PLEASE REF CIRCLE QUADRATURE VIA QUADRATURE TRIANGLE:

1] https://www.linkedin.com/pulse/circle-circumference-equated-naturally-square-area-pi-stefanides?trk=mp-reader-card

2] https://www.linkedin.com/pulse/eppur-si-muove-circolo-la-quadratura-irrespettivo-del-stefanides?trk=mp-reader-card

3] https://www.linkedin.com/pulse/polygonal-method-compared-triangular-quadrature-panagiotis-stefanides?trk=mp-reader-card

4] https://www.linkedin.com/pulse/dialogo-2014-paper-pstefanides-panagiotis-stefanides?trk=mp-reader-card

5] https://www.linkedin.com/pulse/20141101102021-77094622-conference-presented-graphics-decoded-concepts-from-classical-philosophy-of-plato-s-timaeus?trk=mp-reader-card

6] https://www.linkedin.com/pulse/20141031145214-77094622-nested-circles-squares-triangles-configuration?trk=mp-reader-card

7] https://www.linkedin.com/pulse/science-art-%CF%80%CE%B1%CE%BD%CE%B1%CE%B3%CE%B9%CF%8E%CF%84%CE%B7%CF%82-%CF%83%CF%84%CE%B5%CF%86%CE%B1%CE%BD%CE%AF%CE%B4%CE%B7%CF%82-panagiotis-stefanides?trk=mp-reader-card

Regards from Athens,

Panagiotis Stefanides

-http://www.stefanides.gr

-https://www.youtube.com/watch?v=XeOjPmKSsOI&feature=em-upload_owner

P.S. Sides of the proposed Scalene Orthogonal:

Hypotenuse is equal to T^3 , Bigger Side is equal to T^2 ,and Smaller Side is equal to T^1.

Ratio of Sides :

T^3 / T^2 = T^2 / T^1 = T

Panagiotis Stefanides

Season's greetings and happy the new year 2014.

From athens,

panagiotis stefanides

PLEASE REF CIRCLE QUADRATURE VIA QUADRATURE TRIANGLE:

1] https://www.linkedin.com/pulse/circle-circumference-equated-naturally-square-area-pi-stefanides?trk=mp-reader-card

2] https://www.linkedin.com/pulse/eppur-si-muove-circolo-la-quadratura-irrespettivo-del-stefanides?trk=mp-reader-card

3] https://www.linkedin.com/pulse/polygonal-method-compared-triangular-quadrature-panagiotis-stefanides?trk=mp-reader-card

4] https://www.linkedin.com/pulse/dialogo-2014-paper-pstefanides-panagiotis-stefanides?trk=mp-reader-card

5] https://www.linkedin.com/pulse/20141101102021-77094622-conference-presented-graphics-decoded-concepts-from-classical-philosophy-of-plato-s-timaeus?trk=mp-reader-card

6] https://www.linkedin.com/pulse/20141031145214-77094622-nested-circles-squares-triangles-configuration?trk=mp-reader-card

7] https://www.linkedin.com/pulse/science-art-%CF%80%CE%B1%CE%BD%CE%B1%CE%B3%CE%B9%CF%8E%CF%84%CE%B7%CF%82-%CF%83%CF%84%CE%B5%CF%86%CE%B1%CE%BD%CE%AF%CE%B4%CE%B7%CF%82-panagiotis-stefanides?trk=mp-reader-card

Regards from Athens,

Panagiotis Stefanides

-http://www.stefanides.gr

-https://www.youtube.com/watch?v=XeOjPmKSsOI&feature=em-upload_owner

Theories that found classical insoluble problems by ruler and compass have for solutions methods  involving certain mathematical approaches , by which results may be checked  by feeding  them back.

For instance solving the problem of doubling of the cube and finding that the third root of 2 is1.25992105…

We check back if it is true[ to the involved approximation] we get

[ 1.25992105..]^3 = 2

[ using a hand held scientific computer]

Similarly with the trisection of an angle etc.

 Ref: http://www.stefanides.gr/Html/Classical_Problems_et_Alii_with_Web_Links.htm]

However with circle’s pi we do not have a theoretical “Standard Model Theory” for doing this feeding back checking.

Theorists proved that pi is transcendental and cannot square the circle.

However none of their methods involved the “Quadrature Triangle ” as I call the one I proposed as Plato’s Timaeus “Most Beautiful Triangle” [ PMBT ] similar to which is the “Kepler/Magirus” [ known for more than 400 years- uninvestigated and untaught in general] but not as beautiful as Plato’s [component  triangle of the PMBT.

