Vesica Piscis--Real Sacred Geometry

[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?)

[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]

Originally Posted by 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.

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.

[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.

[C1ay]

Originally Posted by Turtle

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.

[C1ay]

Attached is a construction of a pentagram that begins with the construction of the vesica piscis. Complete circles are drawn and numbered in the order they were drawn to show the construction even though the striking of arcs would be sufficient. The centers of the circles are also marked.

[Christopher]

It appears this type of geometry is the foundation for life itself. A fusion of the micro-universe on the cellular leval to the macro-universe that we relate to on a cognitive one. There is now existing fossil evidence that may explain how these two worlds came together.

[Turtle]

___Thanks guys; I will look into both new references you provide.
___Turning into quite a dynamic shape for so simple an origin. I haven't even started on the rotor aspects, as in the vesica piscis rotating inside a close figure. It will rotate in an equalateral triangle, but never touch the corners, but in a square of sides length = the length of one arc of the vesica piscis, it rotates as to touch every point on the square's inside perimeter. Further, the center of the vesica piscis traces out lemnescates, in the case of the square a lemnescate with 4 lobes.
___Now there's still the little matter of the volume & surface area of the revolved veasica piscis. Oh, ...we probably should count on finding some other oddities; surely we have not adduced them all.