## Abstract

Numerical simulations and physical fragmentation experiments confirm the theoretical prediction that the fractal dimension of a two-dimensional (2-D) cut through a set of three-dimensional objects with fractal dimension *D* is approximately equal to *D*– 1. This leads to a size distribution in two-dimensional cuts that is skewed strongly toward larger objects compared to the three-dimensional distribution. Three-dimensional shape (aspect ratio) does not significantly affect the resulting 2-D size distribution except for highly nonequant objects, such as prolate ellipsoids with aspect ratios of 10 or more. In contrast, fragmentation of an object by breakage along persistent fractures results in a non-fractal distribution of sizes and far fewer small objects than predicted by fractal statistics. Powdering a rock by extensive crushing also results in non-fractal size distributions because particles are reduced to sizes on the order of 1 μm, a comminution limit below which further brittle fracture is difficult. Natural examples of fragmental objects observed in two-dimensional cuts, such as crushed rocks, breccias, and xenoliths, are generally consistent with a three-dimensional fractal dimension near 2.5 over one or two orders of magnitude in size. However, a limestone breccia from Death Valley exhibits a non-fractal size distribution consistent with fragmentation of a strongly jointed rock. Mafic enclaves in Yosemite National Park have a restricted size range of about one order of magnitude and a three-dimensional fractal dimension of ∼3.1, consistent with other enclave swarms. The restricted size range of enclaves may reflect the apertures of mafic dikes that fed them.

## INTRODUCTION

Although geologic objects are three-dimensional (3D), geologists are typically presented with two-dimensional (2D) cuts through them, such as outcrops, road cuts, and thin sections. It is known that such cuts can be misleading; for example, the clasts in a clast-supported conglomerate generally do not appear to touch each other in outcrop because the outcrop surface rarely intersects the points of contact (Boggs, 1992, p. 212–213). Converting 2D analyses to 3D (stereology) is a common practice in crystal size distribution analysis (Chayes, 1950; Higgins, 1994, 2000; Peterson, 1996) and other geologic intersection work (Sahagian and Proussevitch, 1998). The 3D size distribution of clastic material is important because it provides information about the fragmentation processes of the original object.

Three-dimensional analysis of a large variety of materials produced by crushing, blasting, grinding, thermal shock, and other fragmentation processes shows that the resultant particles tend to follow a self-similar (fractal) size distribution (Turcotte, 1986). That is, if particles whose sizes follow a fractal distribution are passed through a sieve and photographed, and the photograph is then scaled so that the largest particles are of a given size, the size distribution in the photograph will look the same regardless of the sieve size. However, dicing a solid by random orthogonal cuts generates a decidedly non-fractal distribution of particle sizes (Glazner and Bartley, 2006). This suggests that size distributions can be useful in determining fragmentation mechanisms. For example, the size distribution of xenoliths in a pluton might be used to infer whether they were incorporated via thermal fragmentation, producing a fractal size distribution, or by a process involving magma injection along preexisting fractures, producing a non-fractal distribution. Similarly, Shimamoto and Nagahama (1992) used fragment size statistics to argue against a fracture origin for the matrix of pseudotachylites. However, such analyses are hindered by the necessity to analyze materials in two dimensions in outcrop or thin section, and the necessity to understand how 2D cuts relate to 3D objects.

*N*is the number of fragments with linear dimension greater than

*r*,

*C*is a constant, and

*D*is the fractal dimension. In such a distribution, a log-log plot of the cumulative number of particles greater than a given size

*r*versus

*r*gives a straight line whose slope is –

*D*. The greater the value of

*D*, the greater the number of smaller particles relative to larger ones; for example, if

*D*= 3, then the number of particles larger than

*X*in linear dimension is 1000 times greater than the number of particles larger than 10

*X*.