  Ref :a) http://www.stefanides.gr/Html/platostriangle.htm

b) http://www.stefanides.gr/Html/GOLDEN_ROOT_SYMMETRIES.htm

c)http://www.stefanides.gr/pdf/GOLDEN_ROOT_SYMMETRIES_OF_GEOMETRIC_FORMS_by_Panagiotis_Stefanides.pdf  . 

It seems that theorists seeked a value of pi that could prove the case of quadrature of circle.

However this is not a requirement, in first case, because as shown below it is shown that this relationship exists by using natural numbers based on the “PROPERTIES” of the “QUADRATURE TRIANGLE”.

Ref : d) https://www.linkedin.com/pulse/circle-circumference-equated-naturally-square-area-pi-stefanides?trk=mp-reader-card

e) https://www.linkedin.com/pulse/eppur-si-muove-circolo-la-quadratura-irrespettivo-del-stefanides

Further to this findings, by contriving certain theorems applicable to this case, we get the value of pi as the ratio of 4/ SQRT [Golden Number] , by ruler and compass [ being irrational], which was anticipated, for many years as an approximate one, by many researchers and book authors.

Ref :  f) http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf

g)   [ compare : http://www.stefanides.gr/pdf/2012_Oct/PHOTO_11.pdf  )

h)  http://www.stefanides.gr/Html/QuadCirc.htm

i)  http://www.stefanides.gr/Html/piquad.htm

j) http://www.stefanides.gr/Html/theo_circle.htm

k) http://www.stefanides.gr/pdf/DIALOGO_2014_PANAGIOTIS_STEFANIDES.pdf

l) https://www.youtube.com/watch?v=XeOjPmKSsOI&feature=em-upload_owner

I thank you,

Regards from Athens,

Panagiotis Stefanides

http://www.stefanides.gr

Plato in his Timaeic Philosophy assumes existence of matter before creation in the form of traces of the “ four elements”, [and the existence of a fifth “ consistency”]:

“The conditions prevailed before the World was Created, while all elements [FIRE, AIR, EARTH and WATER] were “WITHOUT PROPORTION” [alogos] and “WITHOUT MEASURE” [ametros], and only “TRACES” [ i.e materialistic] of them existed, as all things, naturally exist in God’s absence. God, under these conditions, transformed them via “IDEAS” and “NUMBERS”, for them to become “MOST BEAUTIFUL” and “BEST” as possible, contrary to their previous state [Pl. Ti .53B]”.

http://www.stefanides.gr/pdf/GOLDEN_ROOT_SYMMETRIES_OF_GEOMETRIC_FORMS_by_Panagiotis_Stefanides.pdf

 Link attached above is a [book of mine, based on international conference proceedings].

bares work on a very special scalene orthogonal triangle[ quadrature triangle- my interpretation of Plato's Timaeus "most Beautiful Triangle"] similar to ,but not the same as, the Kepler/Magirus triangle[ its qualities are for me of importance]

It relates squares, circles, parallelograms...:

Further Link : http://www.stefanides.gr/html/Root_Geometries.htm ,

shows images of interpreted logic of the geometries structuring the five Platonic solids from lines, triangle surfaces volumes , via structuring a great pyramidal model [ lines assumed to represent combinations of the four elements].

In subjects concerning motion, forces, energy, power etc., I noticed, that, there exist seven (7) basic forms, which appear to derive one from the other and thus related. These forms (relationships) necessary for the creation of a (powerful) work element, from its conceptual idea to materialization are Line, surface, volume per unit time (mass rate of unity water density), momentum, force, work (or energy) and power. They are fractions with numerators powers of space (length) and denominators powers of time:

L^1 /T^0 , L^2 /T^0 , L^3 /T^1 , L^4 /T^1, L^4 /T^2, L^5 /T^2, L^5 /T^3.

Length (L^1 ) and surface (L^2) are timeless (T^0 =1), Encephalic Concepts.

Ref. p13:

http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf

worked lengthily, searching Plato's Timaeus " Most Beautiful Triangle" and derived a an interpretation of mine , different to that of current translations. The triangle [ scalene orthogonal] is a harmonic one, it is similar to the Kepler[ Magirus] but not the same.

All sides of this triangle are in a geometric ratio and each side to the next is the ratio of the squre root of the Golden Ratio [ 1.27201965..]. One of the two non 90 deg angles is not 60 deg [ as current translations give] but arctan[ square root of the Golden Ratio] , equal to 51.82729238...

The specialty of this triangle is that : Product of the small horizontal side by Hypotenuse is equal to the square of the its bigger vertical side. This makes it a very special triangle[ quadrature triangle], actually a unique one.

