Polygonal terrains on Mars:
A contribution to their geometric and topological characterization
Pedro Pina * , José Saraiva, Lourenço Bandeira and Joana Antunes
Instituto Superior Técnico
CERENA – Centro de Recursos Naturais e Ambiente
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
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Abstract: Detailed examination of large extensions of polygonal terrains on the surface of
Mars and extraction of some characteristic geometric and topological parameters is made
possible by the application of image analysis methods to scenes of the Martian surface
acquired from orbit. This is illustrated by the analysis of a set of diverse Martian networks,
clearly visible in MOC/MGS images with high spatial resolution. It is shown that these
networks present, in average, a hexagonal habit, and that they verify two classic laws relative
to 2D random networks (those of Lewis and Aboav-Weaire).
Keywords: Mars; Polygonal terrains; Geometry; Topology
* Corresponding author
The occurrence of small-scale polygonal terrains on the Martian surface has been known since
Viking days. It raises the possibility that freeze-thaw processes take place in some areas of the
planet, an idea based on the similarities between many of the Martian polygons and terrestrial
periglacial analogues. Still, the analysis of small-scale polygons on Mars had to wait until the
last decade, when high (spatial) resolution images of the surface became widely available;
relevant works since published were concerned with different aspects of these features, such
as areas of occurrence [Seibert and Kargel, 2001; Kuzmin and Zabalueva, 2003], correlation
with ground ice [Mangold et al., 2004], dimensions and classification based on morphological
characteristics [Mangold, 2005], evolution along the seasonal cycle [van Gasselt et al., 2005],
and modelling according to the effects of thermal contraction [Mellon, 1997]. The elucidation
of the true nature and origin of the features was dealt with in most of these works, but no
definite answers were put forward. One major drawback in most of these studies is that they
seem to encompass relatively small datasets; this puts into question the statistical significance
of any measurements made on the polygons. The use of methods from the field of image
analysis, however, can easily permit the automated identification of the polygon contours,
thus delineating a network whatever its dimensions; furthermore, it can lead to its
characterization, thanks to the swift and comprehensive measurement of a broad set of
properties of the polygons.
2. Image survey
In order to obtain examples of the widely diverse settings in which polygonal patterns are
present on the surface of Mars, we conducted a survey of images acquired between 1998 and
2006 by the MOC/MGS camera in its narrow-angle mode, subject to two conditions: the
latitude of the centre of the image (higher than 50º, North and South) and its spatial resolution
(better than 6 meters per pixel). This was decided according to previous works about the
distribution of these features and its relation with ice content in the soil. Thus, a total of 15855
images were visually inspected to assess the presence of polygonal terrains, a fact confirmed
in 1184 cases. However, only 188 of these showed adequate extensions of clearly discernible
polygonal networks; 35 of these images (15 from the northern and 20 from the southern
hemisphere), covering every type of visually distinct network, were then selected for this
3. Polygon identification
The 35 images selected for this study were individually processed by an automated procedure
[Pina et al., 2006] that segmented the image and produced a new binary image (with
polygonal cells in black and pixel-thick contours in white). This was confronted with the
correspondent ground-truth (as detected by visual inspection of the original images), and
subject to the adequate corrections. Thus, we ended with digital models of the real networks,
from which the collection of quantified parameters could proceed with ease.
4. Network analysis
For measurement purposes, the networks contain only the polygons that are fully included in
the images, and whose neighbors respect that condition too. Squared samples (with a side of
about 500 m) of the typical visual aspect of each of the selected networks are presented in
Figure 1. On one image (E19-00409) two distinct networks were found, and subject to
individual analysis. It must be stressed that, at this point, no genetic hypothesis is taken into
account: if a polygonal pattern was present in an image, it was processed and analyzed.
The extension of the studied networks is quite variable, covering areas (AN) from a few to
some thousand km 2 ; the number of polygons (N) that constitute them also presents a wide
range, from tens to thousands of cells. A density d (in polygons per km 2 ) was computed,
allowing for a first comparison between the diverse networks. These values are presented in
Table 1, along with identification of the image, location, and other parameters mentioned
Another parameter that was computed for each network was AL, the percentage of the global
area of a network that is occupied by the contour lines. This was plotted against the mean
major axis (average of the major axis of the polygons in each network); the result can be seen
in Figure 2. There is a marked decrease of AL with increase of Lmaj, but this soon levels out,
and AL remains approximately constant irrespective of increasing mean major axis.
5. Polygon parameters and characterization of networks
The measurements individually performed on each polygon of a network included some
geometric parameters such as area, length of major and minor axes, shape factor, compactness
index and orientation. The average values (for each network) of the most relevant of these
parameters can be seen on Table 1.
It can be appreciated that there seems to be no significant gap in the distribution of sizes (as
given by the mean major axis) of the polygons, except between 250 and 350 m; this
somewhat contradicts other reports that separate polygons into well-defined size classes, and
may be explained by a non-exhaustive search for all types of networks, or by the inclusion, in
our case, of networks that may not be of periglacial origin.
Topological parameters have been frequently used on many scientific areas to characterize
and model the patterns that arise from the partitioning of space, from ceramics and
metallurgical alloys [Aboav, 1970, 1980; Weaire, 1974], to biological tissues [Lewis, 1928,
1931; Pina and Fortes, 1996], foams and geologic materials [Weaire and Rivier, 1984] and
even solar granulations [Noever, 1994]. However, when it comes to the topological properties
of polygonal terrains there is only one study in which the number of sides of polygons in a
small number of Martian and terrestrial networks was counted [Yoshikawa, 2003].
