Soil Science Society of America Journal 63:930-933 (1999)
© 1999 Soil Science Society of America
DIVISION S-5-PEDOLOGY
Morphological Analysis of Soil Aggregates Using Euler's Polyeder Formula
Karl Heinrich Hartgea,
Joerg Bachmanna and
Nestor Pescia
a Inst. of Soil Science, Herrenhäuser Str. 2, Univ. of Hanover, 30419 Hannover, Germany
bachmann{at}mbox.ifbk.uni-hannover.de
 |
ABSTRACT
|
|---|
In present morphologic analysis of soil aggregates, a subdivision between polyhedra and prisms is commonly performed using the relation between their axis lengths. Further description is limited to qualitative alternatives like "rounded/not rounded." The objective of this study was to obtain a more detailed subdivision of polyhedra using a quantitative approach, thus satisfying a prerequisite for statistical analysis. A series of experiments were carried out to determine the shape of aggregates using the parameters of Euler's polyhedron formula, which states that the number of corners plus the number of faces is equal to the number of edges plus 2. Results were evaluated from sets of 10 aggregates from each of six sites and from several depths as well as from artificial new loess aggregates made from the material of one of the samples. The results indicate that the number of faces was the least variable term. Most aggregates were found to have five faces, so are called pentahedra. The number of edges is generally smaller on field aggregates than on perfect polyhedra. For artificial fresh loess aggregates, however, the number of edges corresponds more closely to that of perfect polyhedra. The number of corners is similar to those in perfect polyhedra in all cases. Thus the number of edges is the element that is most promising for a more detailed subdivision of soil aggregates. Even though there is a general correlation between number of faces and number of edges, the edge/face ratio can provide independent information on the degree of rounding. Difficulties in discerning faces, edges, and corners, and the individual influence of personal assessment by the observer have shown to be of minor importance.
|
INTRODUCTION
|
|---|
MORPHOLOGIC DESCRIPTION of soil aggregates is usually confined to the classification by geometrical structure such as polyhedra, prisms, crumbs, or plates. Sometimes aggregates that occur after tillage operations have been classified as clods, fragments, or lumps (White, 1979). Subclassification beyond this level and beyond groups for size has been confined to separating among "angular/subangular" or "well marked/poorly marked" edges and faces. These norms have prevailed for several decades in many English, French, German, and Russian textbooks (Vilenski, 1957; Duchaufour, 1965; Bridges, 1970; Brady, 1974; White, 1979; Miller and Donahue, 1990; Scheffer and Schachtschabel, 1992; Mückenhausen, 1993; Kuntze et al., 1994). This type of classification is typically performed visually, does not require any apparatus, and is easily carried out in the field. In the current soil survey in Germany morphologic description is a basic component (AG-Boden, 1994). The drawback of this system is that it does not present the results in a quantitative form, making statistical treatment impossible.
Some methods not based on descriptive categories exist for assessing the shape of aggregates. Quite a few of these methods were reviewed by Dexter (1985). Most were developed to characterize the shape of primary particles. These assessment methods use different geometrical properties that must be measured either directly at the aggregate or from a photograph. The handling of soil aggregates for direct measurements is sometimes difficult, because they may break or become rounded during the procedure. Moreover, measurements obtained from photos are restricted to one plane of observation.
In order to minimize handling of the aggregates in this investigation digital results were obtained by counting easily recognized geometric features. In this investigation the Euler formula (Gellert et al., 1965) was chosen to express morphological aggregate properties and their relation to each other. As can be seen in Eq. [1], this formula describes the relation between numbers of faces (f), edges (e), and corners (c) of regular convex polyhedra, i.e. the number of corners plus the number of faces is equal to the number of edges plus 2.
 | (1) |
Since these three features can be determined by counting it might be possible to obtain a finer subdivision of aggregate forms in a way that is acceptable in field work. There are quite a few methods by which aggregates from different soil samples can be distinguished on the basis of the Euler formula. The following three were investigated:
- Direct comparison of the number of faces. The creation of faces by shrinkage and shearing is considered as a general aggregate forming process. This parameter is particularly significant because it is the number of faces, and not the number of corners or edges, that determine the class of polyhedra (pentahedron, tetrahedron etc.).
- Direct comparison of the number of edges and corners. These are considered as an important feature specifying the pedoturbate development of soil aggregates during or after their formation. Pedoturbation as well as tillage actions destroy corners and edges. Low numbers of these features might indicate the intensity of these processes.
- Interactions between the above parameters.
A major condition for an application of this principle is sufficient reproducibility of the results compared with the range of possible values or the differences that can be detected. In addition a specific difficulty arises from the irregularity of the aggregate forms. This could lead to different counting results due to different discernment, not only by different persons, but also by the same person on different days.
