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Philosophical Magazine Letters (preprint)

On Lewis, Desch and Aboav-Weaire laws on polygonal networks on Mars

José Saraiva, Pedro Pina, Lourenço Bandeira, Joana Antunes

**CERENA**-Centro de Recursos Naturais e Ambiente, Instituto Superior Técnico

Av. Rovisco Pais

1049-001 Lisboa

Portugal

jose.saraiva@ist.utl.pt, ppina@ist.utl.pt, lpcbandeira@ist.utl.pt, jbantunes@gmail.com

Tel: +351-218417425

Fax: +351-218417389

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Philosophical Magazine Letters (preprint)

Abstract: The verification of the applicability of Lewis, Desch and Aboav-Weaire laws to

polygonal networks in terrains from the surface of Mars is reported in this letter. The networks

analysed cover a great variety of small-scale polygonal terrains formed in periglacial regions of

the planet. It is shown that these Martian terrains share some of the same geometrical and

topological properties of other random networks built by materials and processes of diversified

nature and origin.

Keywords: Geometry, Topology, Patterned ground, Mars.

1. Introduction

The existence of a correlation between geometry and topology on polygonal networks of

widely diverse nature has been verified and expressed in a number of laws generally known by

the names of their proponents, such as Lewis, Desch and Aboav-Weaire.

The first of these laws, which resulted from the seminal works of Lewis concerning

cucumber and amnion skins [1-2], states that the average area i of cells (or polygons) with i

neighbors increases linearly with i, a relation that is mathematically expressed as follows:

< A> =< A> [1 + λ(

i−

6)]

(1)

i

where is the average area of all polygons in a network and λ is a constant.

area:

An alternative is the substitution of the perimeter i of polygons with i neighbors for the

< P > =< P > [1 + λ(

i−

6)]

(2)

i

where is the average perimeter of all polygons in a network. This perimeter law is also

referred in the literature as the Desch law [3-4].

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Philosophical Magazine Letters (preprint)

In practice, these two laws mean that polygons that have many neighbors tend to be large,

while polygons with few neighbors are small.

Another law was derived from the works of Aboav [5-6] and the contributions of Weaire [7].

It describes the correlation between i, the number of neighbors of a polygon, and mi, the average

number of neighbors of its adjacent polygons. This is expressed by the equation:

m = 6 − a+ (6 a+ μ )/ i

(3)

i

2

where a is a constant and μ2 is the second moment of the distribution of i.

The Aboav-Weaire’s law means, in practice, that polygons with fewer neighbors tend to have

neighbors with more polygons in their neighborhood, and vice-versa.

These laws have been checked and verified by many authors analyzing a large variety of 2D

polygonal networks. Their works deal with biological tissues [8-12], liquid foams [13-17],

polycrystals [18-20], flame cells [21], solar granulations [22] and even galactic-scale bubbles

near a supernova [23].

In this letter we check on the application of these three laws to a varied set of polygonal

networks detected on images of the surface of the planet Mars and make a first evaluation of

their possible role in the characterization of the networks.

2. Polygonal networks on Mars

The polygonal terrains (also called patterned ground) we are interested in are

geomorphological features organized into interconnected networks; their presence is long known

in some parts of the Earth, especially in periglacial and arid regions. They have also been found

on other planetary surfaces, namely on Mars. In the decade of 1970, orbital images acquired by

the Viking probes led to the detection of large-scale geometric patterns on Utopia Planitia, in the

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Philosophical Magazine Letters (preprint)

northern plains of the planet. These presented dimensions in the kilometer range, and their origin

is now considered to be most likely related to tectonics [24]. Later, in the 1990s, the higher

magnifications provided by the Mars Orbiter Camera (MOC), on the Mars Global Surveyor

(MGS) probe, allowed for the detection of smaller features (polygons with dimensions below the

hundreds of meters). These are thought to be mainly linked to periglacial seasonal processes,

such as ice wedge contraction; this is based on the similarities they show to terrestrial polygons

of that origin. Works focused on the study of these networks were concerned with the areas of

occurrence [25], correlation with ground ice [26], dimensions and classification based on

morphological characteristics [27], evolution along the seasonal cycle [28], and modeling

according to the effects of thermal contraction [29]. Only one study [30] presented some

topological measurements (number of neighbors) on few polygons. Recent studies point to the

detection of this kind of features both far from polar regions [31] and (in great quantity) around

the Phoenix probe landing site [32]; however, none has provided a systematic analysis of the

polygonal characteristics that could lead to the verification of geometrical and topological

relations in the respective networks.

