Sabine Grunwald Soil Classification Classification of natural phenomena like soils is typically done for the following purposes: Organize knowledge about the subject to enable investigation and communication to be both logical and comprehensive (structure/organization for scientific/technical development). Provide a framework for establishing relationships among soils and their environment that leads to advancement of theoretical and experimental aspects of soil and related science (academic focus). Establish groupings of soils for which useful and reliable interpretations can be made (utilitarian focus). For example: optimal use(s), hazard/limitation/remediation assessment, potential productivity, framework for technology transfer/information dissemination. There are two different classification approaches, i.e., technical vs. natural classification. Technical classifications are designed for specific applied purposes (e.g., soil engineering classifications, based largely on physical properties), whereas natural classifications attempt to organize the divisions of soils from a more holistic appraisal of soil attributes. The general principles of natural soil classification systems are: A natural system of classification should express general or universal relationships that exist in nature. One should be able to understand, remember, generalize, or predict from information obtained. The scheme should be based on characteristics or attributes of things classified as related to their genesis. It should place similar things together on the basis of their properties.

It is technically impossible to use all of the properties of soils to classify them. Judgment based on existing knowledge must be used to determine which properties are most important. Historical Perspective One of the earliest land evaluation systems that incorporated a soil classification was established during the Vao dynasty (2357-2261 B.C.) in China. Soils were graded into nine classes, based on their productivity. It has been suggested that property taxes were based on the size of the individual land holding and soil productivity. In former times (< 1600 A.C.), soil was solely considered as a medium for plant growth. Knowledge of soil behavior and crop growth was passed from generation to generation gained by observation. For example, in the Middle Ages it was well know that manure applied to soils improved crop growth. For instance, the 'Plaggen cultivation' was practiced for a long time in Europe, which left 'Plaggen soils': The top of grassland was peeled off and used as litter in the stables. This material mixed with manure was applied to arable land to improve crop production. In 1840, the German chemist Justus von Liebig initiated a revolution in soil science and agriculture.

He proved that plants assimilate mineral nutrients from the soil and proposed the use of mineral fertilizers to fortify deficient soils. Crop production was increased tremendously using mineral fertilizers. Another effect was the shift from extensive to intensive techniques in agriculture, which influenced soils. Thaer (1853) published a classification that combined texture as a primary subdivision with further subdivisions based on agricultural suitability and productivity. Several classifications based largely on geologic origin of soil material were also proposed in the 19th century (Fallou, 1862; Richtofen, 1886). From the 1660s onwards, various members of The Royal Society of London proposed schemes of soil classification that incorporated elements of a natural or scientific approach in their criteria. From this period on, the disciplines of agricultural chemistry (with a strong focus on soil fertility), geography, and geology provided a broad but somewhat fragmented background from which pedology emerged as a separate discipline in the late 19th century more or less independently in Russia (Dokuchaev and colleagues) and in the United States (Hilgard and colleagues).

In 1883, Dokuchaev carried out a comprehensive field study in Russia, where he described the occurrence of different soils thoroughly using soil morphologic features. Due to his observations in the field he hypothesized that different environmental conditions result in the development of different soils. He defined soil as an independent natural evolutionary body formed under the influence of five factors , of which he considered vegetation and climate the most important. Dokuchaev is generally credited with formalizing the concept of the 'five soil forming factors', which provides a scheme for study of soils as natural phenomena. The soil classification developed by Dokuchaev and his colleagues (Glinka, Neustruyev) was based on the soil forming factors -> soil forming processes -> and diagnostic horizons / soil properties. The focus in his soil classification approach was on soil genesis, therefore the classification system is called 'genetic'. In the United States, Hilgard (1892) emphasized the relationship between soils and climate, which is known as the climatic zone concept. Coffey (1912), produced the first soil classification system for the United States based on the soil genesis principles of Dokuchaev and Glinka.

Marbut (1951) introduced the concepts of Coffey into soil survey programs in the U.S. carried out by the USDA (United States Department of Agriculture). Between 1912 and the 1960 the soil classification in the U.S. used a genetic approach. Jenny (1941) put together a detailed description of the five soil forming factors responsible for the development of different soils. In 1959, Simonson stressed that many genetic processes are simultaneously and / or sequentially active in any soil. Hence, a soil classification based on principles of soil genesis would not be favorable. Other soil scientists in the U.S., like Smith (director of Soil Survey Investigations for the USDA Soil Conservation Service) agreed, that soil genesis is very important for soil classification, but that genesis itself cannot be used as a basis for soil classification because the genetic processes can rarely be quantified or actually observed in the field. References Coffey G.N., 1912. A Study of the Soils of the United States. U.S. Dept. of Agriculture Bur., Soils Bull. 85. U.S. Govt. Printing Office, Washington D.C.

Dokuchaev V.V., 1883. Russian Chernozems (Russkii Chernozems). Israel Prog. Sci. Trans., Jerusalem, 1967. Transl. from Russian by N. Kaner. Available form U.S. Dept. of Commerce, Springfield, VA. Fallou F.A., 1862. Pedologie oder allgemeine und besondere Bodenkunde, Dresden, Germany. Hilgard E.W., 1892. A Report on the Relations of Soil to Climate. U.S. Dept. of Agriculture. Weather Bull. 3: 1-59. Jenny H., 1941. Factors of Soil Formation. McGraw-Hill, New York. Liebig J. von, 1840. Chemistry in its Application to Agriculture and Physiologie. Playfaeir, Taylor and Waton, London. Marbut C.F., 1951. Soils: Their Genesis and Classification. A Memorial Volume of Lectures given in the Graduate School of the United States Dept. of Agriculture in 1928. Publ. 1951 by Soil Sci. Soc.

