Vesica Piscis--real Sacred Geometry

[Turtle]

___I mentioned this shape & attachment in another thread, but it deserves its own for a proper exploration. I will just start by attaching my exposition, & after you look it over maybe Google Vesica Piscis or check in a dictionary. :Alien:

 

[Qfwfq]

The story behind the Fish Bladder seems quite interesting. In the last paragraph at http://www.philomuse.com/kingfisher/lab/vp.htm I found:

 

"Mary herself can be equated with the goddess Aphrodite Marina"

 

It made me wonder if the Semitic 'Miryam' has anything to do with 'sea' in that group of languages. I looked it up and found it means 'princess'. Funny coincidence, I guess, only in Latin languages. Maria-Marina, they are both used as girl's names here.

[Turtle]

___Very interesting stuff! In my research I had not encountered the term 'mandorla'. I wonder how that may be related to a 'Mandala' as in sacred meditative image? Also, I never saw a pentagram laid out on the vesica piscis as in the drawing at this page:

http://www.philomuse.com/kingfisher/north/vpgeo.htm

I will have to check it out more closely.

___ Speaking of coincidence, I ran across the term vesica piscis in relation to one of the ancient circles laid out on Glastonbury & which is an astronomical observatory. I wish I could remember the book title...something like the "little Book of the Sun & Moon'? Anyway, they related the shape to astronomy.

___Well, I'm off to read up. Great stuff Q! :)

[Turtle]

___So far, the drawing with the petagram on the vesica piscis seems arbitrary, ie. I have not yet found a compass & straightedge construction that follows.

___I did negelect to point out in the original drawing that the vesica piscis also bounds 2 equalteral traingles, & inasmuch as they are the first regular polygon, this is another example of sacred (or significant) geometry in the shape.

___I have found however that a rectangle of sides 2 & 4 bounds the vesica piscis & the diagonal of this rectangle is the 2*SQR5 (sacred), & half that rectangle is a square with diagonal 2*SQR2 (sacred). I will show this in a construction when I have completed it.

___ :cup:

[[email protected]]

Got interested in your vesica piscis, thought it ran a bell, so I looked around and found a reference in John Michell's 'The New View over Atlantis'. They are talking about alchemy and sacred numbers. The number 1746 is mentioned as a 'number of fusion.' I'll quote, "... the profile of the Pyramid is placed in association with that basic igure of sacred geometry and architecture known as the 'vesica piscis' or vessel of the fish. ... It is not merely a figure of abstract geometry, for it occurs in many of the patterns of nature. It represents a state of perfect equilibrium between two equal forces, and in the symbolic vocabulary of the old geometers it was an image of the interpenetrating worlds of heaven and earth, spirit and matter, and other such complementary elements." He also says the shape also taked the form of a magnetic field, of which I have no knowledge.

Anyway, its a neat shape.

[Qfwfq]

I wouldn't bet my savings on there being a connection between Mandala and mandorla, though you never know... Probably coincidence, like Mary and sea/marine, which are very close in Italian and Latin; as a name, Marina may well originate as a diminutive of Maria and just happen to coincide with the adjective, I'm not sure, I would need to look it up. As for the origin of the name, it seems to be not so much Semitic but Egyptian: Myrhiam, meaning princess.

[Turtle]

[email protected] quoted Mitchell saying, "... It is not merely a figure of abstract geometry, ..."

___This comment is very much to the point, ie. this simple to construct shape 'encodes' some very complex mathematical values. As to magnetic fields, other than in general exhibiting curves, I don't know of any specific magnetic field the shape represents. If you find more on this I'd love to hear about it.

___So Q, are you saying you think 'mandora' is related to 'Myrhiam', or that you found since that 'Mary, Maria' is related to 'Myrhiam'? I love etymology hunts!

___I haven't done any more with the construction yet as far as finding a pentagram encoded in the vesica piscis. While it's possible, it seems unlikely as the principle angles at work in a pentagram are 72deg & 36deg, & the principle angles in the vesica piscis are 90deg, 60deg, 45deg, &30deg;inasmuch as it is impossible to trisect an angle with compass & straightedge, no bisections of the angles present generate the angles needed.

___Now if the pentagram were to derive, we would have the sacred geometry of phi, the Golden Ratio/Mean in the vesica piscis as well as SQR2, SQR3, & SQR5.

___Great comments folks; thanks. :cup:

[Turtle]

___Here is my construction relating SQR's of 2 & 5 to the vesica piscis. Please don't hesitate to point out any errors, although in general it is correct. :cup:

[Turtle]

___A short addendum. In post #4 I said the rectangle "bounds" the vesica piscis;the rectangle only "bounds" 2 sides & "surrounds" the other 2.

