The Fibonacci and quasi-Golden spirals are the result of a succession of quarter-circular segments of different radii drawn over templates constituted, respectively, by a finite succession of rotating squares of sides conforming to the Fibonacci sequence, and of an infinite succession of rotating, superposing, and diminishing golden rectangles. 

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Context 1

... closer to  . As this is happening the outer rectangle of the template approaches a golden rectangle. A pretty close template to that in which the Fibonacci spiral of Figure 7(a) has been drawn can be constructed starting not with 13x8 rectangle but with an 8  x 8 golden rectangle or  12 . 95 x 8 , as shown in Figure 7(b). This figure is now divided into a square of side 8, and a golden rectangle of length 8, width 8   8  8 (   1 )  8   8 ( 0 . 618 )  4 . 95 , and ratio of length to width of 8 / 8   1 /    . This last golden rectangle can itself be divided into a square of side 8   4 . 95 and a golden rectangle of length 8   4 . 95 , width 8  2  3 . 05 , and ratio of length to width  . This figure is now divided into a square of side 8  2  3 . 05 and a golden rectangle of length 8  2  3 . 05 , width 8   8  2  8 (    2 )  8  3  1 . 89 , and ratio of length to width of  . The separation in this last golden rectangle of a square of side 8  3  1 . 89 produces a golden rectangle of length 8  3  1 . 89 , width 8 (  2   3 )  8  4  1 . 17 , and ratio of length to width of  . Obviously, the repetitive process just performed consisting in dividing each golden rectangle into a square and a smaller rectangle can be carried on indefinitely, this of course means that there is no way to draw the curve all the way to the pole. Even so, the pole can be located via the simple expedient of drawing “...the intersecting diagonals of the Golden section rectangles.” [7 ] The template coming out of the previous operations will be pretty close to that of the Fibonacci spiral. The difference being that instead of rectangles of dimensions 13x8, 8x5, 5x3, 3x2 and 2x1 we will have their respective golden counterparts instead, of dimensions 8  x 8  12 . 95 x 8 , 8x 8   8 x 4 . 95 , 8  x 8  2  4 . 95 x 3 . 05 , 8  2 x 8  3  3 . 05 x 1 . 89 , 8  3 x 8  4  1 . 89 x 1 . 17 , .... The quasi-Golden spiral produced by this template is remarkably similar in shape to the Fibonacci spiral. This notwithstanding, they differ in the fact that in the former every 90 o degrees or π/2 radians the radii change according to the following pattern ... 8  4 , 8  3 , 8  2 , 8  , 8  0 . From it we learn that the radii growth or multiplication factor, quantified by the ratio of consecutive radii, is constant and equal to the golden number, i.e., 8 / 8   8  / 8  2  8  2 / 8  3  8  3 / 8  4 ...  1 /    . This fact is the one that makes the quasi-golden spiral a closer approximation to the Golden spiral than the Fibonacci spiral. The closeness in shape of the quasi-Golden and Golden spirals can be appreciated in [8,9] As already noted, it was the appearance in Figure 3 of unit squares OJFEO and EFLHE (or, equivalently, of rectangle OJLHO), and of golden rectangle OCDEO, as well as of the infinite succession of squares and rectangles, what brought the ‘ spiral figures ’ subject to our minds. The fact that the construction of the template for the Fibonacci spiral – also known as th e ‘whirling squares diagram’ [9 ]- starts with two unit squares, while that of the quasi-Golden spiral starts with a golden rectangle, and both of them respectively followed with a succession of squares and rectangles serve to explain the connection made by us. At this point it was clear that if we were to draw a spiral we needed to build a template out of the figures available in Figure 3. By trial and error we managed to build two templates out of the infinity of squares. To the description of these two folding paths and the spirals coming out of them is what the next two subsections deal with. The first template coming out of the succession of squares EDABE, AKLGA, LNPQL,...was obtained through a succession of reflections and has been graphically represented in Figure 8. As there depicted, square UXYZU – there referred to as square 1- is reflected over the line which perpendicular to the common diagonal, pass through its lower left vertex U. The result of this operation is composite-square 2 which as seen there, is the unit formed with square UXYZU subsumed in PTUVP. The following reflection is that of composite-square 2 over the line which perpendicular to the common diagonal, pass through its lower left vertex P. The result of this operation is composite- square 3 which as seen there, is the unit formed by composite-square 2 subsumed in LNPQL. The reiterated performance of this operation ends up with composite-square 4 subsumed in EDABE. Even if the starting point of this procedure is arbitrary, the relative position of the squares in the final template is not but actually dictated by the reflection process. This template allowed us to draw the spiral shown in Figures 9 as well as its ‘compressed’ version shown in Figure 11. The following two particularities of these spirals need to be highlighted. 