Am., Madison, WI. Richtofen F.F. von, 1886. Fuehrer fuer Forschungsreisende. Berlin. Simonson R.W., 1959. Outline of a Generalized Theory of Soil Genesis. Soil Sci. Soc. Am. Proc. 23: 152-156. Present Perspective in the U.S. The soil classification based on external environmental factors and an assumed genesis as differentiating characteristics led to dissatisfaction among U.S. soil scientists in the 1950's and 1960's. It was stressed that there is a lot of uncertainty involved dealing with soil forming factors and changing environmental conditions in a landscape. Many processes go on in any soil, often offsetting one another. Hence, it is often difficult to identify the processes in soils because most soils are polygenetic. Furthermore, it is difficult to assess the relative importance of each soil forming factor contributing to the development of a distinct soil class. There was much concern of not being able to classify certain soils with adequate agreement because of uncertainties or disagreement concerning their genesis. The knowledge about processes was and still is limited, because we do not completely understand soil processes.

It became obvious that a more 'objective' soil classification scheme should be developed to replace the 'subjective' one. As a consequence, a completely new system of soil classification was developed: The U.S. Soil Taxonomy . It is termed a 'natural' soil classification system, which attempts to organize the division of soils from a more holistic appraisal of soil attributes. The pedon is used as the sampling unit for soil classification and mapping in schemes employing the U.S. Soil Taxonomy. Diagnostic horizons are used to define different soil taxa. Since that time, the Keys to Soil Taxonomy have undergone six published revisions. The principles of Soil Taxonomy are: Classify soils on basis of properties Soil properties should be readily observable and / or measurable Soil properties should either affect soil genesis or result from soil genesis Soil processes are not refused in the Soil Taxonomy but the focus is on soil properties rather than on processes. The rationale behind this is that processes are important to produce soil properties and that those are used for classification, i.e., genesis 'lies behind' the soil taxa. Soil processes provide a framework for understanding the Soil Taxonomy. For example, the soil forming process of decomposition and humification forms an A horizons.

The soil properties recognizable in the field are a dark color of the A horizon with a granular soil structure. The soil properties selected should be observable or measurable, though instruments may be required for observation or measurement. Properties that can be measured quantitatively are to be preferred to those that can be determined only qualitatively. Other Perspectives There are many diverse approaches to soil classification in different countries. Most schemes have in common a hierarchical structure, i.e. a structure in which individuals are collected in small groups, the small groups belonging to larger groups, a.s.o. The criteria to be used in establishing the different categories are: The inferred genesis of the soil, on the assumption that the soil is in equilibrium with its environment and reflects the influence of prevailing soil forming factors, or: The soil properties being the outward expression of the processes which shaped the soil. No matter how divergent current soil classifications are in principle, language and format, two main levels of generalization stand out: A lower level encompassing the surveyor's series, which are narrow classes in harmony with the landscape. A higher level of broad classes (conceptual classes) with somewhat overlapping boundaries, each of which embodies a central, usually pedogenetic concept.

Examples for modern soil classifications used in different countries: Strunz Mineralogical Tables, Balogh International Hugo Strunz; Ernest Nickel: Strunz Mineralogical Tables. Ninth Edition. 2001. 9th edition X, 870 pages, 262 figures, 24x17cm. ISBN 3-510-65188-X bound, $158.00 Mineralogical Tables, first established as Mineralogische Tabellen by HUGO STRUNZ in 1941, have gone through eight editions, numerous reprints and translations. This ninth, completely new English-language edition is fully revised and updated and incorporates a large number of data on new minerals and new data on previously described minerals. Mineralogical Tables uses a chemical-structural mineral classification system based on the same general principles as those embodied in earlier editions: All minerals are apportioned into one of ten classes according to their principal anionic constituents. Each of these ten classes is then broken down into divisions, subdivisions and groups, based on chemical composition and crystal structure. A simple alphanumeric coding system is applied to the various categories.

This coding system has intentional gaps that permit the insertion of new categories of minerals yet to be discovered without disrupting the general classification scheme. The data are presented in tabular form, with minerals grouped by structure type, and include the chemical formula and crystallographic parameters for each mineral, as well as the authors and reference to the original mineral description. Literature references to reports of crystal structure determinations and to the more recent descriptive papers are also provided. Each group is accompanied by a brief description of the crystal structures of the minerals comprising the group, and crystal-structure drawings are given for many of the important structure types. Audience: Mineralogists, Earth Scientists, serious mineral collectors Contents: Introduction Historical Development 1 Chemical Bonding and Crystal Structures 7 Definitions 13 The Data in the Tabulations 14 The Classification System 14 Summary of System 17 Periodicals 28 1.

ELEMENTS (Metals and intermetallic alloys; metalloids and nonmetals; carbides, silicides, nitrides, phosphides) 1.A: Metals and Intermetallic Alloys 33 1.B: Metallic Carbides, Silicides, Nitrides and Phosphides 46 1.C: Metalloids and Nonmetals 50 1.D: Nonmetallic Carbides and Nitrides 54 2. SULFIDES and SULFOSALTS (sulfides, selenides, tellurides; arsenides, antimonides, bismuthides; sulfarsenites, sulfantimonites, sulfbismuthites, etc.) Sulfides, etc.

2.A: Metal/metalloid alloys 56 2.B: Metal Sulfides, M: S > 1: 1 (mainly 2: 1) 61 2.C: Metal Sulfides, M: S = 1: 1 (and similar) 75 2.D: Metal Sulfides, M: S = 3 :4 and 2:3 93 2.E: Metal Sulfides, M: S <= 1:2 100 2.F: Sulfides of arsenic, alkalies; sulfides with halide, oxide, hydroxide, H2O 110 Sulfosalts 2.G: Sulfarsenites, sulfantimonites, sulfbismuthites 118 2.H: Sulfosalts of SnS archetype 128 2.J: Sulfosalts of PbS archetype 140 2.K: Sulfarsenates 147 3.