___I realized the error just now, when I saw that the beginning circle bounds a hexagon. Extend the sides of said hexagon & you have the 6-pointed star. I botched up my drawing so I have to redo it; it will of course have the proof.

___PS Feel free to post it before I (challenge?) :cup:

 

Any geometers out there? :cup:

[Turtle]

___The promised construction demonstrating the six-pointed star stemming from the vesica piscis. B)

 

[Turtle]

___By the word 'real' in the title I mean that this is secular scaredness, ie. the vesica piscis is unique all of its own properties. Any religous or similar symbolic attachments are secondary tack-ons.

___I note that the equalateral triangle, square, & hexagon encoded by the vesoca piscis are the only regular polygons which tile the plane; another unique quality.

___I also note this all derives from the simple compass & straightedge, & in the pursuit of multifunctionalism, if one leg of the compass is a straightedge, only one tool suffices to draw out the whole lot in the dirt. From the simplest beginning, all this complexity derives.

___I don't have much skill in the Calculus (well, less than that even) but i seem to recall that using integration you can find the volume & surface area of a plane curve rotated in space on an axis. This means lathe work to me & if you rotate the vesica piscis on the long axis you get a football shape & rotate it on the short axis you get a double saucer shape. My question is, how do the volumes & surface area of these two shapes compare? (No, really; I want to know & can't figure how to do the calculus. Little help?)

:) B) :)

[C1ay]

I noticed the attachment in post #1 conatins an error in frame 2. It states:

 

Circle A

Radius = 1

Circumference = pi

Area = pi

 

The only circle with a circumference of pi is one with a radius of one half. I realize it matters not to in the construction of Vesica Piscis, only that the rest of George Chapman's work may require a closer look before using it as the foundation of further constructions. :)

[C1ay]

___So far, the drawing with the petagram on the vesica piscis seems arbitrary, ie. I have not yet found a compass & straightedge construction that follows.

 

See the attached image for reference.

 

From the initial construction I drew vertical diameters for the 2 circles and a vertical construction line through the vesica piscis. Next I added construction lines through CF and DE. These created the points G and H. I then projected lines from each of the circle centers A and B through both G and H. All that remained was to draw the horizontal line.

[Turtle]

___Nice C1ay! Im' looking it over. What a dynamic figure eh?

___On the mistaken value in panel 1 of my first diagram, I have removed it; fortunately I did not invoke it in the construction that foloowed. I also removed the value in panel 1 for the area as it is not invoked in the construction either.

___I found an error of sign in panel 12 as well, & corrected it on the diagram; for those of you that have downloaded it, you may make the following correction or download the corrected image. In the 3rd line of text in panel 12, the original read "=1 1/3 *pi*sqr3":

it should read "=1 1/3*pi-sqr3"

___Thanks also for examing my work closely enough to find errors & demonstrate a new construction. Hypography seems quite unique for this time of wide exchange.

___Now I have to get my drafting instruments out & try that construction. B)

[Turtle]

___Ok...I am obsessed with uderstanding this pentagram logically. C1ay's construction is the similar construction to the one from the link in post#3 which prompted my inquiry.

___Here is my stumbling block; how to establish a geometric proof of the angles in the construction. Keeping in mind that the construction is on its own a proposition & that a proof by the theorems of geometry (adjacency, supplement, trigonometric table, etc.) is required to establish the truth of the construction's proposition. In my first construction at the beginning of this thread, the principle results are those in sections 8 & 9, wherin the angles bounded by the vesica piscis are proved 30-60-90.

___In my construction in post #8, my line JI is C1ay's line ed, & I proved it's length is 2*sqr5. What I can't seem to prove is the 'location' of points g & h, that is those points' angle relation to the horizontal axis in a similar manner to the 30-60-90 angles I just referenced. If this is not clear, please ask for further clarification. B)

[C1ay]

C1ay's construction is the similar construction to the one from the link in post#3 which prompted my inquiry.

 

I used that construction as a starting point because of the dashed line that indicated a construction line in that image. With a strict construction through the points G and H produced by that method I find the pentagram that is produced is not an equilateral figure. The included angle of the 3 upper points is nearly 37° while that of the 2 bottom points is just under 35°. This has piqued my interest though so I will try to see if I can generate a construction that will produce a true equilateral pentagram.

[Turtle]

Exactly! Using your diagram's notation for a moment, I tried to find angle e using tan 2/4; the angle equivalent however lies between 26 & 27 degrees and we need to establish 36 degrees, 72 degrees or one of their even divisors or a multiple of them.

___It now occurs to me that maybe we should/can prove the contradiction true, & by that prove the diagram is nothing more than artistic liscence. B)