Map of Walter Christaller, Poland 1941
Part I: Benchmarking the Map
Sandra Lach Arlinghaus
Adjunct Professor of Mathematical
Geography and Population-Environment
School of Natural Resources and Environment, The University of
Michigan, Ann Arbor
Please set screen to highest
resolution and use a high speed internet connection.
Please download the most recent free version of Google Earth®. Make sure the "Terrain"
box in Google Earth® is checked.
Bringing an Historical Map Across the
One complex system
from the early
20th century in the history of
geography is the development, by Walter Christaller, of a theory of
settlement locations: central place theory. The
communication of his ideas is in the printed format of the times.
There are black and white maps of complex systems of pattern; they tell
one story. What might they look
like, however, when recast using contemporary capability? How
this capability expand the research frontier? The sequence of images
text below will examine a single map of Christaller, from a
1941 document, and bring it into the virtual reality of Google
Earth®. When the reader
downloads the files above, the power of the Internet is harnessed to
permit him/her to replicate the results of the article while reading it
and to experiment with related ideas at the same time. Such
capability is an important aspect of scientific communication.
Settlements in Eastern Europe:
Figure 1 shows
Christaller's 1941 map of a proposed settlement pattern
in Eastern Europe (in the western and central parts of what is today,
Poland). Cities, towns, or villages are marked with circles of
varying size where the size of the circle represents the number of
inhabitants proposed to make up the population. The largest
circles represent cities of 450,000 inhabitants, the next
largest 100,000, the third largest 30,000 and so on according to the
legend. The regional boundaries of varying line weight are drawn
also to include a fixed number of inhabitants: the largest region
is to include 2.7 million inhabitants, the next largest 210,000
inhabitants, and so on according to the
legend. The map from 1941 is a remarkable cartographic
effort: layer upon layer is meticulously drawn and labelled by
hand. One might speculate in various ways about adjacency
patterns, spacing patterns, or others on the existing map.
such patterns from maps is often aided by having the full picture on
the map: terrain, physical features, three dimensional effects,
and so forth. The map that Christaller drew is already
complicated; introducing physical or other features would
clutter this two dimensional map and destroy its legibility.
1. Christaller's map of proposed settlement patterns in Eastern
Europe, 1941. Both city/town and regional values are determined
by prescribed number of inhabitants to occupy the city or region.
[See reference at end].
Figure 2 shows the map
from Figure 1 brought directly into Google
Earth® where one sees
immediately the possibility of visualizing the map in relation to the
terrain. When the opaque
map is placed on the surface, it is
difficult to align the map with the globe. Activating the
"populated places" checkbox in Google Earth® brings up a set of
to use as established positions to see if the alignment of the paper
with the software is reasonable. For additional context in the
virtual environment of Google Earth® current subnational
boundary files are introduced [see reference to Valery35 and Barmigan
for link]. The paper map is made semi-transparent to see
simultaneously both the original map and
the globe under it. The paper map is manipulated in various
ways, suggested in the animation sequence, to improve the
alignment. Despite considerable maneuvering, the paper map does
not line up very well with the Google Earth® image. Bydgoszcz on the globe
should line up with Bromberg on the map; Torun on the globe
should line up with Thorn on the map; Lodz on the globe should line up
with Litzmannstadt on the map; Poznan on the globe should line up with
Posen on the map; Wroclaw on the globe should line up with Breslau on
the map; and so forth. The needed alignment is not present
and cannot be made to work simply by importing the map and adjusting
its position in relation to known positions. The reader wishing
to try may do so using basemaps contained in the second downloaded file
from the top of this article.