HALIDES 3.A: Simple halides, without H2O 148 3.B: Simple halides, with H2O 155 3.C: Complex halides 159 3.D: Oxyhalides, hydroxyhalides and related double halides 170 4.

OXIDES (Hydroxides, V[5,6] vanadates, arsenites, antimonites, bismuthites, sulfites, selenites, tellurites, iodates) 4.A: Metal: 0xygen = 2:1 and 1:1 182 4.B: Metal: Oxygen = 3:4 and similar 187 4.C: Metal: 0xygen = 2: 3,3: 5, and similar 192 4.D: Metal: Oxygen = 1:2 and similar 203 4.E: Metal: 0xygen = < 1 :2 229 4.F: Hydroxides (without V or U) 230 4.G: Uranyl Hydroxides 248 4.H: V[5,6] Vanadates 254 4.J: Arsenites, antimonites, bismuthites, sulfites, selenites, tellurites; iodates 262 4.K: Iodates 281 5.

CARBONATES (+ NITRATES) 5.A: Carbonates without additional anions, without H2O 284 5.B: Carbonates with additional anions, without H2O 292 5.C: Carbonates without additional anions, with H2O 303 5.D: Carbonates with additional anions, with H2O 309 5.E: Uranyl Carbonates 318 5.N: NITRATES 324 6. BORATES 6.A: Monoborates 328 6.B: Diborates 338 6.C: Triborates 341 6.D: Tetraborates 345 6.E: Pentaborates 349 6.F: Hexaborates 354 6.G: Heptaborates and other megaborates 358 6.H: Unclassified borates 361 7.

SULFATES (selenates, tellurates, chromates, molybdates, wolframates) 7.A: Sulfates (selenates, etc.) without additional anions, without H2O 363 7.B: Sulfates (selenates, etc.) with additional anions, without H2O 370 7.C: Sulfates (selenates, etc.) without additional anions, with H2O 379 7.D: Sulfates (selenates, etc.) with additional anions, with H2O 395 7.E: Uranyl sulfates 413 7.F: Chromates 415 7.G: Molybdates end wolframates 418 7.H: Uranium and uranyl molybdates and wolframates 422 8. PHOSPHATES, ARSENATES, VANADATES 8.A: Phosphates, etc.

without additional anions, without H2O 424 8.B: Phosphates, etc., with additional anions, without H2O 439 8.C: Phosphates without additional anions, with H2O 470 8.D: Phosphates, etc. with additional anions, with H2O 491 8.E: Uranyl phosphates and arsenates 521 8.F: Polyphosphates, Polyarsenates, [4]-Polyvanadates 530 9.

SILICATES (Germanates) 9.A: Nesosilicates 534 9.B: Sorosilicates 567 9.C: Cyclosilicates 594 9.D: Inosilicates 617 9.E: Phyllosilicates 657 9.F: Tektosilicates without zeolitic H2O 691 9.G: Tektosilicates with zeolitic H2O; zeolite family 701 9.H: Unclassified silicates 712 9.J: Germanates 716 10. ORGANIC COMPOUNDS 10.A: Salts of organic acids 717 10.B: Hydrocarbons 722 10.C: Miscellaneous Organic Minerals 724 Appendix (New minerals, structures, reclassification) 727 Alphabetic index of mineral names 737 32.91 · Budapest University of Technology and Economics Abstract The most widespread classification system for pebble shapes in geology is the Zingg system which relies on several length measurements.

Here we propose a completely different classification system which involves counting static equilibria. We show that our system is practically applicable: simple hand experiments are suitable and easy to use to determine equilibrium classes. We also propose a simplified classification scheme called E-classification which is considerably faster in practice than the classical Zingg method. Based on statistical results of 1000 pebbles from several different geologic locations we show that E-classes are closely related to the geometric shape of pebbles. We compared E-classes to the Zingg classes, and we found that all the information contained in Zingg classification can be extracted from equilibrium classification. However, the new method is more sophisticated: it may help to identify shape attributes not discovered so far and it is able to store information on special geometries, e.g. on crystal shapes. Discover the world's research The most widespread classification system for pebble shapes in geology is the Zingg system which relies on several length measurements. Here we propose a completely different classification system which involves counting static equilibria. We show that our system is practically applicable: simple hand experiments are suitable and easy to use to determine equilibrium classes.

We also propose a simplified classification scheme called E-classification which is considerably faster in practice than the classical Zingg method. Based on statistical results of 1000 pebbles from several different geologic locations we show that E-classes are closely related to the geometric shape of pebbles. We compared E-classes to the Zingg classes, and we found that all the information contained in Zingg classification can be extracted from equilibrium classification. However, the new method is more sophisticated: it may help to identify shape attributes not discovered so far and it is able to store information on special geometries, e.g. on crystal shapes. Key words: pebble, shape indices, Zingg classes, static equilibria, crystal Introduction The shape of sedimentary particles is an important physical property that facilitates facies differentiation, provides information about the history of the sediment and helps to characterize depositional environments (Graham and Midgley 2000). Analysis of the shape is also a key element in understanding bedload transport and sorting of sedimentary particles, since shape has an influence on the hydrodynamic behavior of particles in the transporting medium (Oakey et al. 2005).

However, shape identification may also help in other fields, Addresses: T. Szabó, G. Domokos: H-1111 Budapest, Mûegyetem rkp. 3, K242, Hungary e-mails: tszabo@szt.bme.hu, domokos@iit.bme.hu Received: May 28, 2010; accepted: October 10, 2010 1788-2281/$ 20.00 © 2010 Akadémiai Kiadó, Budapest Central European Geology, Vol. 53/1, pp.

