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Lattice structures of the earth magnetic field
 
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4 - The angle approach

4.1 - Angle differences

However, temporary, a quantitative analysis can be reached also by other methods. The angle approach takes the co-ordinates of the extreme values (total intensity) and forms the differences first between the latitudes and then with the longitudes.
For simplicity only the extreme values 1 (chapter 2.4) have been illustrated for the following analysis.
Der Einfachheit halber sind im Folgenden nur die Extremwerte 1 aus zur Auswertung abgebildet worden.
 

forming the difference of the latitude coordinates for the extreme values

Name North-Maximum South-Maximum Anomaly Minimum
South-Maximum 118,8625      
Anomaly 2,65 121,5125    
Minimum 82,9375 35,925 85,5875  
Saddle point 50,3875 55,475 53,0375 32,55

 

forming the difference of the longitude coordinates for the extreme values

Name North-Maximum South-Maximum Anomaly Minimum
South-Maximum 118,925      
Anomaly 156,75 37,825    
Minimum 45,2875 164,2125 157,9625  
Saddle point 164,5625 45,6375 7, 8125 7, 8125

 

If one looks these angle differences more exactly, than can be shown that practically all appearing angles are multiples of certain basic angles.
Most angles lie close to values which are a multiple of 7.5 degrees. And one receives 7.5 degrees, if the full circle is disassembled in 48 sections. The number 48 again is a multiple of the number 3 as well as the number 4.
Already from it will consist evidently that here oscillation figures can appear with the number 3 or 4 or a combination of both numbers.

 

 

4.2 - Angle forming

To allow a better comparison, the angle spectra are constructed at first for the 3rd, 4th and 5th partition from 0 degrees up to 180 degrees.

4-8-16-32-64-128 corners
Multiple of 2,8125

2,8125 5,625 8,4375 11,25 14,0625 16,875 19,6875 22,5
25,3125 28,125 30,9375 33,75 36,5625 39,375 42,1875 45
47,8125 50,625 53,4375 56,25 59,0625 61,875 64,6875 67,5
70,3125 73,125 75,9375 78,25 81,5625 84.375 87,1875 90
92,8125 95,625 98,4375 101,25 104,0625 106,875 109,6875 112,5
115,3125 118,125 120,9375 123,75 126,5625 129,375 132,1875 135
137,8125 140,625 143,4375 146,25 149,0625 151,875 154,6875 157,5
160,3125 163,125 165,9375 168,75 171,5625 174,375 177,1875 180

 

3-6-12-24-48-96 corners
Multiple of 3,75

3,75 7,5 11,25 15 18,75 22,5 26,25 30
33,75 37,5 41,25 45 48,75 52,5 56,25 60
63,75 67,5 71,25 75 78,75 82,5 86,25 90
93,75 97,5 101,25 105 108,75 112,5 116,25 120
123,75 127,5 131,25 135 138,75 142,5 146,25 150
153,75 157,5 161,25 165 168,75 172,5 176,25 180

 

5-10-20-40-80-160 corners
Multiple of 2,25

2,25 4,5 6,75 9 11,25 13,5 15,75 18 20,25 22,5
24,75 27 29,25 31,5 33,75 36 38,25 40,5 42,75 45
47,25 49,5 51,75 54 56,25 58,5 60,75 63 65,25 67,5
69,75 72 74,25 76,5 78,75 81 83,25 85,5 87,75 90
92,25 94,5 96,75 99 101,25 103,5 105,75 108 110,25 112,5
114,75 117 119,25 121,5 123,75 126 128,25 130,5 132,75 135
137,25 139,5 141,75 144 146,25 148,5 150,75 153 155,25 157,5
159,75 162 164,25 166,5 168,75 171 173,25 175,5 177,75 180

 

 

4.3 - Evaluation of the differences

The ascertained differences between the magnetic extreme values are now compared with the angle spectra constructed on top.
Only the values from the 3rd, 4th or 5th partition, whose difference to the extreme value differences are smaller than ± 1 degree, are put down in a table.
 