1) That the template on which they are drawn do not correspond with any of the two templates shown in Figure 7; 2) That the spiral of Figure 9 is not composed by a succession of quarter-circular segments, but by a succession of semi-circles circumscribed on the squares, and on reason of this the change in radii takes place not every 90 o or π/2 radians, but every 180 o or π radi ans; and 3) That the growth or multiplication factor displayed by it is not constant; while extremely large close to the pole, it settles to a value of  in its last two π rotations before coming to its end at E. This last point can be exemplified by remembering that the sides of the succession of squares vary as  ,  2 ,  3 ,  5 ,  8 ,  13 ,  21 ,  34 ,  55 ,  89 ... If we omit the ( 2 / 2 ) multiplying factor transforming the side of any of these squares into the radii of their corresponding semicircular segments, the previous sequence doubles as the sequence of radii. From them, and working from the inside of the spiral out i.e. from close to the pole to the spiral’s conclusion, we can get, via the quotient of successive radii, the sequence of growing or multiplying factors here operating, as follows  55 /  89 ,  34 /  55 ,  21 /  34 ,  13 /  21 ,  8 /  13 ,  5 /  8 ,  3 /  5 ,  2 /  3 ,  2 /  . Once simplified, the sequence of growing factors takes the following form: , , , , , , , , . From the outstandingly large multiplication factor operating at its initial stages, the spiral, eventually settle to a multiplication factor of  in its last two π rotations (from L to A, and from A to E). In other words the spiral settles for a Golden-spiral behavior at its conclusion. The segment of the spiral (L-A-E) with constant  per π growth factor has been graphed in Figure 9. The angle   81 . 29  operating in this section was calculated with the equation provided by Sharp [10]:   cot  1 [(ln M ) / 2  ] in which M represents the growth factor for a 2π rotation. Here M =  2 ; (ln 1 . 6180 2 ) / 2   0 . 1532 ;   cot  1 ( 0 . 1532 )  81 . 29  . In view of the previous data equation r  a e  cot  becomes r  0 . 20774 e 0 . 1532  . The factor a  0 . 20774 was obtained from a simple proportion between the scale of the plotter for a  1 and the actual 2  2 / 2 length of the radius of semicircle L-A. The graph of this equation appears in Figure 10. In it the horizontal axis intersections correspond, from smaller to larger radius, to points L, A, and E as they appear in Figure 9. The figure below depicts what we have called the ‘compressed’ or ‘cusped’ version of the spiral shown in Figure 9. Here the spiral is constituted by a succession of inscribed quarter-circles. The same variation of multiplication factors operating in spiral 9 is found ruling here with the difference that the change from one to the next takes place every π/2 r adians and if so its multiplication or growth factor per π/2 rotation varies – from inside out of the spiral- as follows ...  34 ,  21 ,  13 ,  8 ,  5 ,  3 ,  2 ,  ,  . The reason for this is the fact that while the combined lengths of sides BA and AK is one (    2  1 ) , that of ED, TP, and PQ is less than one (    3   5  1 ) . Even so the squares positions are such that allows the drawing of the quarter-circles spiral shown below. The radii of these quarter-circular segments, measured from their respective centers, corresp ond to the square’s sides and if so a change of radius takes place , just like in spiral 11, at π/2 intervals, and just like in that spiral the growth or multiplication factor varies – from inside out- as follows ...  34 ,  21 ,  13 ,  8 ,  5 ,  3 ,  2 ,  ,  . Just like Spirals 9 and 11, this also shows an extremely large growth at its beginnings close to its pole- only to settle to a Golden spiral with a growth or multiplication factor M of  per π/2 , or  4 per 2 π. The Golden-spiral corresponding to the L-A-E rotations is shown in Figure 14. A first revision of the pertinent literature produced no references to spirals of the characteristics displayed by those of Figures 9, 11, and 13. The sequence of squares forming part of the Fibonacci staircase can be immediately associated with both, a sequence of inscribed and a sequence of circumscribed circles. The former is shown below. From it, via elimination of the appropriate sections we obtained a figure remarkably close to what is known as the caduceus [10]. Apparently in this field you can find even that which you weren’t looking for. In a patent display of the multiple and interconnected layers of order permeating the field of knowledge that has the Golden mean, the Golden quadratic, the Fibonacci sequence and the Golden spirals as central concepts, we witnessed along the three papers of this series the Fibonacci sequence subsumed in the geometry of all the re-expressions of the Golden quadratic. The accomplishment of this, the original goal of our effort, came much to our surprise accompanied -in the analysis of the systems studied in the present paper- with us ...

Context 2

... 11. The cusped spiral shown above, drawn over composite-square 5 of Figure 7, is the ‘compressed’ version of that shown in Figure 9. Instead of a succession of circumscribed semicircles, this one is constituted by a succession of inscribed quarter- circles. The radial angle concept is not applicable to this spiral on reason of its cusps or sharp corners. 11b The second folding path The mechanics of this folding can be easily understood by looking at Figure 12. The end result of this process is an incomplete Golden rectangle.  ...