1–19 (2010) DOI: 10.1556/CEuGeol.53.2010.1.1 A new classification system for pebble and crystal shapes based on static equilibrium points Tímea Szabó, Gábor Domokos Department of Mechanics, Materials and Structures Budapest University of Technology and Economics, Budapest as particle shape is a complex function of lithology, particle size, the way and duration of transport, the energy of the transporting medium, the nature and extent of post-depositional weathering, and the history of sediment transport and deposition. Although shape is a basic attribute of all objects, including pebbles, complete characterization of such three-dimensional shapes poses formidable difficulties. Sedimentologists have developed a number of standardized shape indices to roughly estimate and classify pebble shapes; these shape indices are derived from measurements of the length, breadth and thickness of a pebble. We will briefly review these classical pebble shape categories in the next section. In this paper we propose a completely different classification scheme which does not rely on length measurements and shape indices; rather, it involves counting static equilibria.

Static equilibria are points of the surface where the pebble is at rest when placed on a horizontal, frictionless support surface. While our previous study (Domokos et al. 2010) provided a theoretical basis for equilibrium classification, here we concentrate on applicability in field measurements as well as geologic verification. We demonstrate that the new system is faster than traditional length measurements, that it is readily applicable in field work, and we give practical instructions for simple hand experiments. We also show how equilibrium classification can be naturally applied for crystal shapes; the new system is a clear indicator of crystal geometry. One of our central objectives is to show that the number and type of static equilibria is closely related to the geometric shape of a pebble, i.e. the equilibrium class is a natural property which describes pebble geometry well. We present detailed statistical data of 1000 pebbles from different depositional environments, abrasion processes and pebble lithology. Different statistical results of different geologic locations suggest that the latter could be distinguished based on equilibrium classification, e.g. the new system is characteristic for a given geologic location.

We compared the new equilibrium classification system to a well-known classical scheme (Zingg classes) and we found that equilibrium classification contains all the information contained in the classical method. However, the new system provides more detailed data and it may shed light on special shape features not discovered so far. Shape indices One of the important tasks of sedimentology is to sort the infinite number of conceivable particle shapes into a finite number of classes, because well-chosen classes can carry important information on the history of the sediment. Despite the extensive literature on the topic, there is little agreement on the best classification method for pebble shape analysis. A variety of shape indices and diagrammatical presentations of grain shape have been proposed in the past 2T. Szabó, G. Domokos Central European Geology 53, 2010 (Wentworth 1922; Zingg 1935; Krumbein 1941; Sneed and Folk 1958; Aschenbrenner 1956; Smalley 1967; Dobkins and Folk 1970).

These methods for quantitative characterization of pebble shapes have been the subject of lively discussions recent years (e.g. Illenberger 1991, 1992a, 1992b; Benn and Ballantyne 1992; Woronow 1992; Graham and Midgley 2000; Oakey et al. 2005). These classification systems require the measurement of the three orthogonal axis lengths L > I > S of the approximating three-axial ellipsoid, where L is the longest, I the intermediate and S the shortest axis of the pebble. Based on these axis lengths a number of shape indices have been used, some of them to predict the hydraulic behavior of sediment grains. A thorough summary of shape indices is given by Blott and Pye (2008) and Illenberger (1991); see also Table 1. A new classification system for pebble and crystal shapes based on static equilibrium points 3 Central European Geology 53, 2010 Table 1 The most frequently used and most widespread shape indices The first-order ratios of the three orthogonal axes are the easiest shape indices to conceptualize.

Allowing reciprocal functions, second-order indices are products of two first-order indices. Most of the proposed shape indices can be related to these first- and second-order indices and two independent shape indices are required to produce a particle-form diagram. Most researchers use either the Zingg (1935) or the Sneed and Folk (1958) methods to graphically represent and classify pebble shapes. Zingg proposed a Cartesian coordinate system with S/I and I/L as indices for the shape diagram, where S/I is a measure of flatness and I/L is a measure of elongation (Fig. 1a). The Zingg diagram is a simple and clear classification system; however, Sneed and Folk (1958) concluded that the Zingg classification is an inadequate tool, as it only contains four classes. They suggested that pebble shapes should be plotted on a triangular diagram, where S/L is plotted against the Disc-rod index (Fig. 1b, see also Table 1), and they divided the diagram into 10 shape classes.

Several authors have argued whether the Zingg or the Sneed and Folk system is the most suitable classification scheme; Blott and Pye (2008) concluded that the Zingg diagram provides a more even distribution of the shape continuum than the triangular Folk diagram. Zingg classification is probably the easiest and most logical way to estimate pebble shapes; therefore in this paper we will use Zingg classes as a basis for comparison to our new classification system. However, connection with other classification methods could also be demonstrated. 4T. Szabó, G. Domokos Central European Geology 53, 2010 Fig.

1 a) Zingg diagram with 4 classes: Disc (I), Sphere (II), Blade (III), Rod (IV) b) Triangular shape classification diagram by Sneed and Folk; the classes are: Compact (C), Compact- platy (CP), Compact-bladed (CB), Compact-elongated (CE), Platy (P), Bladed (B), Elongated (E), Very platy (VP), Very bladed (VB), Very elongated (VE) The above-mentioned classical pebble categories rely on length measurements, which inevitably cause inaccuracies in the classification method because it is often uncertain as how to L, I and S should be defined and also because measurements involve a degree of error and they might be quite tedious. Most researchers have agreed that L, I and S should be perpendicular to each other, but they do not need to intersect at a common point. However, as shown by Blott and Pye (2008), the exact directions of the three axes are uncertain even in the case of a simple cube.

