The Circumferential Charts
Some Rules Followed in Drawing the Circumferential Charts
In the first division of charts, we find 41 circumferential charts for the 30 highs and lows I have selected for detailed analysis. More than one chart has been drawn for some of the 30 vortexes. Some of the charts have dashed lines, dotted lines, and/or solid lines to indicate various types of symmetry. The particular type of line, whether solid, dashed, or dotted has no special significance. Two or more types of lines on the same chart are used solely to highlight or separate two or more different types of phenomena. The type of line used may change from chart to chart. The charts were drawn over a considerable period of time, during which, Dan and I changed certain drafting techniques. These drafting changes have no effect on the quality of any given chart.
Each chart is labeled with the number of the high or low that is the central or nuclear vortex around which a symmetry pattern is drawn. Since this group of charts show circumferential patterns, we will be looking for patterns with rays that radiate around like the spokes of a bicycle wheel. We will be looking for symmetry in the spacing of the rays from the nuclear vortex. The spacing of the spokes on a wheel can be described by the angular separation between the spokes. The dashed, dotted, or solid lines can be considered as the spokes of a wheel and the angular separation of the spokes or rays is given by a whole number called the angular number, which is an exact multiple of 1.875° (see Table VI, page 101). Therefore, the angular separation between any two rays can be described as any number from 1 through 192. The value of this angular number can be read in angular degrees from Table VI.
I will now describe a little detail of this set of charts that could be confusing. The end of a ray starting from a nuclear center does not always terminate at the exact center of the high, low, or col at which it points. Frequently it ends a little to the right or a little to the left of the true position of the chosen point. These lines were drawn with the aid of a transparency of Figure 10-7. The transparency was centered over the point to be used as a nucleus, and it was immediately obvious which other points were lying exactly on or very close to any one of the 192 numbered lines. If one of these lines fell slightly to the right or left of a chosen point, then the ray from the nuclear vortex was also drawn slightly to the right or left-exactly as the transparency indicated.
Two or more of the rays are drawn so that they pass through the center of the nuclear vortex. This creates a crossing point which serves as a registration mark for the exact location of the center of the nuclear vortex. The rays that are chosen to pass through the center have no special significance. All the rays were not passed through the center so that they would not obstruct the center under a mass of lines passing through a single point. The lines used to identify the rays were made thicker than the lines of the transparency, in order to make the lines on the printed charts easier to see.
I have also entered the great circle distances between the nuclear center and the ends of the rays in each of the first 9 circumferential charts. This radial distance is also given in the same type of numbers as shown in Table VI (remember, distance on the hemisphere can also be entered as degrees of latitude). The circumferential numbers are entered near the center of the nuclear vortex inside the angle formed by any two rays radiating out from the nuclear center. The radial numbers, on the other hand, are entered at the ends of the rays.
A little additional clarification is now added to reduce the chance for any misinterpretation of what is being done. The angular numbers near the center of the nucleus represent the angles (when converted to degrees) between any two adjacent straight lines (each of which represents a small circle on the globe). What we will see are symmetry patterns involving arcs or portions of small circles. Hereafter, I will use the abbreviation "cu" when referring to the angular number units that can be measured between any two rays when working with circumferential type patterns; i.e. 14 circumferential angular units will be called 14 cu (14 times 1.875°=26.25°}.
The radial distance, however, is given in angular numbers (which can be converted to degrees of latitude) that are a measure of the great circle distance between the central vortex and the point at the end of the ray. A line drawn to represent the great circle distance should be slightly curved on these polar stereographic maps, and could not be represented by the straight line shown as a ray (except when the line passes through the North or South pole). The distance along the straight line of a ray is also slightly larger in value than the great circle distance--therefore the straight line is intended to only identify the end points between which there is a certain great circle distance. Hereafter, I will use the abbreviation "ru" when referring to the units of the radial angular number that represents the true radial great circle distance.
In the charts that now follow, I start with points near the center of the map (near the North Pole). The succeeding points are located at lower latitudes. There is no special significance in this order, other than the fact that points near the map center are more completely surrounded by other identifiable points.
You may have been "unnerved" occasionally in the previous chapters by remarks such as: "as will be shown in the charts that follow" or "the charts at the end of the book"--well, congratulations--we will now really investigate "the charts at the end of the book".
First the obvious symmetry. On one side of the straight line joining #81 and #40, there are two angular numbers of 14 cu that are opposite each other and two angular numbers of 34 cu in the middle. Could you get this pattern by throwing darts at a dart board? Look again; #40 has a radial distance of 11.3 ru, #39 has 12.4 ru, while #81 has 24 ru, which gives a ratio close to 2:1, if we use the average value of #39 and #40. That's not the end of it; #55 has a radial distance of approximately 6.5 ru, while #60 has 12.7 ru-this is almost 2:1 again.
