Grain shape analysis,

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Grain shape analysis, including shape entropy
function H.J. Hofmann 1994, 1995 (Jour. Sed. Res.
v. A64, p. 916-920 v. A65, p. 721-723)
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ABSTRACT Plotting of grain-shape data on a
conventional ternary composition diagram, with
long (L), intermediate (I), and short (S) axes of
grains as end members, has advantages over
triangular diagrams that employ geometric ratios
as primary scales. The isometric (linear) nature
of the conventional ternary scale facilitates
plotting and interpretation, and the plot
provides a basis for comparing the three
components over a common denominator (100).
Consequently, the graph preserves the information
on individual axial values (as a percentage of
their sum). The triangle serves as a
compilation diagram on which the L-I-S phase
space can be subdivided in any number of ways,
one of which is the use of selected contour lines
of percentage values. In addition, the graph is
capable of accommodating the various grain-shape
diagrams (e.g., those of Zingg, Krumbein, and
Sneed and Folk) by superposition this allows for
direct, graphic comparison of all such diagrams,
and evaluation of their relative efficacy for
shape classification. The percentage triangle
can be subdivided using a variety of additional
functions, such as those that express relative
axial uniformity (an entropy-like function), or
any other parameter derived from the basic axial
values. Employing a scale of percentage values of
S versus the form index (L-I) in Cartesian
coordinates, superimposed on a conventional
isometric composition diagram, yields a ternary
plot of great simplicity and high information
content.
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Conditions for axes L I S L - I - S

L I S 100
Long 55
Intermed. 35
Short 10
100
0
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Basic ternary composition diagram for compilation
of data
Hofmann, 1994, J. Sed. Res., vol. A64916-920
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Grain shapes plotted on ternary composition
diagram
Hofmann, 1994, J. Sed. Res., vol. A64916-920
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Zingg (1935) classification (replotted)
Hofmann, 1994, J. Sed. Res., vol. A64916-920
Hofmann, 1994, J. Sed. Res., vol. A64916-920
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Snead Folk (1958) classification (replotted)
Hofmann, 1994, J. Sed. Res., vol. A64916-920
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Hofmann (1994) classification (L-I form index)
Hofmann, 1994, J. Sed. Res., vol. A64916-920
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RELATIVE AXIAL UNIFORMITY Diagram (SHAPE
ENTROPY) Another index that can be used for
expressing the relative similarity or
dissimilarity of all three axes is an
entropy-like function from communications theory
(Shannon and Weaver 1949). The relative entropy
value Hr for a three-component system is
Hr - ?(pi ln pi)/Hm It is obtained by
multiplying the proportion, pi, of each component
axis (i L, I, S) by its natural logarithm, ln
pi, obtaining the sum of the resultant products,
and dividing by the maximum relative entropy Hm
possible for a three-component system, which is
Hm - 3 0.333 ln 0.333 1.0986 To
express this function on a percentage basis, the
value is multiplied by 100. The relative axial
uniformity (shape entropy) for a given grain thus
is Hr - 100 (pLong ln pLong)
(pIntermediate ln pIntermediate) (pShort ln
pShort)/1.0986 Values of this function can be
contoured at intervals, forming concentric lines
around the focus at the apex of the basal
right-angled compilation triangle, the point that
was selected as the standard for comparison. The
higher the entropy value, the more the three axes
are alike, reaching a maximum of 100 where all
three axes are equal. Shannon, C.E., and
Weaver, W., 1949, The Mathematical Theory of
Communication. Urbana, University of Illinois
Press, 125 p.
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Hofmann (1994) classification
(shape entropy function)
Hofmann, 1994, J. Sed. Res., vol. A64916-920
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APPLICATION The Hofmann shape entropy function
has been found to provide not only a more
accurate prediction of the settling rate of
ellipsoidal particles than previous equations,
but is also applicable to a much wider range of
Reynolds numbers. Le Roux, 1996, Sed. Geol.,
v. 101, p. 15-20 (p. 19).
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APPLICATION the shape entropy of Hofmann
(1994) appears to be the most representative
single measure of sphericity as applied to
predictions of settling velocity, because it is
valid for a wide range of particle shapes and
grain Reynolds numbers. Le Roux, 1997, J. Sed.
Res., v. 67, p. 527-530 (p. 530).
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APPLICATION The Hofmann shape entropy factor
is not only suitable for determining the settling
velocity of smooth, ellipsoidal objects (Le Roux
1996), or very irregular grains at low Reynolds
numbers (Le Roux 2002), but can also be used to
determine the settling velocity of natural
particles, yielding a mean accuracy of 94.9.
Le Roux, 2002, Jour. Sedimentary Research, v.
101, p. 363-366 (p. 366). See also Le Roux,
2002, Sedimentary Geology, v. 149, p. 237-243.
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APPLICATION These data sets, along with other
data recorded on placer gold grains, provide
interesting insight on the study of placer gold.
More than 7500 grains from 53 locations have been
analysed using this program thus far, as compared
to the most extensive previous quantitative
morphological study in the literature which
measured only the three axis lengths on 1502
grains from 60 locations (Townley et al., 2003).
This study uses the Hofmann (Hofmann, 1994) shape
parameter (or Hofmann shape entropy) as the
primary descriptor of shape. Crawford, E. K.,
Chapman, R. K., LaBerge, W. P., and Mortensen, J.
K., 2007, Developing a new method to identify
previously unrecognized geochemical and
morphological complexity in placer gold deposits
in western Yukon, in Emond, D. S., Lewis, L. L.,
and Weston, L. H., eds., Yukon Exploration and
Geology 2006, Yukon Geological Survey, p.
139-148. http//www.geology.gov.yk.ca/publications
/yeg/yeg06/10_crawford.pdf
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Conversion scale (ratios-to-percentages)
S
1
L
I
0
Hofmann, 1995, J. Sed. Res., vol. A65721-723
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GRAIN SHAPE Summary
Conditions for axes L I S L - I - S
S

L I S 100
I
L
L 55
I 35
S 10
100
0
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Stacked bar plot showing total range of axial
dimensions on percentage triangle extruded into
prism. The front and top panels are transparent
the dark sloping surfaces represent limiting
values for axes.
Vertical lines piercing the two dark boundary
surfaces represent the axial proportions of two
selected grain shapes whose values are also given
quantitatively in the basal triangle.
Hofmann, 1995, J. Sed. Res., vol. A65721-723