The aim of this investigation was to demonstrate the potential of Euler's formula as a basis for the subdivision of polyhedric aggregate forms using an easily applied field method.
|
Materials and methods
|
|---|
Sets of 10 aggregates between 0.5 and 5 cm in diameter were obtained from field samples of 
1 L volume, collected from six sites at one or more soil depths. Table 1
shows the location, soil use, soil group, pedologic horizon, sampling depth, numbers of sets per sample, number of assessments per set, and also the persons involved.
View this table:
[in this window]
[in a new window]
|
Table 1 Survey of soils and assessments, and the number of the repeated procedure of assessment and assessing persons (a, b, and c)
|
|
Most samples fragmented easily during preparation, forming aggregates at natural cracks. Nonfragmenting samples were dropped from 80 cm height before drying out. Since all investigations had to be carried out on the same samples, they were allowed to air dry before work was started.
The loess material from Bv horizon was allowed to dry and crushed to pass through a 2-mm sieve, then moistened with deionized water and remolded to form a stiff slurry. This was left to dry again. The dry blocks were crushed by dropping as explained above.
For assessment of polyhedron forms, 10 aggregates were chosen randomly from each soil sample to form a set. The number of 10 aggregates was considered as a compromise between the requirement of data for statistical use and feasibility in field work. The sets were placed on a sheet of paper and the numbers of faces (f), edges (e), and corners (c) were noted. Three persons carried out the counting procedure. Additionally the counts for randomly selected sets were repeated up to three times by one person. No special instruction was given to the participating soil scientists who had no experience in field work or in soil morphology. The decision whether a particular feature should be counted or not was assumed to be a major source of error. For this reason an "either/or" decision was admitted in some sets. For example, in a case where there was doubt concerning whether a value was 5 or 6, a number of 5.5 was to be recorded. It was considered that this procedure could facilitate assessment and thus decrease deviation within one set of aggregates.
In many cases the aggregates broke or became rounded by handling during morphometric assessment. In other cases not all three investigators participated in the assessment. Finally, a total of 59 sets of values was obtained from 24 sets of aggregates.
|
Results and discussion
|
|---|
The results are summarized in Fig. 1
and Table 2
. The plot shows values from the first set of counts representing every set of aggregates from each of the different soils and from the newly produced sets of loess aggregates. Table 2 gives the arithmetic mean, the standard deviation, and the range of values for all investigated sets of aggregates and assessments. As can be seen, the range of faces is smaller than the one of edges and corners (Fig. 1) and the standard deviation of the faces is also smallest (Table 2). The mean of all field soil samples is 4.89 faces. Accordingly the soil aggregates are predominantly considered as pentahedra. There were, however, always some tetrahedra and hexahedra in each set of 10 aggregates. The range of numbers of corners was wider than that of the number of faces, as is also the case with the variations (Table 2). The number of edges per aggregate and the standard deviation within the individual sets varied most (Table 2). It can be seen further from this table that the number of faces, edges, and corners of the newly prepared loess polyhedra was more similar to those of geometrically perfect polyhedra than the corresponding values of the soil aggregates. The number of edges in soil aggregates in particular, is small compared with the corresponding number from newly prepared loess aggregates and from perfect polyhedra.
View larger version (20K):
[in this window]
[in a new window]
|
Fig. 1 Histogram of numbers of faces, corners and edges obtained from first assessment of every set of aggregates. Means of 10 aggregates. Ticks on the abscissa represent class limits. All dots that fall into one class are depicted in the center of their class. For sample numbers 6, 7, and 22, see Fig. 2
|
|
View this table:
[in this window]
[in a new window]
|
Table 2 Arithmetic means, standard deviations and ranges of the form elements of polyeders according to Euler's equation of sets of 10 aggregates from field soils, remolded loess (A) and regular polyeders (B)
|
|
The irregular shape of field soil aggregates makes it frequently difficult to identify the geometric elements. Comparison of results from six sets, where either integers or averaged values between neighboring numbers were used for the initial assessment, did not show the expected decrease of standard deviation within a set. The mean values for all three elements (f, e, and c) of both procedures were positively correlated at a level of significance
. Therefore only the procedure using integers was applied for further assessment. In order to determine the influence of the personal factor, counts were repeated by the same and by different persons. To show the results of this procedure the no. 7 aggregate was selected randomly from six sets. The accuracy of the procedure was expressed by the difference of numbers of counted elements from the most frequent or from the intermediate one. As shown in Table 3
, faces were the easiest to count precisely. Next were edges, with corners the least reliable to count.
