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Introduction:
A
Cemetery Inside the Grounds of an Auto Plant
A
number of years ago, in the mid-1980s, General Motors Corporation built
the Detroit/Hamtramck Assembly plant near the intersections of major
Detroit freeways and major rail lines. Proximity to
transportation links made sense from a variety of viewpoints. To
acquire the land for the large new plant (eventually to cover 362
acres), a combination of deals were employed (eminent domain, purchase,
and so forth); some met with more favor than did others
(Wikipedia).
The
Detroit/Hamtramck Assembly
Plant, has extensive security surrounding it. Figure 1 shows a
secured entrance gate. Figure 2 shows the general location of the
plant, at the north end of Chene Street, in the contemporary context of
Google Earth.
Figure
1. Plant entrance and security. Photo courtesy of Chene
Street History Study archive.
Figure
2. Plant site at the north end of Chene Street and adjacent
to freeways and railroad tracks.
Take a closer look; the area to the north end of the plant
contains quite a bit of grass adjacent to the giant parking lot.
Figure 3 shows a
small patch of trees that appear more mature than the others on the
plant site. The trees appear walled into a rectangular area.
Figure 3. Rectangular patch
of mature trees behind a wall.
The patch of trees is, in fact, part of a cemetery that predated, by
almost a century, the Detroit/Hamtramck Assembly Plant. General
Motors was not able
to acquire that small patch of land because of zoning and easement
restrictions already in place in association with the cemetery.
Figure 4 shows a closer look at the cemetery.
Figure
4. Cemetery on the grounds of the Detroit/Hamtramck Assembly
Plant. Note
tombstones. Entrance gate is to the left of the white car.
Records
in the Chene Street History Study (CSHS) and elsewhere show that this
cemetery is named Beth
Olem and that it is a Jewish cemetery that is one of the oldest in
Michigan. It is open for only a few hours a year, in association
with selected Jewish holidays. To visit the grave of a loved one,
it is required to enter through GM security first (Figure 1) and
then through cemetery security which requires the gates of the walled
cemetery to be open. The walls are 8 feet tall. Naturally,
this high level of security makes it difficult for visitors to gain
access.
Comtemporary Visualization: Virtual Beth Olem Cemetery
Google
Earth or other contemporary visualization technology could make it
possible, however, to overcome the frustrating security
situation. Imagine a 3D model of the cemetery, complete with
geo-referenced images/models of tombstones. Click on a grave
marker and get taken to materials from the archive (insofar as privacy
concerns permit). Link from the tombstone to a blog of associated
materials. The process of
building a virtual Beth Olem is underway. When complete, it will
serve not only to overcome access and distance issues for loved ones to
visit 24/7, but it will also serve as a basic study in the systematic
use (by blog associations) of the CSHS archive, added to the present
'GEOMAT' (Geographic
Events Ordering: Maps, Archives, Timelines; Arlinghaus, Haug, and
Larimore) methodology.
The archives of the Chene Street History Study have many photos taken
from inside Beth Olem. The image in Figure 5 is one example that
shows clearly the proximity of the industrial complex to the otherwise
peaceful resting ground; the juxtaposition of the different worlds is
really quite startling.
Figure 5. Beth Olem cemetery.
Small white circles may be golf balls. Cemetery maintenance crews
collect golf balls from the grounds that executives apparently hit at
lunchtime into the cemetery from nearby parking lots. Photo
courtesy of Chene Street History Study archives.
The cemetery is no longer taking new 'residents.' In that regard,
it offers to researchers an advantage similar to the opportunity
offered to foreign language students who begin by studying Latin (or
another 'dead' language). There is no (or little) change--the
'syntax' and
'grammar' of the situation are frozen. These are true anchors for
process and a fine place to begin study, prior to moving out, in this
case to the more dynamic setting of the changing urban Chene Street
scene.