#60, with a radial distance of 12.7 ru, compares closely with #39 at 12.5 ru, while both are symmetrical around point #34 at angles of 11 cu! We can't expect everything to be perfectly neat, since we do have a dynamic system that is not in perfect balance or equilibrium.
It must be emphasized that this example was selected to show circumferential symmetry. A simultaneous radial symmetry of any kind is a revelation.
Now, a word about the exact point chosen as the center of #58. It has the shape of two separate ellipses joined together. Here we have a case of multiple centers with the dominating center located at or near the spot where the "H" is marked.
A complete analysis of #58 would require that at least two and probably three centers be chosen around which the various symmetry patternscould be exposed. This type of analysis would give the maximum number of symmetry relations for the conglomerate vortex identified simply by the number 58. The point I chose gave the best overall symmetry.
We will now recall the discussion of Figure 7-23, where the most regular symmetry patterns surrounding any given point will occur when we choose the preferred positions in any system where there is some type of regular spacing of the points. Remember, that any position chosen, even if not a preferred one, will show some type of symmetry, only if the points in the surrounding area have any type of regularity.
Whenever there was any uncertainty as to the location of the center for any given vortex in the charts, the point that was finally chosen, was the point that gave the maximum number of symmetry relationships with the surrounding points. This, of course, was not the only factor considered in locating a center, but nevertheless, it was the most important factor. Important enough to state flatly that: identifying the maximum number of symmetry patterns surrounding a low or high will register the exact center of a low or high where the lines cross.
I chose not to break up #58 into two different sections with different centers, since there are enough other examples to establish the nature of the symmetry pervading the weather map.
If you are not 100 % satisfied with the claim of great order in the arrangement of all points over a hemisphere (from this first chart}, please look at the next chart.
In the symmetry pattern for point #55, we find that 6 is the common denominator of the angular numbers of 12, 18, and 36. Ths circumferential pattern has four 18's, one 36, two 12's balanced by one 24, and two irregular angles (14 plus 22 equal 36) to complete the ring of 192 numbers. The solid lines were drawn to three points to show that they are related by the angular numbers of 48, 64, and 80. These translate into 90°, 120°, and 150° which are in the ratio of 3:4:5.
This chart was drawn to show a circumferential symmetry pattern of points around point #55, with no consideration of any kind as to the radial distance of each of these points from the center. But, careful scrutiny will show that point #58 is a great circle distance of 6.5 ru, #51 which is opposite is a distance of 12.5 ru, while point #76 is a distance of 13.3 ru; point #26 is a distance of 18.9 ru which is 3 times 6.5 approximately.
Point #46 is a radial distance of 11.2 ru from point #55, while point #86 is a radial distance of 21.6 ru (almost 2 to 1) from point #55. This is on a straight line joining three points. In addition, point #99 is 32.8 ru away, which is almost 3 to 1. Point #45 (8.1 ru) and point #77 (16.8 ru) are nearly 2 to 1. Point #6 (40.6 ru) is related to point #45 (8.1 ru) in the distance ratio of 5:1.
A symmetrical circumferential pattern is simultaneously locked into simple radial relationships. In legal terms, I rest my case that there is great order in the arrangement of all points of interest on a hemispheric weather map. However, if there are any doubters left at this point kindly take a look at the next chart. 120
This is another symmetry pattern showing the angular number 48 balanced by an angular number sum of 40+8. An angular number of 48 represents an angle of 90°. What we see is a pattern of rays forming two right angles next to each other, with a spray of two rays spaced at 8 angular units away from one of the legs of one of the right angles. This leg which is the ray to #86, is shown as a solid line.
The symmetries involve cu units of 16 and 17, with four 16's balanced on each side by two 17's each. There is a 33 left over, which is of course 16+17, and magically, the 11 that is needed to complete the sum of 192 cu of circumference, is one third of 33, for a ratio of 3:1. Very neat!
Just for the sake of curiosity let us analyze a few of the radial distances. #72 (17.6 ru) is nearly opposite #8 (35.1 ru) for a ratio of 2:1. #78 with 28.2 ru is almost opposite #12 with 28.3 ru, for a 1:1 ratio. #60 with 15.1 ru is almost opposite to #26, with 16 ru, and #18 with 15.5 ru, which are close to a 1:1 ratio. #49 with 20.8 ru and #76 with 21.1 ru are virtually equal to 3x7; while #78 with 28.2 ru and #12 with 28.3 ru are about 4x7; which leaves #8 with 35.1 ru at 5X7. A mere coincidence?