Aligning the Paper Map on
the Virtual Earth
set of cities already present in Google Earth®
was used in Figure 2 as a set of known positions against
which to test imported map position. There are two sets of
from the intersection of these two sets, all Christaller cities and
towns in the three largest categories. Find their corresponding
positions on the Google Earth®
globe. All of those Christaller cities and towns do appear in the
set although one may need to do a bit of research on place names to
translate the 1941 place names to the corresponding 2006 place
are carefully positioned reference points from which to infer,
interpolate, other positions. The set of locations just
identified in Google Earth®,
as the virtual locations corresponding to the top three point
categories in the Christaller hierarchy, will serve as a set of
benchmarks in the virtual world against which to test position in that
world. The image in Figure 3 shows these benchmarks portrayed as
rods planted on the globe with rod height corresponding to Christaller
in the virtual world--cities and towns in Google Earth®
in a map from the physical world--cities and towns depicted on
largest Christaller point locations are represented by the blue rods
next largest Christaller point locations are represented by the red
third largest Christaller point locations are represented by the gold
rods emphasize benchmark position. They are translucent so one
can see the terrain through them. Structures such as this are
easy to create in either Google SketchUp®
(free software) that can then be imported to Google Earth®
(free software) [see Appendix to
Part I of article by Arlinghaus and Batty in this journal]. Or,
they can be created directly in Google Earth Pro®
3. Benchmarks. The blue rods represent locations for cities
in the top Christaller category; the red one in the second; and, the
gold ones in the third.
of the benchmarks for map alignment
maps in Figure 3 show the position of a subset of Christaller
points as benchmarks for extracting the rest of the
the map. The remaining images in this section suggest ways to use
these benchmarks to improve the fit of the map with the surface of
the virtual Earth. Figure 4 illustrates the location of the flat
respect to the benchmarks: clearly, the benchmarks in the virtual
world cannot be made to line up with the existing map. One
way to improve the fit may be to disassemble the Christaller map into
smaller regions, fit thesmaller regions to the benchmarks, and then
regions assigned to benchmarks produce a better fit of benchmarks to
the map. Such an assignment strategy also spreads the
across the map, away from the benchmarks. Thus, while there are
particular standards for accuracy associated with this sort of mapping
in the virtual world, the same ideas apply as when mapping the physical
world. Figure out where the error is and tell the reader about
possible, develop a quantitative measure to ensure replicable
communication (often, when using control points to digitize a map in
Geographic Information Systems software, one finds a Root Mean Square
0.004 as a default setting).
Disassembly: Use of the Christaller 2.7 million regions
Christaller hierarchy associated with place size was used to create a
set of mapping benchmarks. It is natural, then, to use the
regions in the Christaller hierarchy as the regions in which to
disassemble the map. The largest regions in the Christaller map
are those designed for 2.7 million inhabitants. Will these
regions be small enough? Figure 5 shows the results of using the
three largest 2.7 million regions: only the full regions within
the map (with Danzig, Litzmannstadt, and Posen as largest
cities). The fit of benchmarks in
the virtual world to this set of smaller maps is better than it is
using the entire
map. Nonetheless, there is still much room for improvement.
The blue rods necessarily fit, as the foci of the 2.7 million regions,
but many of the red rods and gold rods clearly miss the
mark. The reader wishing to experiment with alignment may do so,
as well. These files are contained in the files at the top of
5. Alignment of Christaller map with underlying Earth
image. The blue rods necessarily fit, as the foci of the 2.7
million regions, but many of the red and gold
rods clearly miss the mark.
Disassembly: Use of the Christaller 210,000 regions
transparency in Google Earth®
is helpful in seeing, simultaneously, both the map and
what is under the map. Another approach that is also
useful, especially when looking at detail, is first to remove the
interiors from the map. This procedure is simple to
execute: save the map pieces in .gif format and assign
transparency to white colors. Figure
6 shows an animated sequence of Christaller 2.7 million full regions
(Danzig, Litzmannstadt, and Posen)
disassembled into the smaller Christaller 210,000 regions and
saved as transparent .gifs. (One advance-reader noted the
peculiarity that Danzig, as a highest order central place, is not
in the center of its apparently "complete" region.)
6.a. Danzig--2.7 million region disassembled into smaller 210,000
million region disassembled into smaller 210,000 regions.
million region disassembled into smaller 210,000 regions.
focal point of each of the 210,000 regions is assigned to the
corrresponding benchmark. One of these regions has a blue rod as
focal point, others have red rods as focal points, and yet others have
gold rods as focal points. There is no instance, in the case of
the full regions, of assigning
more than one rod to a 210,000 region; in addition, the entire set is
used. Thus, all blue rods, all red rods, and all gold rods (with
none omitted) necessarily fit these three reassembled
2.7 million regions. They are shown in Figure 7: the fit is true
on the rods with distortion and error increasing away from them.