A First Step in Creating the Virtual Beth Olem: The Walls
The
cemetery is a compact entity that is easy to deal with
geometrically: it is a rectangle. The walls around it
delineate it clearly and make it quickly recognizable. In terms
of creating a virtual cemetery, the walls serve as a good starting
point. Once the walled boundary is created, then infill can
proceed with the walls as guides to reduce placement error.
Accuracy in placement of the walls is straightforward: it is easy
to read off the latitude and longitude from a smartphone camera used to
take a photo next to the wall. General placement is
straightforward from tracing the footprint in Google Earth. What
is a challenge with modeling the walls is getting the surface to look
correct so that the created visualization is realistic.
Surface Pattern
It
is a simple matter to capture a swatch of the pattern on the walls from
a photograph. However, it is not possible to use that swatch,
only, to create the full wall--at least not in a realistic
manner. In Figure 6a, a single swatch of an arbitrary pattern is
used to tile a broad area; the visual effect is not satisfactory.
One has a sense that the single tile might be employed to greater
advantage; blue and orange pattern do not align as the walls are tiled
(Figure 6b). The tiling of a plane using geometric shapes is
called a tessellation (see Wikipedia reference).
Figure 6a:
single pattern tile, based on a background from MicroSoft PowerPoint.
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Figure 6b.
Pattern of 8a used to tile walls.
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To improve alignment and consequent appearance and visual impression,
one might flip the single tile or rotate it to create different
patterns and then align it with the base tile of Figure 6a to create a
larger single tile to tile the walls with. Figure 7 shows a flip
about a vertical axis in animated fashion. Figure 8a shows the
flipped tile appended to the base tile; Figures 8b, and 8c illustrate
the resulting pattern when 8a is applied to the walls of a box.
Figure 7. Flip
of tile about a vertical axis.
In the case of the walls at Beth Olem, the situation of Figure 8c
prevails; it is possible to find a swatch from top to bottom.
Figure 9 shows the results of a model created in Trimble
SketchUp. The edges along the tops of the walls, as well as the
dots in the walls, align across the entire wall. Look at the
grass stains on the bottom to see where the vertical flip was
made. Figure 10 confronts the model with the reality of a
photograph. There are no grass stains on the outsides of the
walls. The reason there are grass stains on the modeled walls is
that images from the interior side of the walls were used as textures
on the outsides; use of actual images of the exterior required
excessive removal of tree limbs not present when the interior images
were used. Evidently,
there are varying degrees of wetness at
different times of the year. A similar strategy of using a view
from the inside, and then flipping it, was employed with the sign for
the cemetery, again so that it too might be disentangled from the tree
limbs.
One might further refine the detail of images; that action, however,
has nothing to do with establishing process.
Figure 9. Beth Olem walls, model.
Figure 10. Beth Olem Cemetery entrance.
Photo courtesy of Chene Street History Study archives.
In the situation above, a flipped tile was appended to one side of a
base tile. Naturally, the flipped tile might also be applied to
each of the other three sides to create other tiling patterns.
The one selected to be exhibited is one that works well for modeling
the Beth Olem walls. Thus, the first step in wall completion is
solved. But, to learn more from this 'anchor' case, consider
other possibilities.
The Klein 4 Group: Pattern Alignment Issues
The
case above employed a vertical flip of a rectangular (non-square) base
tile to create a new larger tile by appending the flipped tile to one
side of the base tile; it was an exercise in 'spatial mathematics'
(Arlinghaus and Kerski). What other transformations of the base
tile might be employed to create other larger tiles that improve tiling
alignment issues on a wall? Clearly, one might flip the base tile
about a horizontal axis. Further, one might rotate the base tile
through 180 degrees and still maintain tile orientation. Figure
11a-d illustrates these possibilities.