Topological characterization relies, in fact, on the quantification of neighborhood relations of
the polygons. A link can be established between the number of neighbors of a polygon and its
dimensions (Lewis law), and the number of neighbors of its first-order neighbors (Aboav-
Weaire law). The experimental verifications of these laws yield parameters that may be used
for the characterization of the networks.
To collect this topological data we employed an improved algorithm based on mathematical
morphology, developed recently [Bandeira et al., 2008]. The average number of neighbors
per polygon () for the 35 networks was close to 6 (one exception is 4.89, a value obtained
for a network with only 9 polygons; in fact, this number becomes closer to 6 with increasing
number of polygons in a network). This was expected, since that is the value determined for
an indefinite trivalent network (with three edges at each vertex) [Weaire, 1974], and our
networks are mostly of that nature (though there is a number of tetravalent vertices in
particular regions of some networks). Furthermore, the minimum number of neighbors (imin)
detected was 3 for a vast majority of the networks (only one 2-sided polygon and three 4-
sided polygons were detected as minimum values, all in networks with less than 10 cells),
whereas the maximum number of neighbors (imax) ranges from 7 to 13. In accordance with
these low dispersions, the corresponding variances (μ2) are distributed along a narrow range,
between 1.20 and 3.46.
The Lewis law, derived from the study of biological tissues [Lewis, 1928, 1931], combines
dimensional and topological parameters and describes a linear relation between the average
area (i) of polygons with i neighbors and i:
< A> =< A> [1 + λ(
where is the average area of all polygons in a network and λ is a constant. This relation
reflects topological constraints on the polygonal structure imposed by the need to completely
fill the space; the slope of the line expressing this law is related to the topological aging or
coarsening in a polygonal network [Rivier, 1985].
Plotting the computed average area of polygons with i neighbors as a function of i (example
in Figure 3), we verified there was an approximately linear relation for all networks (the worst
least-square fitting yielded R 2 =0.7015, but 28 out of 35 networks have a value above 0.85).
Hence, the Lewis law is verified, and λ can be computed from equation (1) for the 35
networks (the values obtained are presented in Table 1). Its range is limited (from 0.26 to
0.51), and this parameter shows small promise for the distinction between networks. Note
however that the highest values of λ correspond to networks with worse linear fitting of the
Lewis law (R 2 lower than 0.85).
The Aboav-Weaire law [Aboav, 1970, 1980; Weaire, 1974], describes the correlation between
i, the number of neighbors of a polygon, and mi, the average number of neighbors of its
m = 6 − a+ (6 a+ μ )/ i
where a is a constant and μ2 is the variance of i. This is a structural equation of state for
polygonal networks, describing a statistical equilibrium and assuming that there is no
correlation in polygon neighborhood beyond the nearest neighbors [Weaire and Rivier, 1984].
Again, if the relation between i.mi and i is approximately linear we can compute a, and use it
as a parameter to globally characterize a network. After plotting the corresponding values for
the 35 networks, we confirmed a remarkable linear least-square fitting of our data to the
Aboav-Weaire law (the worst R 2 value is around 0.95, but 28 out of 35 are better than 0.99).
An example is presented in Figure 4.
6. Conclusions and future work
The need for an automated tool capable of collecting parameters that can lead to a quantified
and objective characterization of polygonal networks on Mars is clearly felt. This stems from
the presence of this type of features in many different areas of the surface of Mars, including
low latitudes [Page, 2007], and that occurrences at ever smaller scales are currently being
uncovered (see the HiRISE images of the Phoenix probe landing area) [Levy et al., 2008].
The use of image analysis in the acquisition of geometric and topological parameters from
Martian polygonal networks turns this task into a simple, fast and reproducible procedure,
allowing for the complete analysis of extensive areas, such as some of the examples shown in
this work; the results, while mostly confirming earlier studies, present an improved statistical
significance, since they are based on the analysis of large numbers of polygons. Some
topological results are reported for the first time for Martian polygonal terrains, and the
measured parameters confirm the general applicability of laws applying to 2D random
networks: the average global hexagonal pattern is confirmed for every network (the average
number of neighbors per polygon is close to 6), and both Lewis and Aboav-Weaire laws are
The next phase in this work will include multivariate analysis of the whole set of parameters
extracted, and its combination with ancillary data related to topography, ground ice content,
and regional geology. The aims are a full characterization of the networks, an exploration of
possible classification schemes, and taking steps in understanding origin and dynamics of the
diverse types of polygonal networks on Mars. To achieve this, we plan to analyze an enlarged
set of networks, to extend our research into smaller scales (using HiRISE images) and to
probe terrestrial analogues of this type of features.
Acknowledgements: This work was developed within project TERPOLI (PTDC/CTE-
SPA/65092/2006), funded by FCT, Portugal, which also financially supported JS
(SFRH/BD/37735/2007) and LB (SFRH/BD/40395/2007).
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List of figure captions
Figure 1. Samples from the polygonal networks studied, extracted from MOC/MGS images
(in some of the images the contrast was enhanced for visualization purposes) [image credits:
Figure 2. Plot of AL against Lmaj.
Figure 3. Verification of Lewis law (image M02-04505).
Figure 4. Verification of Aboav-Weaire law (image M02-04505).
List of tables
Table 1 – Geometrical and topological features measured on polygonal networks.