If we define the L dimension as the maximum 'caliper' dimension (as most authors do), then the longest dimension is the body diagonal of the cube and the two axes perpendicular to this are not of equal length. Blott and Pye proposes that the L, I and S dimensions should be defined as the side lengths of the smallest imaginary box which can contain the particle. This definition leads to an accurate calculation in the case of cube; the three axis lengths are equal to each other (and to the side lengths of the cube). However, this method is rather complicated to achieve: the examiner must orientate the particle by eye so that S and I prescribe the smallest projected area. It is also apparent that classical pebble-shape categories involve arbitrarily chosen constants to separate shape classes from each other. The Zingg system uses an axis ratio value of 2/3 to discriminate the four classes. The classical Zingg system can be unified if an internal parameter 0≤p≤1 is introduced.

In the classical Zingg system of p=2/3, the generalized Zingg classes can be given as Zp(I): Disc I/L ≥p and S/I ≤p (1) Zp(II): Sphere I/L ≥p and S/I ≥p (2) Zp(III): Blade I/L ≤p and S/I ≤p (3) Zp(IV): Rod I/L ≤p and S/I ≥p (4) Although this not was explicitly defined, Zingg assumed that the p=2/3 value is universally optimal; however, later we will show that the optimal choice of p depends on the investigated sample. The value of p has a great influence on the result of the classification, i.e. classification schemes derived from length measurements are also ambiguous in that sense. In this paper we propose an alternative classification system which does not suffer from these ambiguities (while inevitably, it includes others!). Our approach is very speedy and does not contain any arbitrarily introduced constants or directions. Our system relies on the (integer) number and type of static equilibria which we outline below. In this section we summarize in a nutshell the results of previous studies, focusing on the practical side of equilibrium classification system.

For a detailed mathematical background, see Várkonyi and Domokos (2006); also Domokos et al. (2010). A new classification system for pebble and crystal shapes based on static equilibrium points 5 Central European Geology 53, 2010 Static equilibria are points of the surface where the pebble is at rest on a horizontal, frictionless support. When placed on a horizontal surface, a homogeneous, convex, rigid body will come to rest at a stable point of equilibrium and it will return there after a small perturbation. For example, a cube will have 6 stable equilibria on the 6 faces. Unstable points of equilibria appear still as balance points when placed on a horizontal surface; however, the body will not return there after small perturbations. There are two generic versions of unstable equilibrium points: unstable maxima and saddles. For example, the cube has 8 unstable points (maxima) on the 8 vertices and 12 saddle type equilibria on the 12 edges.

A typical convex body may have three types of equilibria: stable, unstable and saddle type equilibria and our approach relies on counting these equilibrium points. If equilibrium points are well isolated, i.e. they are sufficiently apart, they can be easily identified by hand experiments. In such experiments small perturbations, realized by slightly tossing the object in one direction or the other, play a key role. Typical points of a surface are non-equilibrium points. When placed on such a point on a horizontal surface, the body will roll away, always in the same direction, even if we toss it slightly in a different way. Stable points are most easily identified in a hand experiment, as stable points behave as attractors (Fig. 2a), i.e. the body will always roll back to a stable point from a nearby location, no matter in which direction it is tossed away. Unstable points act as repellors, i.e. the body will always roll away from an unstable point; 6T. Szabó, G. Domokos Central European Geology 53, 2010 Fig. 2 Counting equilibrium points of a flat pebble in a hand experiment.

Arrows show if pebble rolls away from the equilibrium point or rolls back to the equilibrium point if we toss it slightly away in that direction. a) when placed on a horizontal surface, the pebble will come to rest at a stable point of equilibrium and it will return there after small perturbation b) constrained in the principal plane, vertical position, one of the maxima along the edge still appears as unstable equilibrium: pebble is stable laterally (constrained by hand, see b2 side view); however, unstable in the longitudinal direction (b1 frontal view) c) constrained in the principal plane, vertical position, one of the pebble's saddle points along the edge appears as a stable equilibrium: pebble is stable both laterally (constrained by hand, c2 side view) and longitudinally (c1 frontal view) however, unlike the case of non-equilibrium points, the direction of the roll is ambiguous; it depends entirely on the direction in which it is tossed. Differentiation between unstable points and saddle points requires more attention from the examiner. In hand experiments saddle points behave similarly to repellors; nevertheless, in the case of flat, disc-like objects they can be much more easily identified.

In such cases saddles and unstable maxima are placed along the large perimeter of the flat object. By constraining the object vertically in the plane of the large perimeter, as the object rolls around the perimeter, saddles appear as attractors in this constrained problem, while unstable points remain repellors (Fig. 2b, c). Since most pebbles tend to be flat, this is probably the easiest way to count unstable and saddle-type equilibria. Static equilibrium points can be counted in such simple hand experiments. Let us denote the number of stable equilibrium points by S, the number of unstable equilibria by U and the number of saddle type equilibria by H. The Poincaré-Hopf Theorem (Arnold 1998) establishes the relationship S+U–H=2. Based on this result, we can define a unique classification for convex, rigid bodies based on the number and type of their equilibria (Várkonyi and Domokos 2006): class {S,U} contains all convex, rigid bodies with S stable and U unstable equilibria.

Thus, only two types of equilibria need to be counted in a hand experiment; the third type can be either computed or used to verify the experiment. For example, a cube is in class {6,8} with S=6 stable and U=8 unstable equilibria. Although in Várkonyi and Domokos (2006) it was shown that all classes are non-empty, most classes are not easy to visualize. Figure 3 illustrates some equilibrium classes by characteristic shapes. (The illustrations do not represent all characteristic shapes of the given class.) The Gömböc (Várkonyi and Domokos 2006) is in class {1,1}; a characteristic example for class {2,2} is a tri-axial ellipsoid. Observe that polyhedral shapes appear in the lower right corner of the table in the shaded wedge-shaped segment. Probably the most obvious examples for polyhedral shapes in geology are crystals. This indicates that equilibrium classification is not only a natural tool for classifying pebbles (as we later show), but it also can be applied to describe crystal shapes.

