How does the map construction of Arno Peters work ?
1. The shape of the earth
2. The division of the earth's surface
3. A special feature of the circle of latitude
4. Just a short review of mathematics
5. The vision of Arno Peters
6. Maps made by projections
7. The grid mesh system
8. Insert coordinates on the map
9. The decimal map
Preface
In the documents that I have the map construction of Arno Peters is
described. But unfortunately, as often in special literature, very hard to understand for
someone not involved with the subject.
I tried to review some basics and describe the steps of the map construction in a way
that hopefully most people can understand.
The shape of the earth
With the realization of the fact that the earth is not a disk but must be of the shape of a sphere, a significant change in the view of the world took place.
If you cut through the center of a sphere, the cut's surface will always be a circle. A circle is a geometric figure which is easy to handle with the circle constant p = 4*arctan(1) = 3,14... .
But unfortunately the earth is not a sphere. If you cut the earth through the center, the cut's surface will be an ellipse. This is an annoying fact, because an ellipse is much more difficult to handle. Compared to a circle an ellipse has two radius. The major semiaxis a and the minor semiaxis b.
The earth is rotation relatively regular around the polaxis. During one day any point
on the earth's surface runs on a circular orbit around the polaxis. So if you cut the
earth along the equator through the center the cut's surface will be a circle. That means
that the shape of the earth is  even very slightly  a flattened rotating ellipsoid,
called spheroid.
In the past centuries almost unlimited so called "Reference Spheroids" have
been surveyed. The flattening f is expressed through the following formula:
f = (a  b ) / a
"e" expresses the value of the deviation of an ellipse from a circle and is called "numeric eccentricity". The smaller e the more circular is the ellipse.
Reference Spheroid 
major semiaxis a (m) 
minor semiaxis b (m) 
Flattening f 
1 / f 
e 
Airy 1830 
6377563,396 
6356256,909 
0,003340850641 
299,324964600 
0,0816733739 
Modified Airy 
6377340,189 
6356034,448 
0,003340850641 
299,324964600 
0,0816733739 
Australian National 
6378160,000 
6356774,719 
0,003352891869 
298,250000000 
0,0818201800 
Bessel 1841 (Namibia) 
6377483,865 
6356165,383 
0,003342773182 
299,152812800 
0,0816968312 
Bessel 1841 
6377397,155 
6356078,963 
0,003342773182 
299,152812800 
0,0816968312 
Clarke 1866 
6378206,400 
6356583,800 
0,003390075304 
294,978698200 
0,0822718542 
Clarke 1880 
6378249,145 
6356514,870 
0,003407561379 
293,465000000 
0,0824834000 
Everest (India 1830) 
6377276,345 
6356075,413 
0,003324449297 
300,801700000 
0,0814729810 
Everest (Sabah Sarawak) 
6377298,556 
6356097,550 
0,003324449297 
300,801700000 
0,0814729810 
Everest (India 1956) 
6377301,243 
6356100,228 
0,003324449297 
300,801700000 
0,0814729810 
Everest (Malaysia 1969) 
6377295,664 
6356094,668 
0,003324449297 
300,801700000 
0,0814729810 
Everest (Malay, & Sing) 
6377304,063 
6356103,039 
0,003324449297 
300,801700000 
0,0814729810 
Everest (Pakistan) 
6377309,613 
6356108,571 
0,003324449297 
300,801700000 
0,0814729810 
Modified Fischer 1960 
6378155,000 
6356773,320 
0,003352329869 
298,300000000 
0,0818133340 
Helmert 1906 
6378200,000 
6356818,170 
0,003352329869 
298,300000000 
0,0818133340 
Hough 1960 
6378270,000 
6356794,343 
0,003367003367 
297,000000000 
0,0819918900 
Indonesian 1974 
6378160,000 
6356774,504 
0,003352925595 
298,247000000 
0,0818205908 
Hayford (International) 1924 
6378388,000 
6356911,946 
0,003367003367 
297,000000000 
0,0819918900 
Krassovsky 1940 
6378245,000 
6356863,019 
0,003352329869 
298,300000000 
0,0818133340 
GRS 80 
6378137,000 
6356752,314 
0,003352810681 
298,257222101 
0,0818191910 
South American 1969 
6378160,000 
6356774,719 
0,003352891869 
298,250000000 
0,0818201800 
WGS 72 
6378135,000 
6356750,520 
0,003352779454 
298,260000000 
0,0818188107 
WGS 84 
6378137,000 
6356752,314 
0,003352810665 
298,257223563 
0,0818191908 
The International Civil Aviation Organization (ICAO) prescribed the Reference Spheroid WGS84 in 1998 for all geographical coordinates related to aviation.
All following calculations refer to the Reference Spheroid WGS84.
The division of the earth' surface
Considering the earth to be a sphere it made sense to divide the two
circles which are rectangular to each other into angles. So the equator has 360° and the
cut through the poles as well. But where to begin counting. Anyway, one has agreed
someday.
The Englishmen asserted themselves, they defined arbitrarily the meridian, that ran
through their observatory of Greenwich, as 0°. From here it is counted 180° to both
sides eastward and westward. The longitude 180° is also known as date line. Longitudes
are marked with the Greek letter of l
(lambda).
The referencelatitude 0° is pretended physically from the equator. From the equator
it is counted 90° to both sides northward and southward. The Greek letter for the
geographical latitude is j (phi).
Circles of latitude 

