Chapter 13
The Radial Charts
Radial Spacing, General
This chapter of the charts is drawn to illustrate the symmetry of the radial spacing of highs, lows, and cols at 1230 G.M.T. on December 7, 1950.
What do you hear when a symphony orchestra plays? You may hear several different tunes or melodies simultaneously; occasionally you may make out an individual instrument, but most of the time you can not make out the individual notes of the individual players. You can subject the entire orchestral music to what is called a harmonic analysis (first developed by Fourier) to separate the various instruments and sounds, or you can simply pluck each player out individually and listen to him or her play the individual melody. Fourier analysis requires some complicated mathematics-whereas listening to the individual player is simple enough for a child to understand. It was necessary to use complicated Fourier analysis to solve some of the principles of Singer's Lock, but once the principles were established, the complicated mathematics became redundant. Each of the following charts represents an individual player or note of the symphony of weather incidents occurring simultaneously over the Northern Hemisphere.
The lines that are drawn on the charts are used only to identify the points we are dealing with-nothing else-therefore they are drawn a little thicker and with less care than the lines on the circumferential series of charts. These lines were not used to calculate the distances. The latitude and longitude of each usable high, low, and col was carefully determined and then entered in Table IX, page 110. This information was used to calculate the great circle distances between the points I used in the charts.
These calculated values, expressed in angular numbers, are entered on the map, next to each point where the measurment applies. The other numbers which are circled are used to identify the points being analyzed.
In this set of charts, I am emphasizing the radial spacing, but I will throw in some significant circumferential spacing similar to the first set of charts.
It is desireable to keep in mind that these maps (which are only a small sample of the total that can be drawn) are not merely a group of detailed examples of previously unknown facts. These maps represent the proof, for the first time, that there are simple whole number relationships in both a radial and circumferential direction that join all significant features on a weather map, similar to the Chladni Plates shown in Figure 9-4, page 93.It may be disappointing to some, that the mathematics of this first part of Singer's Lock is as simple as that. The truth is usually simple.
Chart #20A
The strong feature on this map is that #27 is 17.3 ru from #20, while #67 is 52.1 ru (3x 17.3 = 51.9) from #20-all three points (#20, #27, and #67) are roughly on a straight line. Continuing, we find that #58 is 34.7ru (2x17.3=34.6), and #87 is 52.1 ru (3x17.3=51.9)-they are also on a different straight line with #20-in addition, there is a slight offset from the straight line for #100 at 69.2 ru (4x17.3). Lastly, #23 at 8.6 ru, appears to be the lowest common denominator (referred to hereafter as LCD) for this group of points since 2 x 8.6 = 17.2, which is quite close to 17.3. There is also a symmetry in the angles between the rays of the pattern that seems visually obvious, but we will not go into those additional details for this map.
The strong feature on this map is the lining up along an almost straight
line of the centers of #43, #37, and #31; all have an LCD of about 12.1
ru, in the ratio of 2:3:4. We find that #80 also has 48 ru, with the added
curiosity that the angular measurement between the two rays terminating
at #80 and #31 has a value of 35 cu (almost 3 x 12). In addition, the great
circle distance between #80 and #31 i s 47.4 ru, which makes an almost
perfect equilateral triangle formed by #20, #31.and #80.
There are three different clusters of points shown on this single map.
Does point #3 exist? It is 8.8 ru from #20. #19 is 17.7 ru (2 x 8.8
= 17.6). #39 is 35.4 ru (4 x 8.8 = 35.2). #69 is at a distance of 43.8
ru (5 x 8.8 = 44). Even if #3 was eliminated, we find that there is a common
denominator of 8.8 ru for the other three points of this set. The final
intriguing fact is that #3 is a great circle distance of 35.1 ru (4 x 8.8
= 35.2) from #39. A dashed line is drawn to indicate this relationship
between #3 and #39. 
#20 still has a few more tricks up its sleeve. #78 (28.4 ru) and
#77 (28.5 ru) make one side of an "arrowhead", while #56 (28.4
ru) and #45 (28.7 ru) make up the other side of the arrowhead-- while #96at
57.1 ru (2 x 28.5 = 57) is the central shaft of the "arrow".
The visual symmetry is simple. The piece de resistance in this whole
affair is #21 at 5.7 ru, which apparently is the LCD for this group, since
5 x 5.7 = 28.5 exactly. Continuing, we find that #6, . which is almost
in a straight line with #20 and #21, is 23.3 ru (4 x 5.7=22.8). 
Here we start with the obscure #25 at a distance of 14.6 ru. Does
this point really exist? Two times 14.6 equals 29.2. #55 at 29.1 ru and
#41 at 29 ru are close, with #45 at 28.7 ru, not bad either. Three times
14.6 equals 43.8. Guess what! #69 and #34 are both 43.8 exactly with #60
close (in distance) at 43.5. Note that #60 and #45, both nearly in a straight
line, are the two that are slightly below the value of the majority. Lastly,
we note that #91, which is almost in a straight line with #55,is almost
2:1 (2 x 29.1 = 58.2). Even if #25 did not exist, the symmetry pattern
would still be there. Since we can see that the symmetry pattern does indeed
exist, then likewise it is easy to conclude that point #25is real (and
accurate). 
In this chart, we find that 9 ru is the LCD. The most significant
feature is #82 at 45 ru (5X9) and #31 at 36.5 ru (4X9= 36). The great circle
distance between #31 and #82 is shown by the dashed line and has a measure
of 45.1 ru. Thus we see that #20, #31, and #82 form an isoceles triangle
with the sides in the ratio of 5:5:4. Next, we find that #10 (an obscure
point) at 36.4 ru (4X9= 36), is also 63.5 ru (7X9 =63) from #82. Thus we
see that #20,#10, and #82 make another triangle with sides in the ratio
of 4:5:7. Next, we have #13 (an obscure point) at 27 ru (3X9). Lastly,
we have #80 at 45.3 ru and #49 at 17.9 ru, both of which are close multiples
of 9. 
This one is a simple gem. #58 is 23.2 ru and #37 is 23.1 ru from
#26. The angle formed by their rays is 23.4 cu. Similarly, the angle formed
by the rays of #6 and #37 is 45.7 cu (2 x 23.1 = 46). One last touch, the
great circle distance between #6 and #37 is 30.4 ru (4 x 7.7 = 30.8); and
since 23.1 ru equals 3 x 7.7 exactly, we have an isoceles triangle with
the sides in the ratio of 3:3:4. 