3. Experimental results

We conducted a survey of the MOC/MGS image catalogue to search for polygonal networks

[33] above 50º in latitude and with a spatial resolution better than 6 metres/pixel; a set of 33

images judged to be representative of the variability of Martian networks was selected and

analysed in this work. Some examples are shown in Fig. 1; note the variation on the dimension

of the polygons, the roughness of edges or the type of terrain. The corresponding binary

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Philosophical Magazine Letters (preprint)

networks were obtained through the application of an automated image analysis procedure [34].

A set of measurements made on these networks is presented in table I.

The area occupied by a network varies greatly (from less than one to about 20 km 2 ), as does

the number of polygons it includes (from around 20 to some thousand). The average size (major

axis) of the polygons ranges between 30 and 370 meters.

The distribution of the number of neighbors follows a well-established pattern, known from

previous studies: the mode is almost always six (there is one network with a mode of five), and

the fraction of cells with five neighbors is always larger than the one with seven neighbors (the

global behavior of distributions is shown in Fig. 2). The number of neighbors ranges between 2

and 13, but a vast majority of networks presents values between 3 and 11.

The average number of neighbors per polygon is also close to 6 in all networks (minimum is

5.58); this value increases when the number of polygons in a network grows (Fig. 3a). The

polygon density d (the number of polygons per unit of surface area) shows a clear decrease as a

function of L (the average major axis of the polygons), following a power law:

3b).

d k L −

2

1 .

k

= (Fig.

The experimental data obtained for each of the 33 networks was projected to test their fitting

to the three mentioned laws (Lewis, Desch and Aboav-Weaire). In each case, the complete range

of the number of neighbors determined was used: no outliers were filtered out. There are some

points that deserve to be discussed.

In what concerns the Lewis’ law, 28 of the networks show a very good fitting, with

2

R L >0.85, and the lowest value close to 0.75. The parameter λL obtained from this exercise varies

between 0.26 and 0.51, with an average of 0.35. These values show no relation with other

parameters such as quality of fitting, slope of the line, average area or areal range of polygons in

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Philosophical Magazine Letters (preprint)

a network. There is also no relation between quality of the fitting and dimensions of the

polygons. The behavior of the lines follows a common trend, with a few exceptions, in which the

line is steeper or flatter (Fig. 4a). In this instance, slope is strictly related to the mean area of

polygons: the larger this is, the steeper the respective line.

The linear fitting to Desch’s law is even better than for Lewis’ law - in this case, 27 of the

2

2

networks have R D >0.90, which confirms that higher values of D

R are generally achieved in this

case, as reported for other networks [16]. The extracted parameter λD varies between 0.13 and

0.35 with an average of 0.19 (see Table I and Fig. 4b). This can be related to the quality of the

experimental data: in some images the spatial resolution is not enough to give an accurate

representation of the edges of the polygons, and this can lead, for instance, to situations in which

two small and adjacent polygons are counted and analysed as a single, larger one. This can, in

turn, introduce a higher bias on the measurements of areas (Lewis’ law) than on those of

perimeters (Desch’s law).

There is a proportional relation between the parameters of the two laws; this can be seen in

Fig.5. The values for the Desch’s law are close to being half of those for Lewis’ law, more

precisely 0.54.