An analysis occurs at first which regard to the latitude coordinates:

Difference evaluation of the latitude coordinates for the extreme values

Name Difference 3-corner 4-corner 5-corner
Nmax-Smax 118,8625 - 118,125 119,25
Nmax-Anomaly 2,65 - 2,8125 2,25
Nmax-Minimum 82,9375 82,5 - 83,25
Nmax-Saddle point 50,3875 - 50,625 -
Smax-Anomaly 121,5125 - 120,9375 121,5
Smax-Minimum 35,925 - 36,5625 36
Smax-Saddle point 55,475 56,25 56,25 56,25
Anomaly-Minimum 85,5875 86,25 - 85,5
Anomaly-Saddle point 53,0375 52,5 53,4375 -
Minimum-Saddle point 32,55 33,75 33,75 33,75

 

And afterwards an analysis occurs with regard to the longitude coordinates:

Difference evaluation of the longitude coordinates for the extreme values

Name Difference 3-corner 4-corner 5-corner
Nmax-Smax 118,925 - 118,125 119,25
Nmax-Anomaly 156,75 - - -
Nmax-Minimum 45,2875 45 45 45
Nmax-Saddle point 164,5625 165 - 164,25
Smax-Anomaly 37,825 37,5 - 38,25
Smax-Minimum 164,2125 165 - 164,25
Smax-Saddle point 45,6375 45 45 45
Anomaly-Minimum 157,9625 157,5 157,5 157,5
Anomaly-Saddle point 7, 8125 7,5 8,4375 -
Minimum-Saddle point 7, 8125 7,5 8,4375 -

 

How is to be seen from both tables, the 3rd 4th and 5th partition with enough exactness (smaller than ±1 degrees) appears. In the latitude, as well as in the longitude. However, the geometrical or stereometrical consequence from this is, that all Platonic solids can appear as oscillation figures. And just with thus becomes tetrahedron- against dodecahedron-discussion (what concerns fields) simply superfluously.
The consequence is likewise that the pure tetrahedron-, octahedron or dodecahedron and icosahedron models deliver only partial views of the complete oscillation field, and, hence, are not entire. To the treatment of the concerning geologic models see
chapter 13.

 

 

4.4 - The zero point

Most angles of the extreme values lie close to values which show a multiple of 7.5 degrees.
The question which rises here, if within this oscillation structure a zero point exists, from which the whole field becomes representable?
To answer this question it must be checked, how often 7.5 is included in the respective longitude data and which rest appears.

The longitude coordinates for the extreme values 1

Name Longitude Longitude devide by 7,5 Rest
North-Maximum - 96,125 degrees West -12 6,125
South-Maximum +144,95 degrees East 19 2,45
Great Anomaly +107,125 degrees East 14 2,125
Minimum - 50,8375 degrees West -6 5,8375
Saddle point +99,3125 degrees East 13 1,8125

 

If a zero point exists, than the rest shows the longitude of the zero point. It is the matter tf find factors (m) which deliver the identical rest for all points. It turns out the following result:

The coordinates for the extreme values 1

Name Longitude m Rest
North-Maximum - 96,125 degrees West -11 -13,625
South-Maximum +144,95 degrees East 21 -12,55
Great Anomaly +107,125 degrees East 16 -12,875
Minimum - 50,8375 degrees West -5 -13,3375
Saddle point +99,3125 degrees East 15 -13,1875

 

The coordinates for the extreme values 2

Name Longitude m Rest
North-Maximum - 96,04 degrees West -11 -13,54
South-Maximum +144 degrees East 21 -13,5
Great Anomaly +106,5 degrees East 16 -13,5
Minimum - 50,8145 degrees West -5 -13,3145
Saddle point +99 degrees East 15 -13,5

 

The average from all rest corners amounts to -13,29 degrees. Up to 2 values all rests lie close to -13,5 degrees. Hence -13,5 degrees can be accepted here as longitude position of the zero point.
 
In the consequence the longitude positions of the extreme values of the total intensity from the earth magnetic field are representable by the following formula:
 
 longitude position formula
 
m is an element of the integers (...-3,-2,-1,0,1,2,3...) and λ0 = -13,5 degrees of west.
 
From now on this equation is called longitude position formula for the magnetic extreme values.
 
With the help of satellite geodesy in 1966 by C.A.Lundquist and G. Veis determined the following parametres:

(see in addition „Geodetic parameters for a 1966 Smithsonian Institution Standard Earth" von Lundquist, C.A., Veis, G.
See also Torges „Geodäsie&" page 77.)
 
a1-a2 = 69 Meter
λo = -14° 45` (longitude West)
 

a1 is the great equator half-axis, a2 the small equator half-axis and λo the longitude of the great half-axis.
If one takes now the value of Lundquist and Veis and compares this with the zero point - seen on a global scale - a good correspondence is ascertained.

 
With thus the longitude positions of the extreme values from the earth magnetic field
stand in relation with the equator axes of a three-axle ellipsoid.
 
See also chapter 17.

Also in modern publications is still maintained that the magnetic field disposes of no symmetrical structures. As to be seen in this chapter angle approach, however, this is right only the first sight and does not withstand a more precise examination.

 

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