Hartmann (1969) summarized data on fragmentation by various processes such as crushing, blasting, grinding, and explosive volcanic eruption, and showed that the data generally fit a power-law distribution. Turcotte (1986) showed that these data are consistent with a fractal distribution. Figure 1 and Table 1 summarize 319 published *D* values determined for a variety of fragmented geologic materials in both 3D and 2D. The 3D values have an overall mean of 2.2 ± 0.35 (1 standard deviation), or 2.4 ± 0.3 without the values of Perugini et al. (2011), a large data set that is a distinct outlier in Figure 1. Two-dimensional D values from previous studies of broken solids (Fig. 1) are noticeably lower (1. 5 ± 0.4). Allègre et al. (1982), Turcotte (1986), and Sammis et al. (1987) presented a variety of mechanical and mathematical fragmentation processes that produce particle arrays with *D* in the range of 2.5 to 2.9. In particular, Sammis et al. (1987) developed a model showing that if nearest neighbors of the same size are preferentially broken, as is commonly found in experiments, then any initial distribution of particles will tend toward a self-similar distribution having a fractal dimension of 2.58.

An alternative description of fragmentation statistics is the lognormal distribution (Hatch and Choate, 1929), which is commonly applied in studies of sedimentary particles (Krumbein and Tisdel, 1940; Folk and Ward, 1957; Rogers, 1965). Hartmann (1969) suggested that the discrepancy between lognormal and power-law fits is in part due to limitations of observational methods, which must always undercount the smallest particles. Some studies have shown that fragmentation processes can produce fragments that are best described as either mixed lognormal–power law or two–power-law populations (e.g., Meibom and Balslev, 1996; Tavassoli and Shirvani, 2000; Yazdi and Esmaeilnia, 2009).

If a fractal array of particles is viewed in a 2D cut, the fractal dimension of the intersected particles, hereafter referred to as *D _{2}*, is less than the fractal dimension of the 3D particles, hereafter

*D*. Mandelbrot (1983) proposed that when an

_{3}*n*-dimensional set of particles with fractal dimension

*D*is cut by an

_{n}*n*-1-dimensional surface, the fractal dimension of the particles viewed in that surface should be

*D*– 1 provided that the particles are isotropic in the missing dimension. Thus, an array of particles with

_{n}*D*= 2.5 cut by a planar surface should have

_{3}*D*= 1.5. This conjecture has been discussed by Sammis et al. (1987), Russ (1994, p. 301ff), and Tang and Marangoni (2006), among others.

_{2}An intuitive explanation of Mandelbrot’s conjecture is shown in Figure 2. In a random cut through a series of spheres of differing diameters, larger spheres are more likely to be intersected than smaller ones. In particular, a sphere of radius *R* is 10 times more likely to be intersected than one of radius *R*/10. Thus, every decrease in size of a factor of 10 leads to a tenfold decrease in the probability that a particular particle will be counted, decreasing the slope of the fractal plot by one.

In this paper, we examine the size distributions of 3D particles in 2D cuts using numerical simulations and experimental studies, and compare them to outcrop data from breccias and xenoliths.

## NUMERICAL SIMULATIONS

Data to be fit by Equation 1 are generally grouped into size classes and then displayed on a plot of log(*N*) versus log(*r*) (e.g., Turcotte, 1986). We use a slightly different but equivalent technique in which the length observations are ranked from largest to smallest and then plotted against s – 1, where *s* is the rank. This method has the advantages of eliminating the arbitrariness of histogram binning and more clearly showing the effects of left-hand and right-hand truncation (see below).

We generated fractally distributed (in edge length or diameter) populations of 3D objects (cubes, tetragonal prisms, spheres, and tetrahedra) with *D* = 2.5 using inverse transform sampling (Devroye, 1986, Chap. 2). If *F* is the cumulative distribution function of a given probability distribution, then inserting uniform random numbers drawn from the interval [0,1] (*U*[0,1]) into the inverse function *F*^{−1} yields random numbers drawn from the probability distribution described by *F*. The fractal distribution of Equation 1 is itself a cumulative probability distribution.

The general algorithm for the simulations is as follows:

(1) Generate a vector

*r*of fractally distributed lengths by inserting a vector of*n*random numbers drawn from*U*[0,1] into the function*x*^{–1/2.5}.(2) Generate a vector

*z*of*n*random numbers drawn from*U*[0,100], and distribute the*n*objects along the*z*-axis using these numbers.(3) Calculate the cross-sectional area of the objects cut by the plane

*z*= 50.

Following Turcotte (1986), we use the square root of the area (referred to below as eigenlength) as a linear measure of the object’s size, both in simulations and with natural data. Calculations were done in Matlab. With this algorithm, the minimum value of *r* approaches 1. Thus, in Equation 1, *C* = *n* – 1. The simulations are insensitive to the size of the interval drawn from (e.g., [0,100]) provided that it is large compared to the objects being studied.