7.a. Christaller's 2.7 million Danzig region formed from 210,000
regions assigned to benchmarks.
7.b. Christaller's 2.7 million region Litzmannstadt formed from
210,000 regions assigned to benchmarks.
7.c. Christaller's 2.7 milion region Posen formed from 210,000
regions assigned to benchmarks.
Reassembled Full Map
In addition to the
three full regions of Danzig, Posen, and Litzmannstadt, there are two
incomplete perimeter regions, one to the east and one to the west, as
well as regions
centered on Breslau, the Katowice region, Krakow, and Stettin.
They too were processed, as above, to force the blue, red, and most
rods to fit the Christaller map. Only in the Katowice region, near the
bottom of the map, was there any lack of fit: in that region a
red rod is the focal point and in addition there are a number of gold
rods also within the same boundary as the red rod. Because that
region is small in extent, the error in gold rod placement is also
small but increases with distance from the red rod. Finally, all
of these regions were
reassembled on the Google Earth globe. The result is shown in
Figure 8. All blue and red fit exactly. Most gold rods also
fit exactly (except those in the Katowice region). Error is
distributed across the map,
away from benchmarks. It is also evident at the edges of the
map. The fit of the map using smaller regions is superior to any
other considered. Again, the reader has all files available to
replicate results: to see the terrain in relation to the
Christaller map, to drive around through it, to study point location
patterns from various perspectives, to visualize the landscape, to turn
layers off and on, and to make history and associated policy issues
8. The reassembled map.
The Christaller map now fits the blue, red, and most gold rods of the
virtual world. Error increases away from these rods and at the
- A logical
step is to use the map of Figure 8, with the benchmarks, to interpolate
intervening Christaller locations (those lower in the hierarchy).
task is completed in Part II of this
- These 3D maps might offer insight into
studies of, or from, the past. Cosgrove notes that "Few
geographers outside Germany who took up spatial science were aware at
that time that this tradition of settlement landscape study was deeply
compromised, not only by its connections with German geopolitics but
through Christaller's work for Himmler. The geographer's theories
were used in planning the resettlement of the eastern Slavic lands
captured after 1939, directly connecting geographical landscape studies
and the Nazi project of spatial domination and population
engineering. The former Polish and Soviet territories were
divided by German geographers into authentically German zones, where
farmers from the Rhineland and other 'crowded' rural regions could be
relocated, and spaces under German conrol but occupied by lesser
(Slavic) races, were to be managed in the interests of the Reich. According to the plan,
the former zones were to be reshaped and redesigned through the
management of field patterns, farmstead architecture, and woodland
planting to resemble an ideal of 'German' landscape, while the latter
regions, cleansed of 'undesirables,' could be treated precisely as an
isotropic plain, a non-place whose landscape design was merely a matter
of managerial efficiency and productivity" [Cosgrove,
2004]. How might one use these maps with enhanced
consider statements such as these? That task is left to others.
- Work with the underlying geometry--outline
of various projects underway:
of geodesic uniqueness from globe (non-unique) to plane (unique).
- The problem of moving from sphere to plane
and back to sphere again is an interesting one that is reminiscent of
creating a globe from flat sections bent to suggest a sphere (globe
gores). What sort of symmetry is there, or is there not, in
taking a map (already formed from the imperfect transferral of a
sphere to a plane) and trying to stretch it in various ways to fit a
importance of the four color theorem (given
that regional adjacency is across non-trivial line segments) and
the proof (based on stereographic projection) that four colors are all
that is ever needed for map coloring on a globe
- Implications of the one-point
compactification theorem (demonstrating that stereographic projection
misses by one point of creating a one-to-one mapping of the sphere to
the plane) and a consideration of mapping in the non-Euclidean
world. For that work, a Non-Euclidean Atlas is underway.
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