Chart #45 B
From the principle of superposition of waves, we know that we can have independent waves occurring simultaneously. Each wave can do its own thing, as if there were no other waves around. In this analysis of point #45 (shown on page 127), we have extracted another symmetry pattern coexisting around the same point. Here we find that a large portion of the atmosphere of the Northern Hemisphere is resonating at an angular number of 10 around point #45. Of the nineteen points that are linked in this angular mode, we find that the centers of nine of the points at the ends of the rays are not controversial as to location and are very good hits. These nine are #42, #37, #34, #55, #90, #88, #93, #78, and #21. If the other ten points were removed, we would still find an unusual, balanced symmetry pattern, with the separation between rays at 10 cu or a whole number multiple of 10 cu. Not willing to leave it at that, I continued the exploration with good results for the 10 extra points numbered: #5, #61, #67, #66, #63, #84, #80, #50, #46, and #25.
The ray that goes to #63 is not a perfect hit, but is slightly off to one side of the "H". This same ray passes right near #66, which is off by a similar amount, but in the opposite direction. What we have is an example where the ray for this angular number of 10 lies exactly between #63 and #66. This type of symmetry on both sides of a line is quite common and you can find it everywhere that you look on a weather map. This is an example of longitudinal glide reflection of the type shown in Figure 7--9. So it is not unreasonable to consider that this ray, which passes between #63 and #66, can be added as an additional good hit to be added to the original eight.
Points #46 and #25 are poorly defined, but their centers appear to be reasonably located on one of the rays for the angular numbers involving 10 cu. Points #5 and #80 are poor hits for the centers, but they have been entered as a matter of interest. #55 and #84 are very close hits, but not perfect. I must add that most meteorologists would consider such small distances (as are involved in the close but not perfect hits) as being not discernible in any type of measurement they usually make. Any slight variance, however, stands out sharply in the light of the rules of Singer's Lock.
You might ask: "How do cols fit into these symmetry patterns?" You will find that cols do indeed fit into the patterns, empirically, as can be
seen in the charts. Point #50 is the col between #49 and #51. Theoretically, there is no difficulty in understanding why. If the centers of highs and lows show an orderly arrangement in space, then the center point in between any two highs or lows must reflect a similar type of symmetry, just from mechanical considerations alone; in the same way as the center point on a bar joining the two bells of a dumbbell.
Now we come to the fascinating points of #61 and #67. They represent the terminal ends of the symmetry pattern with 10 cu as the fundamental unit. When we go (in an easterly direction) 10 cu, exactly, from #63, we end up a slight distance to the east of the center of #61. I have joined the rays for #61 and #63 with a symbolic coil or spring to illuminate the elastic nature of the termination. Likewise, when we go (in a westerly direction) 10 cu, exactly, from #34, we end up a slight distance to the west of the center of #67. In a similar manner, the rays for #34 and #67 are joined by a symbolic spring.
The symmetry of the way the pattern terminates is an interesting feature. There is an overshoot of two tiny remnant vortexes. The 192 cu of a complete circle should have left a remainder of 2 cu when 192 cu is divided by 10. Instead we have an overlapping at each end to eliminate the need for a remainder of two.
The radial distances are entered without comment at the end of each ray. You can investigate these relationships for your entertainment.
You will recall from the discussion of the Chladni plates in Chapter 9 that the particles of sand slide off moving parts of the plate and gather along nodal lines. In a similar manner, we find that the disturbance (of which #45 is the center) has set the atmosphere into a circumferential oscillation with a spacing of 10 cu between components of the configuration. This oscillation tends to force other vortexes to line up along the nodal positions separated by 10 cu, since vortexes that are not exactly on the nodal line tend to slide towards that line in the same manner as the sand particles on the Chladni plates.
Of course, #45 is not the only disturbance in the hemisphere. There are other centers that are growing or decaying at the same moment in time. The ones that are stronger and are growing will tend to force other vortexes to line up into resonant nodes. The weaker disturbances will tend to lose control of the vortexes in their respective nodes. Whether any given low or high is moving towards any special nodal line, or away from any special nodal line is determined by the resultant of the
forces. The predominantly stronger and growing disturbances will naturally have a stronger influence. In summation, any vortex or col that is near any given line of symmetry will tend to move towards or away from that line in accordance with the well established principles of wave interference. If the conflicting forces are in resonance, the nodal lines will strengthen. If the interference is nonresonant, then the nodal lines will be destroyed.
One other point of interest is to be noted: #34, #88, and #78 are all common to charts #45 A and B. This is one way in which different symmetry patterns are linked to each other through the nuclear #45.
These two symmetry patterns for #45 should prove decisively by themselves that there is an ordered arrangement of all the vortexes over the hemisphere. Just to be on the safe side, however, let's look at the next chart.