Figure 3 shows the results of one simulation. One million cubes with fractally distributed edge lengths were placed with a face parallel to the *x-y* plane. Thus, if a given cube is intersected by the plane *z* = 50, the cross-sectional area of the intersection is equal to the full area of a face. The blue curve gives the distribution of edge lengths for the 3D objects, and the red curve for the intersection simulation. The intercept at *r* = 1 for the 3D objects is 10^{6}, since all 10^{6} objects were counted, whereas the corresponding intercept for the 2D intersections is ∼16,500, indicating that <2% were cut by the plane. Linear fits to these arrays have calculated slopes of –2.5 and –1.5, verifying that the generating algorithm is correct and that the Mandelbrot (1983) conjecture works for this system. If the cubes are randomly oriented (green curve), then the calculated fractal dimension is somewhat higher (1.62 in the simulation in Fig. 3) owing to a higher number of small objects that correspond to triangular and rectangular cuts far from the cube center. This value is ∼8% higher than that predicted from the *D* – 1 rule.

Spheres and tetrahedra present slightly different cases (Fig. 4). The linear portion of the 2D sphere curve in Figure 4 has a slope of –1.52, but there is a flat tail at low radii caused by spheres that were intersected far from their equators by the intersection plane. The tetrahedron simulation in Figure 4 consisted of 10^{6} regular tetrahedra oriented with a face parallel to the *x-y* plane. The 2D intersection curve for the tetrahedra also has a linear section with a flat left-hand tail; the linear portion has a slope of –1.55 owing to the bias toward small intersection areas toward the apex. For both of these objects, the *D*_{3} – 1 method is approximately correct.

Figure 5 shows the size distributions of 2D cuts through randomly oriented ellipsoids of varying aspect ratio, both prolate and oblate. For each curve, 10^{5} ellipsoids were generated with fractally distributed equatorial axis *a* and rotational axis *c* = *ka*, where *k* is the aspect ratio. For *k* = 1, the ellipsoids are spheres, and this simulation gives *D _{2}* = 1.50 as expected. For

*k*= 5, the calculated slope is only slightly different from –1.5, but for

*k*= 10 (cigar-like), it is substantially different (–1.69), and for

*k*= 20 (pencil-like), it is even higher (–1.9). For highly oblate ellipsoids with

*k*= 0.1, the calculated

*D*(–1.58) is again slightly greater than 1.5.

^{2}## SAMPLING OF FRACTAL DISTRIBUTIONS AND THE TRUNCATION PROBLEM

Measurement of objects drawn from a population with a fractal distribution is confounded by problems of truncation (Pickering et al., 1995). Left-hand (small-scale) truncation is related to the sensitivity of the measurement process; small objects are increasingly difficult to detect as their size decreases. Right-hand (large-scale) truncation occurs under several circumstances. For example, the maximum xenolith area that can be measured in a given exposure cannot exceed the area of the outcrop, and large objects that intersect the edge of the measurement area (such as the edge of a frame undergoing image analysis) are either omitted from analysis (as in this study), or their size is underestimated. If the larger values of a distribution are not sampled, then an array of particles that should plot as a straight line on a fractal plot will instead be convex up, falling below the nominal slope on the right and leading to an overestimate of the fractal dimension (Fig. 6; Pickering et al., 1995).

## DATA FROM PHYSICAL EXPERIMENTS

### Crushed Rocks and Minerals

As noted earlier, fragmented materials commonly have *D* ∼2.4 ± 0.3 in three dimensions (Fig. 1). Figure 7 shows data from an aphyric, indurated Miocene rhyolite from the western Mojave Desert that was crushed in a jaw crusher such that the largest particles were on the order of 1 cm. The resulting particles (72 g) were put through a stack of sieves, and the numbers in each size range were estimated using the method described by Glazner and Bartley (2006) with an assumed particle density of 2600 kg/m^{3}. The data array is linear with a slope of –2.51 (the largest and smallest size classes were discounted to minimize truncation issues). A medium-grained quartz monzonite from the Eocene Mount Princeton batholith of central Colorado was similarly processed, yielding *D _{3}* = 2.37. Thermal fragmentation of granite by Glazner and Bartley (2006) yielded

*D*∼2.2.