The Aboav-Weaire’s law is remarkably verified by the experimental data, with 27 networks

2

2

presenting R AW >0.99 and 31 networks with AW

R >0.98 (Fig. 4c). The parameter a varies between

0.46 and 1.47, with an average of 0.98. Again, it shows no relation with dimension of the

polygons or quality of fitting. Though there are some references to a tendency for the decrease of

a when μ2 grows larger [35, 36], we could not clearly identify this in our set of 33 networks (Fig.

6).

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Philosophical Magazine Letters (preprint)

4. Conclusions

The main conclusions are that the three laws can be applied to the polygonal terrains on the

surface of Mars, and that the topological measurements made fall within the reported range of

variation of parameters for other types of polygonal networks of diverse origin and size. The

processes involved in the formation of the Martian networks can vary from location to location,

along with the lithology of the terrains or the climatic conditions, and this can surely explain the

variation found in the parameters. Still, the verification of these laws points to a universal,

geometric truth: Nature is governed by the same rules everywhere.

There is still no quantitative classification scheme of these Martian networks; we expect that

a more detailed exploration of the parameters of these laws, together with an in-situ analysis of

analogues on Earth, can provide discriminant features that will lead to achieving that goal.

Acknowledgements

We thank FCT (Portugal) for financial support to the project PTDC/CTA/65092/2006, and

also to J. Saraiva (SFRH/BD/37735/2007) and L. Bandeira (SFRH/BD/40395/2007).

References

[1] F.T. Lewis, Anat. Rec. 38 (1928) p. 341.

[2] F.T. Lewis, Anat. Rec. 50 (1931) p. 235.

[3] C.H. Desch, J. Inst. Met. 22 (1919) p. 241.

[4] N. Rivier, Phil. Mag. B 52 (1985) p. 795.

[5] D. Aboav, Metallography 3 (1970) p. 383.

[6] D. Aboav, Metallography 13 (1980) p. 43.

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Philosophical Magazine Letters (preprint)

[7] D. Weaire, Metallography 7 (1974) p. 157.

[8] J.C.M. Mombach, M.A.Z. Vasconcellos and R.M.C. de Almeida, J. Phys. D 23 (1990) p. 600.

[9] J.C.M. Mombach, R.M.C. de Almeida and J.R. Iglesias, Phys. Rev. E 47 (1993) p. 3712.

[10] P. Pina and M.A. Fortes, J Phys. D 29 (1996) p. 2507.

[11] B. Jeune and D. Barabé, Ann. Botany 82 (1998) p. 577.

[12] M.E. Rosa, Phil. Mag. Lett. 88 (2008) p. 637.

[13] D. Weaire and N. Rivier, Contemp. Phys. 25 (1984) p. 59.

[14] J.A. Glazier, S.P. Gross and J. Stavans, Phys. Rev. A 36 (1987) p. 306.

[15] J. Stavans, E. Domany and D. Mukamel, Europhys. Lett. 15 (1991) p. 479.

[16] M.A. Fortes and P.IC. Teixeira, J. Phys. A 36 (2003) p. 5161.

[17] M.F. Vaz, Phil. Mag. Lett. 88 (2008) p. 627.

[18] V.E. Fradkov, L.S. Shvindlerman and D.G. Udler, Scr. Metall. 19 (1985) p. 1285.

[19] V.E. Fradkov, A.S. Kravchenko and L.S. Shvindlerman, Scr. Metall. 19 (1985) p. 1291.

[20] L. Ciupinski, K.J. Kurzydłowski and B. Ralph, Mater. Charact. 40 (1998) p. 215.

[21] D.A. Noever, Phys. Rev. A 44 (1991) p. 968.

[22] D.A. Noever, Astron. Astrophys. 282 (1994) p. 252.

[23] D.A. Noever, Astrophys. Space Sci. 220 (1994) p. 65.

[24] H. Hiesinger and J. Head, J. Geophys. Res. 105 (2000) p.11999.

[25] N.M. Seibert and J.S. Kargel, Geophys. Res. Lett. 28 (2001) p. 899.