Circles of longitude 

A special feature of the circle of latitude
The geographical latitude of j (phi) declares about which angle the local horizon of a location on the earth's surface is inclined against the earth's horizontal axis.
The horizon touches the surface of the ellipse as tangent. The zenith
cuts the horizon perpendicular. This line does not cut through the center of the earth.
The angle between the horizontal axis of the earth and the line from the center of
the earth to the intersection of the zenith with the horizon is called geocentrical
latitude and is marked with the Greek letter y (psi). The angle
of the geocentrical latitude is always smaller than the corresponding geographical
latitude. The geocentrical latitude is expressed with the following formula:
The difference between the geocentrical and the geographical latitude decrease to the poles and to the equator an reaches the value 0. The following diagram illustrates this and shows the largest deviation at 45° geographical latitude.
WGS84 Spheroid 
Just a short review of mathematics
Circle: U = circumference, r = radius 
Ellipse: U = circumference, a =
major semiaxis, 
U = 2pr 
U is the limit value of the row with e < 1 
Equator = 40075,0166855785 km  Ellipse (circle) of longitude = 40007,8629172458 km 
Here only a short hint for the calculation of geometry bodies: There
are different names for the surfaces or parts of the surface of geometry bodies.
The area of a cylinder and the area of a cones normally include the area of the
top and/or bottom disk(s). The unwrapping area M does only consist of the unrolled area of
a cylinder or a cone.
The area of a spherezone does not include the areas of the top and
bottom disks.
The area of a spheroidzone does not include the area of the top and
bottom disks.
Cylinder: M = wrapping area, h = height, r = radius of circle 
M = 2prh 
Spherezone: M = wrapping area, h = height, r = radius of sphere 
M = 2prh 
Spheroidzone: Z = wrapping area, b = minor semiaxis, a = major semiaxis, y = geocentrical latitude 
According to my opinion, the vision of Arno Peters is based, on an
especially distinctive equalitythought and truthfeeling.
It is true; historical events have taken place at different geographical places at the
same time. Therefore, each historical event has the right on an individual place on an
uniform timescale. Arno Peters illustrated this in his " Synchronoptical
Worldhistory" impressively.
All countries, that have a territory, possess also a geographical area. Who wanted to
claim nowadays to pass a judgment on the valence of countries? All countries of the world
have the same right to be portrayed on a global map in accordance with fidelity of area.
Arno Peters has succeeded this representation with the construction of his global map.
Someday it was recognized that it is not possible to unwind the
surface of a sphere in order to be able to represent it twodimensional. Therefore a
procedure was developed to transfer the surface of a sphere on to a geometrical system
that can easily be unwind. The magicword of the cartography is called
"projection".
It is considered that the water on the earth's surface is transparent. Now a lamp
will be illuminated in the center of the earth. A transparent geometrical body is put over
the earth and all to do is, to outline the shadows that the landmasses cast. After
unwrapping the surface of the geometrical body the result is a map on a twodimensional
surface. Simply brilliant.
The question, that poses itself, is: Can a map be produced exclusively through a projection? The answer is: No! Arno Peters has developed a system, with which a map can be produced by purely mathematical calculations. Therefore, the type of mapproduction of Arno Peters is not called projection but "construction". In order to understand the Petersconstruction, one must oust the thoughts of any type of projection of the surface of the world and must turn to the mathematical construction completely.
In the following the system is described how the global map is produced using the Petersconstruction method.
Definition of terms 