_{3}To examine a 2D cut through the crushed rhyolite, an aliquot was passed through a sieve with an opening of 1 mm, and then a small random aliquot was scooped up with a spatula and stirred into UV-curing epoxy (Norland 63) on a glass slide. The epoxy is viscous (∼2.5 Pa·s) and hardened rapidly, ensuring negligible particle settling and size fractionation. After hardening, this mount was polished and imaged on a scanning electron microscope using backscattered electrons, a high probe current, and high contrast. This allowed for unambiguous discrimination of the sectioned particles, which were then counted and measured using ImageJ software (http://imagej.nih.gov/ij).

Figure 8 shows that the largest particles have an eigenlength of ∼0.5 mm, and the total number counted was ∼1200. In the size range between ∼0.03 and 0.3 mm, a line with slope –1.49 fits the data well, but the number of smaller particles is far smaller than described by *D _{2}* = 1.5. Thus, in this example, although the 3D particle array has eigenlengths described by

*D*∼2.5 over a size range of ∼2.5 orders of magnitude, a 2D cut only follows the

_{3}*D*– 1 rule over approximately one order of magnitude.

A crushed calcite crystal displays a somewhat different size distribution. A calcite crystal 2 cm in longest dimension was struck lightly with a hammer three times, passed through a 1-mm sieve, and then mounted, imaged, and counted as described above. The resulting data array (Fig. 8B) is noticeably more convex-up than the crushed rhyolite and lacks the central linear segment displayed by the rhyolite particles. It is also noticeably more convex than a lognormal array. Crushed fluorite, another mineral with excellent cleavages, also shows a rounded size distribution (Fig. 8C), whereas crushed amphibole has a short linear segment with a slope of –2.00. The rounded array shapes for calcite and fluorite resemble those produced by dicing a cube by cuts parallel to the three principal face directions. Figure 9 shows 3D and 2D simulations of this process. The 3D simulation follows the method of Glazner and Bartley (2006); for the 2D simulation, rectangular prisms produced by this algorithm were distributed randomly between *z* = 0 and *z* = 50, and areas intersected by a plane at *z* = 25 were determined. Both arrays are continuously curved.

In contrast to the fractal particle array produced when a rock is crushed in a jaw crusher (Fig. 7), rock powder produced by extensive pulverization shows a curved size distribution (Fig. 10). Basalt sample P-36 (Glazner et al., 1991) was crushed in a jaw crusher and then powdered for 10 min in a tungsten carbide ball mill. A small aliquot of powder was shaken vigorously in water, and a small drop of this dilute suspension was placed on a glass slide for analysis under backscattered electrons, yielding full particle diameters. The resulting size distribution is curved and non-fractal (Fig. 10).

### Xenoliths and Breccias

In order to look for self-similarity in xenolith size distributions, we collected measurements of 950 xenoliths from the outer margin of the Tuolumne Intrusive Suite in Yosemite National Park, California. Abundant xenoliths occur in the granodiorite of Glen Aulin adjacent to metamorphic wall rocks near the southeastern corner of May Lake. These metamorphic and surrounding igneous rocks were mapped by Rose (1957), Bateman et al. (1983), and Taylor (2004), and studied geochemically by Gray et al. (2008) and Mills et al. (2009). As noted by Mills et al. (2009), xenoliths almost exclusively occur where granodiorite is in contact with schistose pelitic quartzite, which is the only foliated rock type in the metamorphic screen, and nearly all measured xenoliths were pelitic quartzite (a few percent were quartzite). Xenoliths with a minor axis length greater than 1 cm were measured in five areas (average area of 50 m^{2}) with abundant xenoliths and near-100% exposure. Eigenlength was calculated for each xenolith as the geometric mean of the long and short axes.

Figure 11 shows the size distribution of these eigenlengths. A line with a slope of –1.44 fits the data well over a size range from 30 cm to ∼4 cm. This indicates that over the relatively restricted size range measured, these xenoliths likely have *D _{3}* ∼2.4, consistent with the fragmentation studies summarized in Figure 1. The largest xenolith measured in this study was 1 m

^{2}in area, although xenoliths over 100 m long occur in the contact zone (Taylor, 2004; Glazner et al., 2004).

