In the video "Phi Ratio Meets Vortex-Based Mathematics", Nassim Haramein can be seen arguing that the off-centre lo! cation of the emanation point of a phi spiral within an enclosing circle is connected to 9-fold 'resonances'. To illustrate this he divides a circle into 9ths and shows a W-shaped set of chords intersecting at a point that appears to be the emanation point of the phi spiral (illustrated above).
In fact it is straightforward to show that they're not related, and the two points are not the same at all.
1. True Phi Spiral
A phi spiral is a logarithmic spiral which increases in radius by a factor Ï•≈1.618 every quarter turn. The properties of logarithmic spirals can be derived with a bit of basic calculus. A full derivation would be a little too long to show in full here, and it turns out to be irrelevant to what was on Nassim's screen anyway, as he wasn't using a true phi spiral.
Briefly, t! hough... The first step is to calculate the radius of curvatur! e of the phi spiral, which is r√{1+(2 lnÏ• /Ï€)²} at a distance r from the emanation point. The second step is to work out the angle between a tangent of the spiral and the radius, which is arctan[(2 lnÏ• /Ï€)]. From these, with a little trigonometry, the distance of the phi spiral emanation point from the centre of its enclosing circle works out as √{1-1/[1+(2 lnÏ• /Ï€)²]} ≈ 0.29291 of the way along a radius from centre to circumference.
2. Intersection of Chords
[a diagram here of: an isosceles triangle with a vertex of 360/9 = 40º, and another with the same base but a lower vertex of 60º. The vertex of the first triangle is at the centre of the circle, for reasons of 9-fold symmetry; the vertex of the second is where the chords cross. What we're calculating is the distance between the two vertices. Wha! t can I use to draw this?]
The distance of the meeting point of the (W-shaped) chords from the centre can be calculated using basic trigonometry. The chords join points 1 & 5 and points 4 & 8 respectively of a regular 9-fold division of the circumference, as can be seen in the screenshot at the top of this page.
The result is {cos(Ï€/9) - √3 sin(Ï€/9)} ≈ 0.34730 of the way from centre to circumference. The 9-fold division of the circumference shown in Nassim's diagram, and which he specifies in the video, is clearly reflected in the formula.
This is nowhere near the emanation point of a phi spiral. The reason it appears close on the presentation is that the spiral used in the video is not a pure spiral at all, but an approximation.
3. Approximated Phi Spiral
Nassim's approximated phi spiral is m! ade from quarter-circles, each increasing in radius by a facto! r of Ï•. The two versions of the spiral are shown in green (pure spiral) and red (approximation using quarter-circles) on the diagram below. Where they overlap, it shows as yellow. They may look similar, but if you plot continuation circles around them they come out markedly different.
Image from wikipedia; creative commons license; modified from original.This approximated spiral is constructed from circles embedded in squares, so we can use some basic geometry to calculate the position of the emanation point of this spiral – it lies √{1 - 2/√5} ≈ 0.32492 of the way from centre to circumference. To calculate this, note that the symmetry of the construction implies that it must lie at the point of intersection of the two diagonal blue lines in the diagram above. The eqations of the lines are shown on the diagram, in cartesian form, relative to an origin as shown on the diagram. They can be solved as standard simultaneous linear equations.
This value is close enough to the position of the intersection of the chords to look almost indistinguishable in the presentation. A careful viewer, however, will note that in Nassim's video, the lines do in! deed cross further from the centre than the emanation point, a! s can be seen in the still below.
A still from the video of Nassim's 'discovery'. In this close-up of an area below the centre of the circle, Nassim's diagonal chords are seen to cross below the emanation point of the spiral, not on it. I've added blue cross-hairs at roughly where the emanation point of this spiral would be.The last two numbers (0.34730 and 0.32492, the ones relevant to Nassim's presentation) can easily be checked by careful construction of the shapes on paper or on a graphics package, and by measurement of the appropriate distances.
Conclusion
There is no mathematical relationship between a phi spiral and a circle split into 9ths. The relationship claimed in Nassim's presentation works only if you don't look too closely, and even then only if the spiral is approximated using quarter-circles.
For him to consequently make claims about the inspiralling of particles into a black hole is preposterous for a number of reasons. Firstly, inspiralling matter doesn't follow any logarithmic spiral (let alone a phi spiral). As matter spirals inwards it is diving in at an ever! steeper angle, even without resistance to motion. A logarithmic spiral, in contrast, approaches the centre at a constant angle. The resulting spirals are markledly different (see images below). Secondly, in the model he presents, as it is not a true spiral, the inspiralling matter would in fact have to follow a circular path for 90º and then, suddenly and discontinuously, follow a different circular path for another 90º, which is even sillier. And thirdly, even if it did, the values are still unrelated aside from their happening to be within about 1/44 of a radius of each other.
In addition, the 'interference patterns' that appear to reveal themselves when the spiral is replicated (in part 2 of this video) are purely an artifact of the approximation that he's used to build the spiral. If he had used a pure logarithmic spiral instead of quarter-circle ! building blocks, no such patterns could occur. On this topic, as with the others, we see his willingness to jump to vague but profound-sounding conclusions rather than investigate what he is thinking.
I suggest that it is clear from the above that the method of his arguments – if indeed there is one beyond spotting something that looks a little bit like something else – is very deeply flawed, very easily shown to be false, and lacking in both a basic understanding of mathematics and the imagination required to properly investigate straightforward mathematical situations. For this reason (indeed plenty more may be given, but this is reason enough), any of his claims to be employing mathematical or scientific methods to explain the underlying nature of the cosmos or anything else of any complexity should be seen as extremely dubious.
Above: spiral path of matter falling into a black hole (Using dr/dt∝1/r³; but assuming ω²∝r³. Can anyone help me out with a more accurate formula?)Below: a phi spiral; clearly very different.
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