Spheroidzone 
Zonestripe 
Zonesegment 
A part of the surface of an spheroid between the equator and any circle of latitude. 
A part of the surface of an spheroid between
two predefined circles of latitude. 
A specific part of the Zonestripe 
For demonstration the circles of latitude are drawn in steps of 10 ° and the circles of longitude are drawn in steps of 20°. 
A grid mesh is a rectangular area from which a grid is constructed.
Starting point for our global map is a horizontal line with a length
of the desired width of our global map. Let's consider our map should be 3,60 m (= 360 cm)
wide. This line represents the equator. Because there are twice a many circles of
longitude than circles of latitude a zonesegment is twice as wide than high. Every
zonesegment is l = 2° wide and j =
1° high.
The width of our basic grid mesh is 360 cm / 180 = 2 cm wide. Because the basic grid
mesh is a square the height is also 2 cm.
First we determine the geocentrical latitude of the geographical latitude j = 1° and the result for y = 0,993306966°. Then we put all values in to the
formula and determine the area of the spheroidzone = 4431047,0014894
km².
Now we divide this area by 180 and the result is the area of the zonesegment 4431047,001489400 / 180 = 24616,927786052 km².
To get a square we draw the square root of this area and we have the basic grid mesh
height Ö24616,927786052
km² = 156,897825944 km. This height is called baseline and will be used
to determine all heights of all following grid meshes.
On our global map this value equals the ratio 2 cm / 156,897825944 km = 0,012747149.
Because of the conversion to a square a distortion in form is the result.
Estuary of the Amazonas j = 0°N to 1°N, l = 51°W to 49°W 

Zonensegment 
Basic grid mesh 
All grid meshes within the grid have the same width (baseline).
All grid meshes between two circles of latitude have the same height.
To get the height of the next grid mesh we first have to determine
for j = 2° equals y =
1,986622004°. The area of the spheroidzone is 8860780,38317385 km². To get the area of
the zonestripe between j = 1° to 2° we have to subtract the
area of the preceding spheroidzone from the actual spheroidzone 8860780,38317385 km² 
4431047,0014894 km² = 4429733,38168445 km². The area of the zonesegment is
4429733,38168445 / 180 = 24609,629898247 km²: The height of this zonesegment equals the
ration between the area of the zonesegment and the baseline 24609,629898247 km² /
156,897825944 km = 156,851312312 km.
On our map this value equals 156,851312312 * 0,012747149 = 1,999407084 cm.
The height of a grid mesh equals the ratio between the area of the zonesegment and the baseline.
All following grid meshes are calculated with the same scheme.
Earth's surface 
Our global map 

j 
y 
Spheroidzone 
Zonestripe 
Zonesegment 
Height  Height  Cumulated Height 
(°)  (°)  Z (km²)  Zx  Zx1 (km²)  A = Zx / 180 (km²) 
h (km)  h (cm)  hcum (cm) 
1 
0,993307 
4431047,001489 
4431047,001489 
24616,927786 
156,897826 
2,000000 
2,000000 
... 

10 
9,934394 
44093962,579799 
4372034,074873 
24289,078194 
154,808252 
1,973364 
19,902277 
20 
19,876630 
86881830,441838 
4183281,499747 
23240,452776 
148,124760 
1,888168 
39,215034 
30 
29,833636 
127088269,973908 
3869280,153550 
21496,000853 
137,006365 
1,746441 
57,362637 
40 
39,810611 
163500855,294729 
3437827,233588 
19099,040187 
121,729158 
1,551700 
73,797843 
50 
49,810390 
195004372,156136 
2900248,357196 
16112,490873 
102,694163 
1,309058 
88,017289 
60 
59,833076 
220616960,330178 
2271530,040678 
12619,611337 
80,432034 
1,025279 
99,577802 
70 
69,875993 
239526470,178184 
1570231,881968 
8723,510455 
55,599945 
0,708741 
108,112809 
80 
79,933979 
251124238,080918 
818078,072684 
4544,878182 
28,967120 
0,369248 
113,347585 
90 
90,000000 
255032810,842458 
39192,005387 
217,733363 
1,387740 
0,017690 
115,111761 
This is just an extract of the complete table 