Crystal shapes Crystal shapes are traditionally classified into the following 7 crystal systems: the triclinic, monoclinic, orthorombic, tetragonal, trigonal, hexagonal and the cubic system. Although there is a detailed and abundant literature on crystal shapes and various methods – including the Zingg method (Mock and Jerram 2005) – are used to describe the geometry and symmetry of crystals (Hahn 2002), equilibrium classification may provide new information on these special shapes. Although equilibrium classification method based on hand experiments can only work on a relatively large single crystal, in fact in most cases the shape of a single crystal is so characteristic that we do not even need to take the crystal into A new classification system for pebble and crystal shapes based on static equilibrium points 7 Central European Geology 53, 2010 our hand, the equilibrium class can be determined just by visual inspection. The shape of a single crystal is usually a minimal polyhedron, i.e. each face of the polyhedron contains a stable equilibrium point, each vertex is an unstable equilibrium point and each edge contains a saddle type equilibrium.

Let us denote the number of faces, vertices and edges by F, V and E, respectively. For minimal polyhedra Euler's formula F+V–E=2 is equivalent to the Poincare-Hopf 8T. Szabó, G. Domokos Central European Geology 53, 2010 Fig. 3 Equilibrium classes for 3D bodies: examples for some characteristic shapes. Rows (S) denote number of stable equilibria, columns (U) denote number of unstable equilibria. Várkonyi and Domokos (2006) established the existence of {1,1} and created an object belonging to this class: it is called the Gömböc. The tri-axial ellipsoid (which is used to approximate pebble shape in the Zingg system) is in {2,2}. Observe that shapes in row 2 (class E(I)) are rather flat whereas shapes in column 2 (class E(IV)) are rather thin. Also, observe that polyhedral (crystal) shapes emerge toward the lower right corner of the table (shaded wedge-shaped segment).

Some typical crystal shapes in the table: pyramids with n- sided base, S=U=n+1: tetrahedron {4,4}, tetragonal pyramid {5,5}, hexagonal pyramid {7,7}, dihexagonal pyramid {13,13}; dipyramids with n-sided base, S=2n, U=n+2: trigonal dipyramid {6,5}, octahedron {8,6}, hexagonal dipyramid {12,8}; n-sided prisms with S=n+2, U=2n: trigonal prism {5,6}, cube {6,8}, hexagonal prism {8,12}; other crystal shapes: cuboctahedron {14,12}, rhombic dodecahedron {12,14} relationship S+U–H=2 because S=F, U=V and H=E. Therefore, to identify E- classes, the only task is to count the faces (F) and the vertices (V) of the polyhedron, determining the equilibrium class {F,V} of the crystal.

Minimal polyhedra are positioned in the shaded wedge-shaped segment of the table in Fig. 3, i.e. outside the wedge no minimal polyhedra exist. This statement can be proven along the following line: as each face is bounded by at least three edges, we have 3F≤2E. From Euler's formula (F+V–E=2) we have 2E=2F+2V–4. The previous two imply 3F≤2F+2V–4 from which F≤2V-4 and, finally, S≤2U–4 follows. Similarly, at least three edges start at each vertex, so we have 3V≤2E which leads to U≤2S–4. Relationships S≤2U–4 and U≤2S–4 determine the shaded wedge-shaped segment of the S–U table. Table 2 illustrates equilibrium classification of crystals with some examples. Equilibrium class is very easy to determine; nevertheless it is a clear indicator of crystal geometry, although it uses only two integer numbers.

A new classification system for pebble and crystal shapes based on static equilibrium points 9 Central European Geology 53, 2010 Table 2 Examples for equilibrium classes of single crystals. Crystal shapes are minimal polyhedra which are positioned in the shaded wedge-shaped segment of the table in Figure 3 E-classification and geometric shapes Equilibrium classification is not only an adequate tool for classifying crystal shapes; it also describes the shape of pebbles well, as we will show it in this section. Our main goal is to show how shape information can be extracted from the new equilibrium classification system and to demonstrate the connection between equilibrium classification and the classical Zingg system. For easier comparison with the Zingg system we introduce the simplified E- classification. The latter is based on equilibrium classes; however, it is radically simplified (see Fig.

3) as follows: E(I): E-classes {2,U}, U > 2 E(II): E-classes {S,U}, S, U >2 E(III): E-class {2,2} E(IV): E-classes {S,2}, S > 2 Pebbles with S,U=1 are extremely rare (Várkonyi and Domokos 2006), so these classes do not appear in the above simplified scheme. E-classification has the additional advantage that it is very easy to apply in hand experiments: the examiner has only to decide whether the pebble has two or more than two (stable and unstable) equilibrium points. This simplified scheme is a very speedy process; in the case of most pebbles it can be done in seconds. Our goal is to show how E-class is related to the geometric shape of the pebble. Although E-class does not exactly determine the geometric shape, in some cases the shape of the pebble is so characteristic that the E-class can be determined even without counting, just by brief visual inspection.

Figure 4 illustrates E- classes with 4 characteristic pebbles: flat objects will typically belong to the simplified E-class E(I); elongated objects will typically belong to the simplified E-class E(IV); flat and elongated objects belong to E(III). Objects neither flat nor elongated typically belong to E(II) (cf. Fig. 3). This indicates that simplified E-classes E(i) are closely related to the generalized Zingg classes Zp(i) as the axis ratio S/I is a measure of flatness and I/L is a measure of elongation. As mentioned before, in the case of E-classification the examiner must make two binary choices by de- termining whether the number of stable points (S) and the number of 10 T. Szabó, G. Domokos Central European Geology 53, 2010 Fig. 4 Four characteristic examples for E-classes. Typically, flat pebbles belong to E(I), elongate pebbles belong to E(IV), flat and elongate pebbles belong to E(III). Pebbles neither flat nor elongate belong to E(II).