We determined *D _{2}* from a photograph of a brecciated fault surface (Fig. 12) and found a somewhat higher fractal dimension. The brecciated surface is from the Castle Mine fault in the Marble Mountains, Mojave Desert, California, a steeply dipping, curved normal fault that places early Miocene rhyolite on Jurassic granodiorite (Glazner and Bartley, 1990). It exhibits abundant polished fault surfaces with steep slickenlines, and brecciated particles are clearly evident on the fault surfaces (Fig. 12). Outlines of particles down to ∼0.5 mm across were traced and the resulting array analyzed as above with ImageJ. The size-distribution array (Fig. 13) has a pronounced linear section from ∼7 mm to 1 mm, with a slope of –1.87. The magnitude of the slope indicates that there are a greater number of small particles in each successively smaller size class than predicted if the

*D*were 2.5.

_{3}Fragments from a limestone breccia in Death Valley National Park, California, exhibit a particle size distribution in 2D that differs from both May Lake xenoliths and the Castle Mine fault breccia. This breccia (Fig. 14) occurs in lower Titus Canyon in limestone of the Cambrian Bonanza King Formation (Norris, 1985). The breccia likely formed by cave collapse based on upward termination of the deposit with no apparent relationship to faults in the range, and concentrically zoned calcite layers that fill voids and are probably speleothems (B.P. Wernicke, 2011, personal commun.). The data array from this breccia (Fig. 15) is conspicuously curved.

### Mafic Magmatic Enclaves

We measured long and short axes of 460 enclaves in a 500 m^{2} area of 100% glacially polished exposure in the Half Dome Granodiorite near Tenaya Lake, Yosemite National Park. All enclaves with a minor axis length of at least 1 cm were measured. The largest enclave measured was 70 × 38 cm in dimension, and the geometric mean of their aspect ratios is 2.4. The data array (Fig. 16) displays a linear section that covers approximately one order of magnitude with a slope yielding *D _{2}* ∼2.1.

## DISCUSSION

### Validity of the Mandelbrot Conjecture

The numerical simulations with simple geometric objects described above support the Mandelbrot conjecture that if a 3D set of objects with fractal length dimension *D _{3}* is cut by a 2D plane,

*D*of the intersected objects will be approximately

_{2}*D*– 1 for most geologic objects other than those that are highly prolate or oblate. Tang and Marangoni (2006) came to a similar conclusion in a numerical and experimental study of 2D sections through networks of crystalline fat.

_{3}### Geologic Objects

Geologic objects studied in 2D present a more complicated picture. A crushed rhyolite with a 3D size distribution (Fig. 7) that is best fit by a fractal dimension of ∼2.5, when viewed in a 2D section, shows the predicted fractal distribution with *D* ∼1.5 over a limited size range of approximately one order of magnitude (Fig. 8). The distribution tails off with a substantially smaller number of small particles than predicted by *D _{2}* = 1.5, and the largest particles are significantly smaller than predicted.

Xenoliths from the outer margin of the Tuolumne Intrusive Suite in Yosemite National Park yield a *D _{2}* of ∼1.44 (Fig. 11). As with the crushed rhyolite, the size-distribution plot is convex upward, tailing off for small particles and exhibiting a maximum particle size smaller than predicted by the linear portion of the curve. This paucity of large xenoliths may be an artifact of the areas chosen for study, which specifically avoided the enormous xenoliths that occur in the area (Taylor, 2004)—another case of right-hand truncation.

### The Comminution Limit

Measuring the abundance of small particles is problematic owing to limitations of observation (e.g., the resolution of optical techniques), the possibility that small particles are lost during sample preparation, and the effects of truncation. However, the smaller-than-expected number of small particles may instead be a result of a fundamental change in the mode of failure as the material deforms by ductile rather than brittle deformation mechanisms.

