New Path details [for Non-Symbol Area V] located diagonally below and
to the left of Seed No. 6.




Posted February 23, 2021 

New Path details [for Non-Symbol Area U] located midway between
[Non-Symbol Area V] and Seed No. 10.



Posted February 23, 2021 

[Note: I cannot take full credit for this particular design.  The
individual leaf growth patterns and upper circular geometry depicted
here were obtained from http://gofiguremath.org/contact/
but the narrative is my own.  Placement of the six leaf sets and 
upper related geometry image were chosen by me.]



New Path details [for Symbol Area BBB] on the left side of the drawing
[to the left of Seed No. 6].  This depicts something referred to as
"The Golden Angle" which is another example found in nature of the
Golden Ratio or Phi.  Here we see [at the top] a circle representing
the relationship of "a" to "b" in the arc distances they make AND how
this ratio satisfies the the Golden Ratio equation. 



(a + b / a) = (a / b) = phi = 1.61803398...


  
As the stem of a sunflower seed begins to develop, each subsequent
leaf will form 137.5 degrees clockwise from its previously formed
leaf.  This "Golden Angle" pattern will continue throughout the life
of the plant.  Likewise, this angle mathematically influences the
whorled seed pattern structure in the head(s) of the flowers formed at
maturity.  Below the large circle, we see the first five examples of
this natural sequence beginning from one tiny leaf. The second leaf
forms 137.5 degrees from leaf 1.  Then we see the third leaf forms
137.5 degrees clockwise from leaf 2.  The fourth leaf 137.5 degrees
from leaf 3, etc.  As you can see, at the bottom, there are 7 patterns
not shown for leaf, 6, 7, 8, 9, 10, 11 and 12.  The Golden Angle would
always be found between each to define their sprouting location.
Finally, we see the 13th leaf location in the sixth pattern at the
bottom.



One life (in one seed) like our own, may seem singular in and of
itself.  Yet, when considered collectively, we see one seed bears many
in its life, and each of those, many more.  So, the "fabric" of our
own life is really a collection of many past and future lives all
within us.  To think mathematics is somehow separate from our own
evolution(s) I think would be an incorrect assumption.




Posted March 5, 2021 

New Path details [for Symbol Area VV] located on an imaginary diagonal
line midway between the previously drawn Phi Symbol [Symbol Area SS]
and a yet to be drawn Symbol below and to the right of this -- which
also will depict the Golden Ratio geometry.  This symbol, shown here
to the left, is a logic math symbol meaning "equivalent to" and is
therefore being drawn in this location to literally point to two
equivalent things.  I have mapped out these symbols [there are 6
remaining to be drawn] to further capture the sunflower's Natural, or
'Sacred' geometry as some refer to it.




Posted March 6, 2021 

Introduction:



Here we see details of how and why two [not one] whorled patterns
develop to determine seed locations within the head of a sunflower. 
To more clearly understand this, you will see a *counter-clockwise*
pattern in the UPPER whorl developing from a spiral #1, then #2, then
#3, #4 and then #5.  To more easily see the seed locations, you can
see dots in the next 4 rows after the #5 "seed row" -- leading to a
solid #10 spiral path designation.  Likewise, we continue with four
more dotted seed location paths. Continuing in similar fashion, we
come to solid path #15, then #20, then #25, then #30, Between each of
these solid paths, we find dotted seed paths. Finally, after 3 more
dotted seed paths, we come to the last spiral #34.



Moving to the lower spiral pattern, we see it at first appears to
be the same.  But it is not.  This series of spirals naturally runs
clockwise to the upper set.  It also will amazingly always have 21
spirals -- which happens to be the sum of the previous two Fibonacci
numbers, 8 and 13.



Since the last spiral in both the upper and lower heads is in the 34th
and 21st position respectively, and since EACH is a Fibonacci number
(next to one another in the consecutive series), we see where they
occur.



0 + 1 = 1

1 + 1 = 2

2 + 1 = 3

3 + 2 = 5

5 + 3 = 8

8 + 5 = 13

13 + 8 = 21

21 + 13 = 34



Finally. I thought it would be fitting to emphasise the all important
a / b = phi ratio occuring here.  Between the two whorled patterns
(which have been separated for clarity) I have written the equation 
34 / 21 = (approximately) phi or, in this case, 1.619047619047619...



These are New Path details [for Symbol Area AAA] located on the far
right (middle) of the drawing.  The two whorl patterns cover 9 inches
in height and 4.25 inches in width.  They are found to the right of
Seed No. 3.  As with many of these designs, neighboring designs have
been Photo-shopped out for clarity.





Posted March 14, 2021 

Path details for [Symbol Area Q] located near the center of the
drawing and just south-east of central Seed No. 13.  The "sacred
geometry" is beautifully described in this construction of the Golden
Ratio.  In studying this I came upon an example of one Fibonacci
spiral as its formation relates to something called the Golden
Triangle.  As discussed in the three accompanying images from
Wikipedia, it has the following attributes.  More could be said
about this, but for now I will simply say one thing (about circles).