Figure
11a. Base tile
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Figure
11b. Vertical flip
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Figure
11c. Horizontal flip
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Figure
11d. 180 degree flip
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Because the base tile is non-square, rotation through 90 degrees will
not maintain tile orientation; the 'landscape' tile will rotate to a
'portrait' tile under such a transformation. Are there, however,
other rigid motions (see Wikipedia reference) of a non-square rectangle
that will yield new
pattern? Intuitively, the answer appears to be 'no'. It is
possible to prove that answer using a structure from a branch of
mathematics called group theory.
To introduce appropriate notation, replace the visual pattern in the
non-square rectangles with numerical pattern, labelling the vertices of
the rectangles as 1, 2, 3, 4. Thus the sequence in Figure 13a-d
is replaced by the sequence in Figure 12a-d.
Figure
12a. Base tile
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Figure
12b. Vertical flip
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Figure
12c. Horizontal flip
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Figure
12d. 180 degree flip
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To
illustrate how to use the numbers, represent the base tile as the
identity permutation on these four numbers: (1)(2)(3)(4).
Represent the vertical flip as: (12)(34), read perhaps as '1 goes
to 2' then once the end of a parenthetical notational phrase is
reached, the last element 'goes' to the first one, so here '2 goes to
1'. Similarly, represent the horizontal flip as:
(14)(23). Finally, represent the 180 rotational flip as
(13)(24). Does this set of permutations form a closed
system? If so, then there are no other possible rigid motions to
use to generate larger tiles from the base tile.
A 'group' is a mathematical structure that may exist on a set of
elements with one operation. When elements are combined using
that operation, the system is said to be a group if it is closed (no
new elements are generated), if it is associative (grouping using
parentheses is clear: a(bc)=(ab)c), if there is an identity
property: a*I = a, and if there is a unique inverse for each
element: a*a(-1) = I. In the case of the
permutations representing rigid motions of a non-square rectangle I =
(1)(2)(3)(4). The operation, *, involves combining permutations
as below.
(12)(34)*(13)(24)
is executed as starting on the left (for example) with 1--in the first
permutation, 1 goes to 2; in the second permutation, 2 goes to 4.
Thus, in the resulting product, 1 goes to 4 or (14. Now, where
does 4 go? Start in the first permutation--4 goes to 3 and in the
second permutation, 3 goes to 1. Thus, in the resulting product,
4 goes to 1 so it is now correct to close the parentheses (14).
Now go back to the first permutation to see where 2 goes. In the
first permutation, 2 goes to 1; in the second permutation, 1 goes to
3. So, in the result, 2 goes to 3: (23 . Then go back
to the first permutation where 3 goes to 4 and then in the second
permutation 4 goes to 2. Thus, 3 goes to 2 and it is correct to
close the parentheses: (23).
Thus, (12)(34)*(13)(24) = (14)(23). With a bit of practice, one
can perform this operation quickly. Look at a table composed of
all possible permutation 'multiplications' (Figure 13). The
column on the left is the set of 'first' permutations; the row across
the top is the set of 'second' permutations
*
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(1)(2)(3)(4)
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(12)(34)
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(14)(23)
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(13)(24)
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(1)(2)(3)(4)
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(1)(2)(3)(4) |
(12)(34) |
(14)(23) |
(13)(24) |
(12)(34)
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(12)(34) |
(1)(2)(3)(4) |
(13)(24) |
(14)(23) |
(14)(23)
|
(14)(23) |
(13)(24) |
(1)(2)(3)(4) |
(12)(34) |
(13)(24)
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(13)(24) |
(14)(23) |
(12)(34) |
(1)(2)(3)(4) |
Figure
13. Group table, Klein 4 Group.
Verify
that no new elements were created: all are displayed in the
table. Verify that the associative law holds: for example,
(12)(34)*[(13)(24)*(14)(23)] is the same as
[(12)(34)*(13)(24)]*(14)(23). Show for each grouping; begin by
working from within sets of parenthetically enclosed
permutations. It is straightforward from the table that
(1)(2)(3)(4) is an identity element; it is also straightforward from
the table that there is no other identity element. Finally, read
the table to see that each element is its own inverse:
(12)(34)*(12)(34) = (1)(2)(3)(4), for example. Thus,
this set of four permutations, representing rigid motions of a
non-square rectangle, forms a group. It was discovered by Felix
Klein and is referred to as the Klein 4-Group (Vierergruppe) and
is often denoted V.
