I thought I'd start this thread for all our Phi related findings, of which I have been finding a few lately. As soon as I started using powers of Phi, I found many things.
What I was, and am looking for, is the connections between Phi/powers of Phi and doubling/halving/powers of 2. I believe that these two functions are recipricol in some way and are both exremely profound ratios. Phi, I believe, is the key to the torus shape and the fractal torus skin iterations/macro-micro expansion contraction...whereas the 64 Tetrahedron grid/Flower of Life/doubling-halving/powers of 2 are all about the sphere - a container, a matrix, a boundary....
I was particularly interested when I found this correlation between the flower of life/vesica/doubling-halving and CUBED Phi, because I am looking for 3D scaling. For instance, as Walter Russell points out also, to make a new "1" (sphere) you need 8 parts, as in, the seed of life in 3D is 8 spheres creating a new equalibriam, or 8 Tetrahedrons (Star Tetrahedron). (1,8,1,8,1,8) (1,8,64,512,4096,32768)
1 and 8 are recipricol in the sense of cubing and are of course mirror pairs (the 2 directions,+/-, on a single axis). I have a strong feeling that this is why, when the Fibonacci series is compresed to single digits, and the VBM circuits are found, there are 2 doubling/halving circuits (in opposite directions, one emanation/spirit circuit and an extra circuit of 1,8,1,8,1,8 that mirrors the emenation circuit. So it is similar to the emanation circuit in some way... I have a feeling it has something to do with a dimensional scaling axis (macro/micro).
@Barbitone, so essentially there are two sizes of circles, the large one, and the two smaller ones that are HALF the diameter, this is the DOUBLING/HALVING circuit and you can scale it in either direction by doubling or halving the circles.
The PHI (1.618...) is a circle or arc that it's center is at the very bottom of the larger circle, in AutoCAD it's called the QUADRANT, or the midpoint of the square. The radius of the PHI circle/arc, runs smack through the center of the smaller circles till it hits the outer edge of this circle, this makes the PHI RATIO (1.618...). In the first few pics I've trimed it to make it more compact, but in the last picture I've shown the full circles.
As for the PENTAGON, it is inscribed in the large circle and intersects perfectly where the LARGE CIRCLE-PHI CIRCLE-CORNER OF PENTAGON all converge.
Let me know if this makes sense, maybe I can explain some of it further as to what you are looking for.
@Barbitone, to draw this picture as above in the steps as I describe.
Draw a CIRCLE from CENTER(1) to its QUADRANT (2) with a radius of 1.0 or diameter of 2.0
Draw a smaller CIRCLE (2P two point method) from the QUADRANT(3) to the CENTER of the larger circle (4)
Draw a smaller CIRCLE (2P two point method) from the CENTER of the larger circle (5) to the QUADRANT of the larger circle (6)
Draw a LINE from the QUADRANT of the larger circle (7) PERPENDICULAR to the small circle (8)
Draw a CIRCLE from the ENDPOINT of the line (9) to the ENDPOINT of the line (10)
This CIRCLE has a radius of PHI = 1.618....
You can skip the line if you prefer and draw the second CIRCLE from the QUADRANT (11) to PERPENDICULAR (12) for the second PHI circle.
For the last step I use the TRIM command to clean it up in the last picture.
All of the BOLDED words are COMMANDS or SNAPS to draw this accurately using AutoCAD, for others reading this these are the steps I used to draw this illustration.
You can open the PROPERTIES dialog box and check the RADIUS property of the largest circles to see that it equals PHI = 1.618... it will show you depending on the precision of your units you have set. Otherwise it may round it on you.
FORMAT-->UNITS... change the precision to EIGHT decimal places if you want to see the most precision.
Hope this helps and wasn't overkill.
To make it myself, how do I do it? I make the two large circles with the Vesica, but then how do you find the centres of the two small circles?
@Barbitone, also to note that that shape isn't exactly a VESICA PISCES, if you drew two circles for a vesica pisces the quadrant (outer edge of the circle) is in the center of the other circle. This isn't the case for this picture, but it probably is close or related.