[26] N. Mangold, S. Maurice, W.C. Feldman, F. Costard and F. Forget, J. Geophys. Res. 109

(2004) p. E08001.

[27] N. Mangold, Icarus 174 (2005) p. 336.

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Philosophical Magazine Letters (preprint)

[28] S. van Gasselt, D. Reiss, A.K. Thorpe and G. Neukum, J. Geophys. Res. 110 (2005) p.

E08002.

[29] M.T. Mellon, J. Geophys. Res. 102 (1997) p. 25617.

[30] K. Yoshikawa, Geophys. Res. Lett. 30 (2003) p. 1603.

[31] D.P. Page, Icarus, 189 (2007) p. 83.

[32] J.S. Levy, J.W. Head, D.R. Marchant, D.E. Kowalewski, Geophys. Res. Lett. 35 (2008) p.

L04202.

[33] P. Pina, J. Saraiva, L. Bandeira and J. Antunes, Planet. Space Sci. 56 (2008) p. 1919.

[34] P. Pina, J. Saraiva, L. Bandeira and T. Barata, Lect. Notes Comp. Sci., 4142 (2006) p. 691.

[35] G. Le Caer and R. Delanney, J. Phys. A 26 (1993) p. 3931.

[36] G. Vincze, I. Zsoldos and A. Szasz, J. Geom. Phys. 51 (2004) p. 1.

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Philosophical Magazine Letters (preprint)

**Tables**

Table I – Range of parameters for Martian networks

Parameter Min-Max

Area of the network (km 2 ) AN 0.14-20.43

# of polygons Np 24-9547

Average major axis (m) L 29.06 - 372.86

# of neighbors i 2 - 13

Average # of neighbors 5.58 - 6.02

Variance of # of neighbors μ2 1.24 - 3.46

Density (# polygons/km 2 ) d 19.14-2665.56

Lewis’ law constant λL 0.26 - 0.51

Lewis’ law fitting

R 0.7477-0.9891

Desch’s law constant λD 0.13-0.35

Desch’s law fitting

2

R D 0.8421-0.9988

Aboav-Weaire’s law constant a 0.46-1.47

Aboav-Weaire’s law fitting R 0.9464-0.9996

2

L

2

AW

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Philosophical Magazine Letters (preprint)

List of figure captions

FIG. 1 – Details of Martian polygonal terrains and respective segmented networks on

MGS/MOC images (sides of images are ± 500 meters): (a) M08-07124; (b) M14-00154; (c) R10-

03862 and (d) R10-04639 [image credits: NASA/JPL/MSSS (complete images can be found on

http://www.msss.com)].

FIG. 2 – Distributions of the number of neighbors in the 33 networks studied. Each point

represents an experimental observation (relative frequency fi) of a given number of neighbors i in

a network. The distributions show a slight positive skewness, which is common for this type of

features.

FIG. 3 – (a) Relation between average number of neighbors and number of polygons (#) in a

network for the complete dataset. The average number of neighbors tends to 6 when the

number of polygons in a network increases; (b) Polygon density d, number of polygons per km 2 ,

as a function of the average longer axis , in km, of the polygons in a network, given by the

equation shown (R 2 gives a measure of the good quality of the fitting).

FIG. 4 – Linear fitting of the law of (a) Lewis ( is the average area (in km 2 ) of polygons

with i neighbors); (b) Desch ( is the average perimeter (in km) of polygons with i neighbors)

and (c) Aboav-Weaire (i.mi is the product of the number of neighbors i of a polygon by the

average number of neighbors of its adjacent polygons mi). Each line corresponds to a different

network. A summary of the quality of the fitting for each law can be consulted in Table I.

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Philosophical Magazine Letters (preprint)

FIG. 5 – Relation between the experimentally determined Desch and Lewis parameters for the 33

network studied: λL vs. λD. The relation is approximately linear.

FIG. 6 – Relation between the experimentally determined Aboav-Weaire parameter a and the

second moment of the distribution of the number of neighbors μ2, for the 33 networks studied.

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