Thus, because the group structure is verified, there are no other
patterns of the sort above, based on a non-square rectangle, that can
be used to generate wall tiling. Of course, one can improve
pattern alignment by using a larger tile, such as the one in Figure 8a,
and flipping that to create an even larger base tile.
Figures 14a-c suggest one such strategy: apply a vertical flip to
the base tile, append that to the base tile (Figure 14a) then apply a
horizontal flip to the tile in 14a to create a larger tile in Figure
14b. This new tile, as shown in Figure 14c, will combine both the
good side-to-side alignment and top-to-bottom alignment of the vertical
and horizontal flips. The tile is new, the alignment pattern of
wall tiling is new and improved; however, there is no new motion
involved, as the Klein 4 Group shows.
Figure
14a. Vertical flip of base tile appended to base tile
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Figure
14b. Horizontal flip of 14a appended to 14a
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Figure
14c. Argyle style of pattern
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Here,
only a simple non-square rectangular tile was considered. One
might
carve out pieces of a rectangular tile and glue them on top or bottom
or sides and create oddly-shaped Escher-like fish that fit together
perfectly in two different directions and at different scales.
The process is similar and employs the same general style of
reasoning. One may use the chain of reasoning for non-Euclidean
as
well as Euclidean objects (see comments in the Escher Wikipedia
reference involving the Escher 'Circle Limit series). The
subjects of group theory and of
tiling are deep ones--group theory lies at the theoretical root not
only of simple tiling such as that shown here but also at that root of
any tiling (see for example, Wikipedia, 'Wallpaper group').
References for further reading are suggested at the
end of this document.
References Cited
- Archive,
Chene Street History Study, The University of Michigan, Ann Arbor.
- Arlinghaus,
Sandra L. and Kerski, Joseph. 2013. Spatial
Mathematics: Theory and Practice through Mapping.
Boca
Raton: CRC Press.
- Arlinghaus,
S., Haug, R., and Larimore, A. Nov. 2011. GEOMAT Guide: Directions for
Building a GEOMAT Web Architecture for any Investigation or Case Study.
http//:www.geomats.org/
- Krzyzowski,
Marian. Director, Chene Street History Study; Director, Institute
for Research on Labor, Employment, and the Economy, The University of
Michigan, Ann Arbor.
Software
Used
Adobe
PhotoShop
Google
Earth
MicroSoft
Office, PowerPoint, Word.
Trimble
SketchUp
References
for further reading
Birkhoff,
G. and Mac Lane, S. 1961. A
Survey of Modern Algebra. New York: MacMillan.
Coxeter, H. S. M., 1961, Introduction
to Geometry. New York: MacMillan.
Coxeter, H. S. M., Emmer, M., Penrose, R., and Teuber, M. L.
eds. 1986. M. C. Escher:
Art and Science. Amsterdam: North-Holland.
Grünbaum,
Branko and Shephard, G. C. 1987. Tilings and Patterns. New
York: Freeman.
Herstein,
I. N. 1975. Topics in
Algebra, New York: Wiley.
Loeb, Arthur. 1976. Space
Structures: Their Harmony and Counterpoint. Reading,
MA: Addison-Wesley.
Schattschneider, Doris
(June/July 2010). "The
Mathematical Side of M. C. Escher"
(PDF). Notices of the
American Mathematical Society
(USA) 57
(6): 706–18.
Weyl, Hermann. 1952. Symmetry.
Princeton: Princeton University Press.
Wikipedia:
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