Kendall (1978) and Hagan (1981) showed that the mechanics of crack nucleation and propagation impose a limit to the size of particles that can be produced by breakage. Using fracture toughness and hardness data from Broz et al. (2006) and the relation of Hagan (1981), we calculate this comminution limit for quartz and orthoclase as ∼0.5 μm. Full physical data are not available for all materials of interest, but similar fracture toughness values for olivine (deMartin et al., 2004) and basalt (Balme et al., 2004) predict comparable comminution limits. For the extensively powdered basalt in Figure 10, the number of particles tails off dramatically below 1 μm even though the resolution of the images used to measure the particles was on the order of 0.1 μm. This result is consistent with a comminution limit comparable to the calculated value for silicates and with the observation that fine rock powders produced by natural processes, such as glacial flour, commonly have size distributions that dramatically tail off below ∼1 μm (Rosenbaum and Reynolds, 2004; however, see Chanudet and Filella, 2008), and with the common observation that powdering of a sample in a ball mill or similar device typically reduces rock fragment size to ∼1 μm regardless of how long the process is continued.

We simulated fractal fragmentation of cubes numerically, with and without a lower comminution limit, as follows. One hundred cubes with edge lengths drawn from *U*[1,100] were generated. At each step, a given cube was fragmented into eight sub-cubes with probability *p*, and this process was repeated for several steps. When no lower size limit is imposed, a fractal size distribution results (Fig. 17); this is the schematic fragmentation process outlined by Turcotte (1989), and it is insensitive to the starting distribution of edge lengths or number of starting cubes. However, if a lower (comminution) limit is imposed below which a cube will no longer fragment, the size distribution steepens dramatically as larger cubes are fragmented down to this limit.

### Dicing versus Fractal Fragmentation

We suggest that the three excellent cleavages of calcite and four of fluorite make their fragmentation process resemble dicing more than the nearest-neighbor crushing process proposed by Sammis et al. (1987). In a true fractal distribution, ever-smaller particles are increasingly abundant, whereas in dicing, ever-smaller particles are less and less likely to be formed because they only form where three pairs of closely placed dicing planes intersect. Thus, through-going cleavages may account for the curved data array determined for the crushed calcite and fluorite crystals (Fig. 8). The cleavages allow the crystal to fragment along intersecting fractures and likely signify a fragmentation mechanism that is fundamentally different from fractal fragmentation. Amphibole fragment data in Figure 8 are more difficult to interpret, but the plot does have a short linear section with *D _{2}* = 2.0. With only two good cleavages, amphibole likely exhibits a mix of fragmentation mechanisms.

The Titus Canyon breccia (Figs. 14 and 15) may exhibit a similar distribution due to pervasive fractures that dissect the western margin of the Grapevine Mountains (e.g., Snow et al., 1989). Rather than having been crushed by faulting, this breccia likely formed by collapse, a process that would exploit pervasive fractures. Alternatively, the Titus Canyon breccia may have originally had a fractal size distribution with *D* ∼2.5, but after formation small particles were washed out of the system. Water flow through a rubble pile preferentially moves the smaller particles, and such a process could lead to convex-up curvature on an otherwise linear size-distribution plot.

### Fragmentation in Igneous Systems

The decrease in fractal dimension caused by viewing particles on cut surfaces has important implications for statistical analysis of such size distributions. Glazner and Bartley (2006) stated that “in a particle group described by a fractal distribution with *D* = 2.5, for every 10 m block there should be >300 1 m blocks, 100,000 10 cm blocks, and >30,000,000 1 cm blocks. We are unaware that any pluton has been observed to contain millions of small xenoliths for every large one.” This reasoning is correct but misleading, because outcrop observations are 2D cuts through these distributions. If *D _{2}* = 1.5, then a relevant restatement would be “for every 10-m block there should be ∼32 1-m blocks, >1000 10-cm blocks, and >30,000 1-cm blocks.” Data in Figure 11 indicate that the xenolith size distribution near May Lake does reflect a fractal size distribution with

*D*∼2.5 over at least a limited size range.

_{3}It is unlikely that we undersampled xenoliths 1–4 cm in eigenlength because measurement was done directly on glacially polished slabs, and xenoliths a centimeter across are easily recognizable by texture and color. Extrapolation of the linear part of the curve in Figure 11 to 1 cm shows that there should be several thousand xenoliths in the data set between 4 cm and 1 cm in eigenlength, if the xenolith population follows the fractal distribution down to that size. Thus, xenoliths of this size range are underrepresented, possibly due to dissolution or disaggregation (e.g., Farris and Paterson, 2007). Figure 18 shows a sphere simulation; spheres were generated with *D _{3}* = 2.5 and were then decreased in radius by a constant value in order to simulate a constant linear rate of melting. Such a process preferentially affects the smaller particles, leading to a left-hand tailing off of the distribution.