All 16200 calculated grid meshes (including the 180 basic grid
meshes) are now mirrored horizontally around the equator. so we get the grid for the
southern hemisphere.
Remark: The grid has to be mirrored because the valid range for the formula to
calculate the area of the spheroidzones goes from y =
0° to y = 90° only.
The total height of our global map is htot = 2 * hNorth =
2 * 115,11176061 cm = 230,223521219 cm. The ratio between width and height equals 360 cm /
230,223521219 cm = 1 : 1,563697741.
For the contemplation of the precision, two aspects are imperative. To the one the user of the map, who expects, that the surface of the earth is represented, as shown on the map. A pilot compares the surface of the earth below him with the map and determines his position. Depending on the use of a map the scale of the map will vary. Therefore, the surveyor who surveys a property will uses a map with a larger scale than the climateresearcher, who presents a report on the global warming of the seas.
On the other side are the cartographers, whose task is it to draw geographical data of the world on a twodimensional map. For the drawing of coordinates predefined lines are required.
Since the width of the Peters global map can be chosen arbitrarily,
all longitudes behave to each other linearly. The referenceline for the drawing of
longitudinalcoordinates is any circle of longitude. No problem.
The referenceline for the drawing of latitudinalcoordinates is the equator. The height
of the grid meshes decreases toward the poles. Latitudinalcoordinates do not
behave linearly within even one grid mesh.
In order to draw any latitudinalcoordinate, the ration of the area of the spheroidzone,
from which the basic grid mesh was developed, to the baseline must be known first.
Area of the spheroidzone / baseline = 4431047,0014894 km² / 156,897825944 km =
28241,608669977 km.
If for example the geographical latitude 30° needs to be drawn, the geocentrical latitude
is calculated first. With this value the area of the spheroidzone is determined. Then,
this area 127088269,973908 km² is divided by 28241,608669977 km divides = 4500,036504968
km.
This value corresponds to on our map 4500,036504968 km * 0,012747149 = 57,362635834 cm.
This result can also be achieved by adding the heights of all 30 grid meshes. However
problems will occur using this method when decimal numbers need to be drawn (e.g. 37,839°
N). For any 1/10 degree precision the number of grid meshes that need to be calculated
will increase by 10.
Simultaneously with the introduction of the mathematical principles of constructing his global map Arno Peters established a decimal grid. He relocated the reference longitude 0° to a position between the continents Russia and America arbitrarily. His new longitude 0° runs almost exclusively over water. From there he counts 100 circle of longitude to the right (east). The reference line for the circle of latitudes is the north pole. From there he counts 100 circles of latitude downwards to the south pole. For expressing any geographical coordinate the terms northern and southern latitude or easterly and westerly longitude is not required anymore.
On the Petersmap the Longitude 0 runs through the BehringSea. 
The conventional longitude 0° runs through the very easterly part of Russia. 
New York has the geographical coordinates » 41° northerly latitude 74° westerly longitude. 
If one intended to establish the decimal grid for the use on maps all geographical data would need to be recalculated worldwide. A hopeless venture.
The conventional 360° grid is based on a long historical development by the observation of the sun and the stars. This has influenced our calendar (365 days for one orbit around sun), as well as also our understanding of the time (24 hours for one rotation around the poleaxis). Should the decimal grid be established our calendar and the definition of time has to be adapted inevitably. In present days this seems to be impossible to me.
In his atlas Arno Peters used the conventional 360° grid. Only in the rear bookcover his global map with the decimal grid is found. Also on his global map "All countries of the world projected with fidelity of area" the conventional grid is used. Only on the maps edges, the division according to his decimal system can be found.
Please report all broken LINKS, thank you.
This is a private homepage with absolutely no commercial intentions.
Copyright © Jürgen Heyn 2001, All rights reserved
Date of last amendment: 01. Dezember 2011