E-classes E(i) are closely related to the generalized Zingg classes Zp(i). Zp-classes of the 4 characteristic pebbles are shown in brackets (cf. Fig. 1a) unstable points (U) is greater or equal to 2. Whether S is greater or equal to 2 can be often be decided without even taking the pebble in hand; as pointed out above, flat pebbles typically have S=2. Similarly, very elongated pebbles typically have U=2. If a pebble is neither very elongated nor very flat then the examiner must begin counting S and U. If any of these numbers exceed 2, counting may cease: from the viewpoint of E-classes no further information is required. However, if we are interested in the exact number of stable and unstable equilibria, it sometimes helps – especially in case of complicated shapes – to mark the identified equilibrium points on the surface (thus avoiding counting them twice). Figure 4 suggest that simplified E-classes E(i) are closely related to the generalized Zingg classes Zp(i). We will validate this statement with quantitative, statistical data in the next section.

Statistical comparison of E-classes and Z-classes We examined the agreement between E-classes and Zp-classes on 20 collected pebble samples (each of them consisting of 50 pebbles). The samples are from A new classification system for pebble and crystal shapes based on static equilibrium points 11 Central European Geology 53, 2010 Fig. 5 Place of origin of the 20 pebble samples. Each sample consists of 50 pebbles. Geologic locations are rather different: abrasion processes (pebbles on the seashore and river-bank) and rock types differ from each other (see also Table 4) several different geologic locations (Fig. 5), representing different depositional environments, abrasion processes and pebble lithology. We determined the E-class of all pebbles as well as their axis ratios S/I and I/L. Based on the latter, the Zp-class can be determined if the parameter p is given (in the classical Zingg case p=2/3 is assumed). Table 3 shows a sample of our table in which all required data can be recorded.

Instead of assuming a single value for p we let p vary between 0 and 1 and for each value we determined the Zp-class. Also, for each value of p we determined the agreement a(p) between Zp- and E-classification: this value is between 0 and 1 and expresses the ratio of pebbles where Zp- and E-classes agreed, i.e. if the pebble was classified in E(i) it was simultaneously classified into Zp(i). The function a(p), drawn in the unit square, we call the Z/E agreement diagram. Figure 6 presents such a Z/E agreement diagram. The diagram has one definite 12 T. Szabó, G. Domokos Central European Geology 53, 2010 Table 3 A simple table for statistical comparison between E-classes and Zp-classes (Sample data refer to the case p=0.84). If time allows it is worth using the "full" counting process, and record the exact number of S and U. If time plays an important role, from the point of E-classification it is enough to record whether S (or U) is greater or equal to 2 Fig.

6 Some typical pebbles and Z/E aggreement diagram for pebble sample V1. Horizontal axis: parameter p of the generalized Zingg classification (cf. equations (1)–(4)). Vertical axis: agreement a(p) between Zp- and E-classification. The diagram has one definite peak at popt. The value a (popt) is the best match between Zp- and E-classes peak, the p-value associated with this peak we denote by popt . The value a(popt) is the best match between Zp- and E-classes. Table 4 summarizes our experimental results of the 20 samples. We found that for all locations and all samples the Z/E diagram has one (and only one) characteristic peak, in the range 0.55 < popt < 0.89. The values popt are listed in a separate column of Table 4. The value a(popt), i.e. the best match between Zp- and E-classes, varies between 0.56–0.96 and is also listed separately, as well as the percentages in all classes at p=popt.

The definite, single peak in the Z/E curves suggests that there exists an optimal Zingg parameter for each sample at which Z-classification works most efficiently. For well-rounded, worn stones the optimal Zingg parameter is higher (~0.8), for more rugged stones it is smaller (~0.6). This shows that the optimal Zingg parameter is not universal; however, Zingg's original guess of p=0.67 was a very good one. It also shows that one can estimate the value of popt pretty accurately based on the pebble's shape. The shape of the Z/E agreement diagrams is very characteristic for the sample. In the case of well-rounded, worn stones (samples 1–11) the curve starts low and has a long monotonic slope to the peak. In case of rugged shapes (samples 12–20) the curve starts high and has only a minor slope before the peak. Based on the values a(popt), we can see that data on Z-classes can be extracted from E-classification with fair (~10–20%) accuracy. In our experiments we only used simplified E-classes since our goal was to compare E- and Zp-classes.

Simplified E-classification contains the same information as the Zingg classes; however, the 'full' equilibrium classification admits more subtle distinction of shapes which cannot be captured by Z-classification. As an example we mention that the Zingg system cannot distinguish between tetrahedral and cubic shapes, while equilibrium classes {4,4} and {6,8} give clear descriptions of this shapes. Statistical results also show that there is a significant difference between different geologic locations; this implies that different locations can be distinguished based on equilibrium classification. The distribution of pebbles in the different E-classes can indicate a special abrasion process, depositional environment and/or pebble lithology. For example, collecting pebble samples of identical rock types can help to distinguish between the abrasion of wave-current on a beach, where pebbles are dominantly sliding on each other, and the abrasion in a riverbed, where the case of rolling is more frequent. Analysis of pebble shape can also characterize the environment, as it can separate sandy beaches from gravelly beaches, high-wave energy beaches form low-wave energy ones (Dobkins and Folk 1970).

Finally, equilibrium classification also can indicate a special pebble lithology (e.g. schist pebbles are dominantly flat because of the lamellar strucure of this rock type; see samples 1–3). In this paper we presented pebble samples from various geologic locations to show the applicability of our new system; however, our future project is a A new classification system for pebble and crystal shapes based on static equilibrium points 13 Central European Geology 53, 2010 14 T. Szabó, G. Domokos Central European Geology 53, 2010 Table 4 Comparison between E-classes and Z-classes. Except for sample size, all data is given in percentages. Z/E diagrams are plotted as agreement 0<a<1 between Z and E classes versus Zingg parameter 0<p<1. Peak of Z/E curve corresponds to best agreement; values are listed in separate columns.