Wow, thanks Riseball! That's great. The hard part now is analysing the procedure to work out exactly how the relationship between the doubling and Phi is occuring etc.... You dont really need the grid of squares, you can do it pure sacred geometry style by finding the quadrants with vesicas... it's a very nice find!!
I found not only Phi but squared Phi and cubed Phi! :) This is how I did mine, and then the Phi powers highlighted... ;
I just realised I buggered up that last graphic where it says squared Phi Ive copied cubed phi over....doh! It should be 2.618033...of course.
I just want to direct your attention (especially Dean) to these graphics I hastly posted...... They are essentially the same as Riseballs graphics earlier on in the thread, but basically, I have just started with the same diameter circle for each but divided the circle by different ratios......this gives me an angle that eventually meets except that it is an infinite point, meaning I could continue to divide by whatever ratio into infinite but I have only done what I could be botherered to.....but you could go on and on into infinity towards the tip - but, the tips of different ratio cascadences are longer or shorter depending on the angle that is dertermined by the diameters of each circle in sequence giving me a ratio between ratio cascades so to speak. There are many interesting co-inciding points which are very simply observed..... I also did this with all the 9 digits divided into 1 (1/2, 1/3, 1/4......1/9) but they are very hard to show because they are on such a small level.....but each increment lines up with the next in a very interesting way.
In these graphics you can see that the infinite tip of 3ness, lines up with the top of the first iteration of halving, thus showing a connection between the doubling/halving and the 3,9,6.......sort of. And the infinite tip of the halving cascade lines up perfectly with the top of the circle at the second iteration of the Phi cascade ratios.....
In the second graphic you can see that; cubed Phis infinite tip lines up with the exact centre of the first Phi iteration, then PPhi squareds' infinite tip lines up with the top of the circle of the first iteration of Phi, then the square root of fives' infinite tip lines up with the centre of the circle at the second iteration of Phi circles, and then......halving! it's infinite tip lines up with the top of the second iteration of Phi circles as already pointed out.....but what is interesting is that it is a perfect progression of centres and edges and halving fits right in there, which is the odd one out....Phi cubed, Phi squared and Square root of 5 (main part of Phi calculation) are all expected to line up, not such a big deal, but halving is the doubling/havling circuit that is meant to be the essence of boundary definition etc, and as Dan Winter points out, is basically the exact opposite function of the Phi ratio. Phi seems to be all about continuam (infinite constant) whereas doubling/halving (powers of 2, 64 Tetra-matrix etc..) are all about boundary definition (infinite cyclic). These are the two major characteristics of the behaviour of infinity/fractals (the All is the One and the One is the All) - how one thing becomes an infinitude and vice-versa, in other words, CREATION.
Hey Rhuben, sorry I haven't responded to your pics, we seem to both be into the geometry which I am fascinated with. Your pics look a bit rough to me but the ideas behind them are right. Essentially what I have been seeing playing with the geometry for a while is that PHI and DOUBLING/HALVING circuits are definitely connect which I've shown in the pics previously, also what you are showing above with the HALVING (1/2) and THIRDS (1/3 are connected, and each to infinity 1/4, 1/5, 1/6... they are connected to infinity.
This brings up a great point regarding the definition for INFINITY? From studying the geometry it can be a PROGRESSION that will continue FOR EVER, and essentially strives to connect to where the TWO LINES CONVERGE, but really never will. So in essence the two lines create a BOUNDARY CONDITION for INFINITY. This makes me go back to the PARALLEL lines and seems to me must have a different meaning than infinity, one I believe we have no word in our language for which is why these concepts are hard to grasp.
As for the above pics, the PHI squared and cubed are apart of the same POWER PROGESSION, just shown independently. Which is probably why they connect to the original PHI progression.
I would check out JANUSZ KAPUSTA's book, it really is a gem and has so much to study. I have been looking at it for over a year now, and keep finding more for myself.
I will post some pics I have been playing with tonight, they are ALL PHI.