis length 1 (ie. a radius of 1/2) & my constructions procede from a radius of 1.

[Turtle]

___I now have in mind to work out a drawing/painting which incorporates all of the geometry we have developed & discovered based on the vesica piscis. Of course a lot of what I have in mind simply leaks out

[Turtle]

___Things of significance leak more slowly; as I suggested, I worked up a simple pencil overlay of the 2 star pentagons & the single star hexagon defined by the vesica piscis. Find it in the Science Gallery:

[CraigD]

Folk interested in the intersection of Math and mysticism might appreciate several of the early 20th century “exposé” of various esoteric societies, especially Israel Regardie’s “The Golden Dawn”.

Though it’s necessary to filter through a lot of magikal ritual many would find ranging from silly to disturbing, a lot of western traditional magik writings contains some very interesting math play – though I’ve encountered nothing in any of it is mathematically profound. A surprising number of very sharp Math folk were/are deeply involved in various esoteric societies, like the Golden Dawn, Masons, the OTO, or the Rosicrucians. In the middle half of of the 20th century, the latter society was well known for advertising in popular science magazines – I still remember some beautifully illustrated ones, such as a depiction of bird-winged spheres flying from a human head, with the caption “thoughts have wings”.

[Turtle]

Like a comfortable slipper, this darn thread is just where we left it.
As nothing is not connected & I have a whole lot of new geometric concepts to reapply, this thread came to mind. My Kingdom for a Field. Anywho, they say it is proven that with compass & straightedge alone it is impossible to trisect an angle; this is further qualified to say that the only addition to these tools necessary for trisecting an angle is to make 1 (one) single mark on the straightedge to transfer a measure.
I seem to recall that in this thread we all together constructed the Vesica Piscis & all its inherant special ratios using naught but compass & straightedge. (granted some used computer drawing, but I did not in my attachments). So, as the intersection that is the Vesica Piscis is 120 degrees of each contributing circle, & 120 degrees is 1/3 of 360, haven't we trisected an angle of 360 degrees?

[C1ay]

Originally Posted by Turtle

Like a comfortable slipper, this darn thread is just where we left it.
As nothing is not connected & I have a whole lot of new geometric concepts to reapply, this thread came to mind. My Kingdom for a Field. Anywho, they say it is proven that with compass & straightedge alone it is impossible to trisect an angle; this is further qualified to say that the only addition to these tools necessary for trisecting an angle is to make 1 (one) single mark on the straightedge to transfer a measure.
I seem to recall that in this thread we all together constructed the Vesica Piscis & all its inherant special ratios using naught but compass & straightedge. (granted some used computer drawing, but I did not in my attachments). So, as the intersection that is the Vesica Piscis is 120 degrees of each contributing circle, & 120 degrees is 1/3 of 360, haven't we trisected an angle of 360 degrees?

Yes. What is claimed I think is that you cannot trisect any arbitrary angle with only a compass and a straight edge.

On any circle, you can easily strike off the circle's radius about the circumference in such a manner as to divide it into 6 pieces the way you would lay out a hexagon. This not only divides the circle by three but each half as well.

A pseudo approach one can use that will approach 1/3 of an arbitrary angle is as follows:

Bisect the angle and bisect again to find 1/4 of the angle.
Bisect the upper quarter to find 3/8.
Bisect the angle between 3/8 and 1/4 to find 5/16.
Bisect the angle between 5/16 and 3/8 to find 11/32...etc

By bisecting alternating remainders of an angle a you generate the series 1/2a-1/4a+1/8a-1/16a+1/32a-1/64a+1/128a-1/256a... At the end of just 8 alternating bisections you will have an angle of 0.332 times the original. You will also be getting close to the point that the next angle to bisect is barely the width of your pencil so as to be practically 1/3 of the original.

Somewhere I have seen a rigorous proof that an actual trisection cannot be performed because a series of bisections cannot generate a cube root. I think it was the work of Wantzel if I recall correctly.

[Turtle]

Originally Posted by C1ay

Yes. What is claimed I think is that you cannot trisect any arbitrary angle with only a compass and a straight edge.

On any circle, you can easily strike off the circle's radius about the circumference in such a manner as to divide it into 6 pieces the way you would lay out a hexagon. This not only divides the circle by three but each half as well.

Ok.Now you have me thinking that if the act of laying out the vesica piscis naturally trisect an angle, and you have an arbitrary angle is it possible to overlay the vesica piscis so as to extend its trisection to the given arbitrary angle?

Originally Posted by C1ay

...Somewhere I have seen a rigorous proof that an actual trisection cannot be performed because a series of bisections cannot generate a cube root. I think it was the work of Wantzel if I recall correctly

___Such a proof may include any such machinations as I suggest. I'll have a further look.
PS post rumination: You said [a series of bisections cannot generate a cube root]. But 1 bisection leaves 2 angles & 2 is the cube root of 8.

[C1ay]

Originally Posted by Turtle

PS post rumination: You said [a series of bisections cannot generate a cube root]. But 1 bisection leaves 2 angles & 2 is the cube root of 8.

In the same way that certain angles like 360° or 180° can be trisected there are some incremental roots that can be found. IIRC the proof actually shows that you cannot trisect an angle of 60° which can be generated itself by trisecting a semicircle. It says that you cannot generate an arbitrary cube root as needed for solving such cubic equations. MathWorld has an interesting paper on it here.

[Turtle]

Originally Posted by C1ay

In the same way that certain angles like 360° or 180° can be trisected there are some incremental roots that can be found. IIRC the proof actually shows that you cannot trisect an angle of 60° which can be generated itself by trisecting a semicircle. It says that you cannot generate an arbitrary cube root as needed for solving such cubic equations. MathWorld has an interesting paper on it here.

So this is what I see now. The arbitrary angle only has meaning in regard to a non-arbitray angle, and the only non- arbitray angle is the right angle, or 90 degrees. The case we agrred on in this last discourse is that yes, you can with compass & straightedge trisect an angle; but only some times. Now when are those times? Why the times when you come at it from the right angle. The first time in other words. And after the first time, you have to wait for the next time you can trisect an angle and the algebraic inconsistencies that forbid the trisection are that interval of time. Zenos paradox.
It is no less or more than squaring the circle; algebra forbids it, Peaucellier's linkage just does it. The algebra is an illusion in that it does not tell the full story - Zenos Paradox. Push me pull you.

http://mathworld.wolfram.com/PeaucellierInversor.html