Please refer to :

http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf

-------------------------------------

My attempting to consider, possible, natural structures, is the following concept:

In a static, but vibrating field [aether- electromagnetic medium], conductive

[ massive] elemental lines with alternating bipolar charges moving in it by the action of the field , should result into alternating currents running within them.

Two such lines could be contacted electrically at the ends of each line, via their + and - charges, and similarly

three lines [ in the correct lengths] could form triangles [ orthogonal according to my theory], and in such forming a surface. Similarly by joining two pairs of such triangular forms [elecromagnetically attracted by the currents running within them] could create materialistic volumes[ tetrahedra].Continuing, by these similar actions of electromagnetic forces, the joining of these materialistic volumes[tetrahedra] could result into further building blocks of matter.

According to my geometric theory [ pure classical geometry, based on the Square Root of the Golden Section ] such materialistic volumes [ tetrahedra ] build a Great Pyramid Model via which the structure of the world of the 5 Platonic [ or Eucleidian] solids are formed.

Ref:

http://www.stefanides.gr/html/Root_Geometries.htm

http://www.youtube.com/watch?v=c39K0iHRtgg&list=UUi4v2r_QyjvOZSOeD6PBuVA&index=1&feature=plcp

http://www.stefanides.gr/html/Root_Geometries.htm

http://www.stefanides.gr/html/All_triangles_derive.htm

I understand that all geometric figures are related together apart the circle which is related to them by the constant π.

As this is an approximate value, by thus, geometry is turned into physics, where constants are used, in the form of approximation. 

Thus we cannot expect any form of unification under this form of π.

I have worked for lengthy periods in my past, searching this question.

I went to Plato [Timaeus], and studied the original statement regarding the two orthogonal triangles as basic for the structure of solids.

These are the orthogonal isosceles and the orthogonal scalene.

I made my interpretation as to the form of the scalene one, which inscribed in a circle [after searching specially contrived conditions for relationships with it], get relationships with parallelogrammes and squares [ complying with Eucleidian theorems]

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf 

http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf .

Similarly this method proves [ in one page the insoluble of 3.141592654…..]

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_11.pdf .

I attach the simplest form of these relationships in the photo below.

Many links may be found giving the solution [ 4/ sqrt{ [sqrt( 5 ) +1 ]/2 }, approx.: 3.14460551…..]

My Links, however, below give the compass and ruler solution, coupled in parallel with AutoCad graphics for checking further.

http://www.stefanides.gr/Html/piquad.htm

https://www.researchgate.net/publication/257469474_Qrt7a_1

https://www.researchgate.net/publication/305346326_MY_PAPER_DIALOGO_2014

https://www.youtube.com/watch?v=XeOjPmKSsOI&feature=em-upload_owner

P.S. a similar triangle [ but not the same ] is the Magirus triangle given to Kepler and known as Kepler triangle :

Data Qrt7a 1

Data MY PAPER DIALOGO 2014

https://www.youtube.com/watch?v=XeOjPmKSsOI&feature=em-upload_owner

Published on 3 Mar 2015

The film was created by Ivanov Max and Proskurina Alina in Lyceum N 6, Voronezh, 2015. Students and teachers of Maths,Biology, IT discuss the qualities of real scientists and researchers.

Category

People & Blogs

Licence

Standard YouTube Licence

http://www.stefanides.gr/Html/QuadCirc.html

Thank you, dear Panagiotis, for your interesting question and materials! I read  and saw them with a great interest! Thank you from our students!Your materials were interesting and useful for them and our colleagues!

Dear Irina,

I thank You, your Students and Colleagues,very much, appreciating your kind message.

From Kifissia,

Panagiotis

Please  Ref. further :

https://www.researchgate.net/publication/316667593_Treatise_on_Circle-Generator_Polyhedron_Harmony_and_Disharmony_Condition_of_Three_Concentric_Circles_in_Common_Ratio

Book Treatise on Circle-Generator Polyhedron Harmony and Disharmo...

Similarly added files: 

https://www.researchgate.net/publication/315801180_GENERATOR_POLYHEDRON_OF_PLATONIC-EUCLEIDEAN_SOLIDS_By_Panagiotis_Stefanides_1A

https://www.researchgate.net/publication/315779314_A_TOOL_TO_DEMONSTRATE_RELATIONSHIPS_BETWEEN_CIRCLE_SQUARE_TRIANGLE_AND_PARALLELOGRAMME_By_Panagiotis_Stefanides

Data GENERATOR POLYHEDRON OF PLATONIC-EUCLEIDEAN SOLIDS By Panagi...

Data A TOOL TO DEMONSTRATE RELATIONSHIPS BETWEEN CIRCLE, SQUARE, ...