Farris and Paterson (2007) studied xenolith size distributions in two Alaskan plutons using outcrop measurements and image analysis, and derived large *D _{2}* values with a mean of 2.3 (Fig. 1). They reasoned that the 2D nature of their measurements does not affect the fractal dimension because random cuts through spheres generally result in an intersection diameter that is a large fraction of the true diameter (McCammon, 1975). This reasoning is faulty for reasons outlined above (e.g., Fig. 2). It is not clear why their estimated

*D*values, which imply

_{2}*D*> 3 for four out of six measurements, are so high. For one pluton (Shaft Peak), the high number may be a result of the image analysis method used. Thresholding grayscale images is highly subjective unless the contrast between particles and background is high, as in Figure 8, and their example thresholded image (their fig. 10) appears to have broken many gray xenoliths into myriad tiny fragments. This would lead to an overestimate of

_{3}*D*.

The size distribution of mafic enclaves we analyzed is consistent with the hypothesis that dispersal of mafic magma into felsic magma is a fractal process, but only over a restricted size range (Holtz et al., 2004; Perugini et al., 2007). Holtz et al. (2004) found *D _{2}* = 1.8, whereas Perugini et al. (2007) derived

*D*∼2.5. This value implies a

_{2}*D*of 3.5, significantly higher than the fragmentation processes discussed above. The high

_{3}*D*values derived from our study (2.1) and by Perugini et al. (2007) indicate that there is a distinct difference between this process and the breakup of solid objects. The sharp decrease in enclave abundance at eigenlengths less than 6 cm is real; as with the xenolith study, enclaves were measured by close inspection of polished surfaces. In our study, nearly all measured enclaves have eigenlengths <50 cm, and the maximum measured eigenlength was ∼2 m, consistent with the Sierra Nevada Range as a whole (Pabst, 1928; Tobisch et al., 1997; Barbarin, 2005). This size distribution is comparable to the distribution of the injections that comprise individual dikes in the Jurassic Independence dike swarm of eastern California (Glazner et al., 2008). In mafic dike swarms worldwide, dikes thinner than ∼10 cm are rarely found (e.g., Jolly and Sanderson, 1995; Glazner et al., 2008), and well-exposed dikes are typically found to be highly composite. We speculate that if mafic enclaves form by disaggregation of mafic dikes into drops (e.g., Vernon, 1984; Barbarin, 2005), then their size distribution has upper and lower limits that are controlled by the aperture of the dikes that form them. Alternatively, the lower size limit may result from recrystallization and reaction with the surrounding granodiorite, making smaller enclaves hard to recognize.

_{2}## CONCLUSIONS

Numerical simulations and physical fragmentation experiments demonstrate that the fractal dimension of an array of particles with fractal dimension *D* is generally *D* – 1 in a two-dimensional cut. This results in a size distribution in outcrop or thin section that is highly skewed compared to the true 3D array. However, in systems where there is a persistent set of fractures, such as a crystal with good cleavage or a well-fractured rock, the resulting particle arrays are continuously curved on a size-distribution plot, and lack the small particles predicted by a fractal distribution. Natural examples of fragmentation always lack the infinite number of particles predicted by a true fractal array but are commonly linear, with *D _{2}* ∼1.5 (corresponding to

*D*∼2.5) over an order of magnitude or more. Extensively fragmented rocks and minerals typically reach a comminution limit on the order of 1 μm, below which further brittle fracture is mechanically unfavorable. The range of lengths covered by observations of natural phenomena is typically limited to a few orders of magnitude. Observations are constrained on the small end of the scale by limitations of optical techniques of observation and sample handling problems, and on the large end of the scale by sampling issues and truncation of large objects by the boundary of the observation area.

_{3}This work was supported by grants from the National Science Foundation to AFG (EAR-8816941, 8917291, 0336070, and 0538129). Chris Glazner helped with mathematical development. Data presented in this paper were collected over the course of several years, with the assistance of John Bartley, Drew Coleman, Kjell Lindgren, Bryan Law, Sam Coleman, Scott Hetzler, and Cecil Patrick, among many others. We thank Associate Editor Michael Cheadle for both an excellent review and efficient editorial handling, and two anonymous reviewers for helpful comments.