Observe that Z/E curves are characteristic of stone shapes: samples 1–11 show similar, smooth shapes and similar Z/E curves, the rest 12–20 show rugged shapes and the Z/E curves are also similar in this group 0.79 purposeful application of the new system, based on geologically interesting pebble samples. In our paper we proposed a new classification scheme for pebble shapes, based on the number and type of static equilibrium points. Although our system is more elaborate in the sense that it includes – in theory – infinitely many classes, in practice only half a dozen play an important role. We introduced a simplified version with just four E-classes which is a rapid and adequate tool in most cases. The new system appears to be a natural choice in the sense that the integer number of static equilibria is encoded in the pebble's shape. There is no need to involve any arbitrary constants or directions in the classification system; the only task is to count static equilibrium points of the surface. We observed that flat shapes always belong to class E(I), elongated shapes to E(IV) and by similar argument flat and elongated shapes with small axis ratios b/a, c/b always fall into class E(III).

This indicates that the number and type of equilibria constrains important shape informations although it does not completely determine the geometric shape. We compared Zingg classification to E-classification. We found that, depending on the choice of the Zingg-parameter p a 80–90% agreement exists, i.e. all the information contained in Zingg classification can be extracted from the simplified equilibrium classification. However, if the "full" counting process is applied, the new system is more sophisticated and is capable of storing information on special (e.g. polyhedral) shapes which cannot be extracted from the Zingg system. In particular, in the case of crystal shapes the new system promises to be useful: equilibrium class is a clear indicator of crystal habit. Our approach appears to be a natural one and possibly in combination with other methods may help to identify statistical shape features not discovered so far, which may add new information about the wear process and the depositional environment of pebbles. The authors are grateful to the staff of the Department of Mineralogy, Eötvös Loránd University, for providing access to their collection of minerals. We are most indebted to Ákos Török for his kind help, in particular with identifying rock types.

The support of OTKA grant 72146 is appreciated. References Arnold, V.I. 1998: Ordinary differential equations. – 10th printing, MIT Press, Cambridge. Aschenbrenner, B.C. 1956: A new method of expressing particle sphericity. – Journal of Sedimentary Petrology, 26, pp. 15–31. 18 T. Szabó, G. Domokos Central European Geology 53, 2010 Benn, D.I., C.K. Ballantyne 1992: Pebble shape (and size!) – discussion. – Journal of Sedimentary Petrology, 62, pp. 1147–1150. Blott, S.J., K. Pye 2008: Particle shape: a review and new methods of characterization and classification. – Sedimentology, 55, pp. 31–63. Corey, A.T. 1949: Influence of shape on fall velocity of sand grains. – unpublished MSc Thesis, Colorado A&M College. Dobkins, J.E., R.L.

Folk 1970: Shape development on Tahiti-Nui. – Journal of Sedimentary Petrology, 40, pp. 1167–1203. Domokos, G., A. Sipos, T. Szabó, P. Várkonyi 2010: Pebbles, shapes, and equilibria. – Mathematical Geosciences, 42, pp. 29–47. Graham, D.J., N.G. Midgley 2000: Graphical representation of particle shape using triangular diagrams: an Excel spreadsheat method. – Earth Surface Processes and Landforms, 25, pp. 1473–1477. Hahn, T. (Ed.) 2002: International Tables for Crystallography, Volume A: Space Group Symmetry – 5th edition, Dordrecht: Kluwer Academic Publishers. Illenberger, W.K. 1991: Pebble shape (and size!). – Journal of Sedimentary Petrology, 61, pp. 756–767. Illenberger, W.K.

1992a: Pebble shape (and size!) – reply. – Journal of Sedimentary Petrology, 62, pp. 538–540. Illenberger, W.K. 1992b: Pebble shape (and size!) – reply. – Journal of Sedimentary Petrology, 62, pp. 1151–1155. Krumbein, W.C. 1941: Measurement and geologic significance of shape and roundness of sedimentary particles. – Journal of Sedimentary Petrology, 11, pp. 64–72. Mock, A., D.A. Jerram 2005: Crystal Size Distributions (CSD) in three dimensions: Insights from the 3D reconstruction of a highly porphyritic rhyolite. – Journal of Petrology, 46, pp. 1525–1541. Oakey, R.J., M. Green, P.A. Carling, M.W.E. Lee, D.A. Sear, J.

Warburton 2005: Grain-shape analysis – a new method for determining representative particle shapes for populations of natural grains. – Journal of Sedimentary Research, 75, pp. 1065–1073. Smalley, I.J. 1967: The presentation of subjective shape and roundness data. – Sedimentology, 8, pp. 35–38. Sneed, E., R.L. Folk 1958: Pebbles in the lower Colorado River, Texas, a study in particle morphogenesis. – Journal of Geology, 66, pp. 114–150. Várkonyi, P., G. Domokos 2006: Static equilibria of rigid bodies: dice, pebbles and the Poincaré-Hopf Theorem. – Journal of Nonlinear Science, 16, pp. 255–281. Wentworth, C.K. 1922: The shapes of beach pebbles. – US Geological Survey Professional Paper, 131- C, pp. 75–83. Woronow, A.

1992: Pebble shape (and size!) – discussion. – Journal of Sedimentary Petrology, 62, pp. 536–537. Zingg, T. 1935: Beitrag zur Schotteranalyse. – Schweizeriscke Mineralogische und Petrologische Mitteilungen, 15, pp. 39–140. A new classification system for pebble and crystal shapes based on static equilibrium points 19 Central European Geology 53, 2010 CitationsCitations2

Question: Developed in 1941, the Nickel-Strunz classification scheme categorizes?
Answer: Minerals