The angle approach - An angle analysis of the earth magnetic field

	The magnetic field of the earth Lattice structures of the earth magnetic field

Impressum	Copyright © Klaus Piontzik


German Version

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:

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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