It is interesting to note that the points of intersection (of THIS
particular spiral) depend of the perfect ratio of two equal side
lengths AND their respective base length (and vise versa in other
triangles whose base forms the side of the first discussed triangle.
Each progressively longer 'chord' (WITHIN A CIRCLE) by definition
works out to be 1.68 units longer than the previous chord.  Similarly,
a new chord within a larger circle dictates the form of another circle
on which those two point are tangential to its circumference.  These
individually drawn ARCS, each one less "tight" than its predecessor,
end up defining the path of the spiral.  So, without actually
mentioning this, there are about 6 circles drawn here, but only
partial circumference arc portions are shown.  Likewise, there are
about six golden triangles shown here, and each can be seen if you
look closely.




Posted March 18, 2021 

Path details for [Symbol Area S and R] located diagonally and midway
between Central Seed No. 13 and the lower-left corner Seed No. 10. 
These 14 numbers form the beginning of an infinite series of numbers
discovered by the Italian mathematician Leonardo Bonacci. In the year
1202 the term “Fibonacci” was popularized to describe this generated
set of numbers in his “Book of the Abacus”, which also popularized
Hindu-Arabic numerals and the decimal number system in Europe. There
is a great deal written about these numbers (also found and discussed
in my painting). For the sake of simplicity, I will describe how they
are created in layman's terms:



Beginning with zero, you add 1 to it (since it is both zero and the
preceding number).  The result is 1 since 0 + 1 = 1.  Following this
same rule of adding the previous number (which has now become 1) to
the last generated number (which is also now 1), we get 1 + 1 = 2. 
Continuing, we add 2 + 1 which is 3.  Continuing we add 3 + 2 which is
5.  Do you see the pattern now? 5 + 3 = 8 ... 8 + 5 = 13 ... 13 + 8 =
21 ... 21 + 13 = 34 ...  



The very interesting thing about this is that EACH successive
resulting pair of Fibonacci numbers (for instance 21 and 34) happen to
also be in the "Golden Ratio." 34/21 = 1.619047.  The higher you go
(in the infinite pairs of Fibonacci numbers) the resulting division
solution will approach phi or 1.61803398875... The three dots indicate
this is an "irrational" number since it never repeats and is
infinitely long. Pi or 3.14159... is also irrational.



The beautiful crisscrossing whorled pattern in both the leaf and seed
formations (in the sunflower head) use both 21 and 34 whorls in their
respective left and right directional whorl count.  This is yet again
another example in nature of the Sacred Geometry occuring.



My brother Lee actually wrote a program in the tiny-c language which
more definitively discusses the finer and finer accuracy of
determining the ratio values for phi based on the user's input of just
one integer.  To read about this, please see the following
discussion.





Posted June 7, 2021 

Path details for [Non-Symbol Area AA] located in the upper-right
corner of the drawing and at its edge.  An arc can be seen running
through it longitudinally.  This curve (as found in many places
throughout the drawing) will later define a color boundary explained
later. These arcs are part of eight concentric circles emanating from
a common origin (Seed No 13). Their radii were arbitrarily chosen. 
The arc segment in this particular shape has a radii 18.25 inches from
the center.  There is only one larger "color circle" in the set of
eight. It is 21 inches in length and defines a diameter of 42 inches for
the full design. The pencil line for this arc can be seen in the
upper-right.



Posted June 17, 2021 

Path details for [Non-Symbol Area BB] located in the upper-center
border area of the drawing. One particular "golden triangle" has been
included (with two 72 degree angles at its base [right-most line] and
one 36 degree angle at its vertex pointing to the left.  If either of
the two equal sides are divided by the base, the solution will be phi
or 1.6180339 ...



Posted June 19, 2021 

Path details for [Non-Symbol Area DD] located just to the left of
center in the upper border area of the drawing.  The 7th of 8 large
concentric [dotted] circle portions is found running through it. 
This, as mentioned earlier, will define a color/tint boundary found in
all areas other than the drawn areas themselves (ie. the
background between shapes).



Posted June 19, 2021 

Path details for [Non-Symbol Area EE] located at the upper left border
of the drawing.  A circle divided into quadrants is found superimposed
within two intersecting designs.  The "path" found in this and
virtually all designs (emanating from the seeds) will show in its
completed form on completion of the drawing.  The entire path is a bit
complex but will show up as a maze with no openings other than those
leading to other contiguous path segments.  It is a closed system of
many paths.



Posted June 29, 2021 

Path details for the small [Non-Symbol Area FF] located at the
upper-left part of the drawing immediately to the left of Seed No. 1.



Posted July 1, 2021 

Path details for the very small [Non-Symbol Area HH] located at the
upper-left part of the drawing and in the 5 o'clock position relative
to [Symbol Area GG].



Posted July 2, 2021 

Path details for [Non-Symbol Area II] located midway between central
[Seed No. 13] and the upper-left corner [Seed No. 7].  It is also
found exactly between [Non-Symbol Areas HH and QQ] (discussed next).



Posted July 5, 2021 

Path details for the just referenced [Non Symbol Area QQ] found midway
between central [Seed No. 13] and [Non-Symbol Area II].



Posted July 5, 2021