View this table:
[in this window]
[in a new window]
|
Table 3 Examples of hit conditions in assessing elements (e = edges, f = faces, c = corners) of aggregates by observers a, b and c. Number 7 aggregate was randomly chosen from each of the 6 sites (10 aggregates per site)
|
|
Another important question is, which of the Euler parameters expresses most clearly differences in aggregate morphology? Initially, the data in Fig. 1 reveals three morphologic groups of samples: New aggregates, field samples from non tilled soil depths and tilled horizons. Closer inspection shows that with the number of edges the difference between the newly formed aggregates and those from freshly tilled soils is greatest. Each of these groups show relative homogeneity. This also applies to the distribution of the number of faces. Both of these groups of samples are somewhat different from the rest, which mainly consist of samples from lower parts of profiles.
A comparison of the numbers of faces as well as of edges of the freshly tilled topsoil samples (Fig 1) with sample JG (no. 22 in Fig. 1) shows that there are appreciable differences in tilled soils. This is recorded by both parameters. In this case, the morphologic change might be due to the long time interval between tillage and sampling. On the other hand, the samples from fresh tidal sediment (TS, no. 6) and from glacial till subsoil (SUBS, no. 7), which could be expected to show the morphology of the newly prepared aggregates, are definitely different (Fig. 1). So within the groups of tilled soil as well as within the ones of fresh and old subsoil, morphologic differences between aggregate sets are recorded by both parameters. These differences are significant at the P < 0.05 level when the standard deviations from Table 2 are considered.
If the rounding of aggregates by tillage or by pedoturbation is to be quantified, a special morphologic feature, the number of destroyed edges per aggregate should be the most promising parameter. This is shown for 24 sets of values in Fig. 2
. It is obvious that differences that were discussed earlier show up here again. New aggregates (Sample no. 1–5) show most edges per face, the fresh tilled ones the least (no. 23 and 24). However, the variation is much larger and overshadows the differences.
Most aggregates within all of our samples were pentahedra. This makes it possible to judge the numbers of faces, edges, and their relationship by comparison with those of perfect pentaeders, which would be
(Table 2). This fact moreover suggests that there is a generally equal distribution of forces that causes failure by cracking and thus aggregation in all of the investigated soil samples.
|
Conclusions
|
|---|
From the results we conclude that the Euler formula can be used to produce a concise description of aggregates and provides a tool to quantify aggregate forms without requiring complicated and expensive laboratory measurements. The comparison of the number of edges and of faces between sets of aggregates shows differences clearer than number of corners. The relation of edges per face seem to be less promising as a means of comparison, because of the high variation.Arbeitsgemeinschaft Boden der Geol. Landesämter der 1994
|
ACKNOWLEDGMENTS
|
|---|
The authors express grateful appreciation to Mr. J.W.E. Henderson for his assistance in preparation of the English text, and to the German Research Foundation (DFG) for financial support of the project Ha 453/39.
Received for publication February 17, 1997.
|
REFERENCES
|
|---|
- Arbeitsgemeinschaft Boden der Geol. Landesämter der B.R.D. Bodenkundliche Kartieranleitung, 4th ed Germany: Schweizerbarthsche Verlagsbuchhandlung Stuttgart, 1994.
- Brady N.C. Nature and properties of soils, 8th ed New York: Macmillan Publ, 1974.
- Bridges E.M. World soils. UK: Cambridge Univ. Press. London, 1970.
- Dexter A.R. Shape of aggregates from tilled layers of some Dutch and Australian soils. Geoderma 1985;35:1-107.
- Duchaufour P. Precis de pedologie. 12.Ed. France: Masson & Cie. Paris, 1965.
- Gellert W., Küstner H., Hellwich M., Kästner H. Kleine Enzyklopädie: Mathematik. Leipzig, Germany: VEB.Verlag Enzyklopädie, 1965.
- Kuntze H., Roeschmann G., Schwerdtfeger G. Bodenkunde, 5th ed Stuttgart, Germany: Ulmer, 1994.
- Miller R.W., Donahue R.L. Soils — An introduction to soil and plant growth, 6th ed New York: Macmillan Publ, 1990.
- Mückenhausen E. Die Bodenkunde und ihre geologischen, geomorphologischen, mineralogischen und petrologischen Grundlagen, 4th ed Frankfurt/Main, Germany: DLG–Verlag, 1993.
- Scheffer F., Schachtschabel P. Lehrbuch der Bodenkunde, 13th ed Stuttgart, Germany: Enke, 1992.
- Vilenski D.G. Soil Science. Jerusalem, Israel: Moscow. Engl. 1963 by Israel progr. of Scientific Transl, 1957.
- White R.E. Introduction to the principles and practices of soil science. UK: Blackwell Scientific Publ. Oxford, 1979.
TOP
ABSTRACT
INTRODUCTION
