Particle Form (Sphericity)

Particle form, commonly expressed as sphericity, describes the gross overall shape of a sediment grain - specifically, how close in proportions its three major axes are to one another. ^[1] It is the first-order aspect of particle shape, meaning it describes the overall outline of the grain before the finer details of corner sharpness or surface texture are considered. A grain with roughly equal long, intermediate, and short axes approaches a sphere and is said to have high sphericity; a grain whose axes differ considerably in length has low sphericity.

Measuring Sphericity

Krumbein’s Formula

Krumbein (1941) developed a mathematical formula that captures the relationship between the three axes and yields a value of 1 for a perfect sphere. ^[1] Less spherical particles receive lower, fractional values. The formula gives a single number that allows quantitative comparison of grain forms across samples, though in practice the measurement is laborious enough that many workers prefer visual estimation.

Zingg Shape Classification

Zingg (1935) proposed a classification based on two shape indices - D_I/D_L and D_S/D_I - plotted against each other on a bivariate diagram to define four shape fields. ^[1]

The four Zingg fields are: ^[1]

• Oblate (disc): short axis much shorter than the other two - flat, disc-like grains.
• Equant (sphere): all three axes roughly equal - compact, blocky grains.
• Bladed: intermediate shape, neither strongly disc-like nor roller-like.
• Prolate (roller): long axis much longer than the other two - elongated, roller-shaped grains.

An important consequence of the Zingg classification is that particles of quite different form can receive the same mathematical sphericity value from the Krumbein formula. Lines of equal sphericity cut across Zingg shape fields, meaning a disc-shaped particle and a roller-shaped particle can both score the same sphericity even though their actual forms are nothing alike. This reveals a limitation of any single-number summary of grain shape: it collapses two distinct geometric properties into one value.

Fourier Shape Analysis

Both sphericity and the Zingg classification are relatively coarse descriptors. A more rigorous approach uses Fourier analysis - a mathematical method that represents a periodic function as an infinite series of summed sine and cosine terms. ^[1]

The grain outline is treated as if it were cut and unrolled into a wave-like trace. That trace is then decomposed into a series of simpler periodic components called harmonics - each with its own amplitude and frequency. Adding the harmonics together reconstructs the full outline shape. In practice, the grain periphery is digitized by projecting it onto a grid and recording intercepts manually or with an automatic digitizer; computer software then reduces the digitized data into the harmonic terms. ^[1]

The results of Fourier analysis capture aspects of both sphericity and roundness simultaneously, making it a more sensitive and complete description of shape than either property alone. Applications include tracing the provenance of sediment grains and characterizing shape signatures of grains from specific depositional environments.

Significance and Limitations

Sphericity in sedimentary deposits is primarily controlled by the original shapes of grains in the source rock. ^[1] Transport can modify the form of gravel-sized grains through abrasion and breakage, but sand-sized grains are affected far less.

Sphericity has two clear physical effects: ^[1]

• Settling velocity: spherical particles settle faster than non-spherical particles of equivalent mass.
• Transportability by traction: spheres and roller-shaped pebbles roll more readily than disc- or bladed-shaped pebbles.

Despite these effects on transport physics, sphericity alone has not proven reliable for identifying depositional environments. Empirical studies show that sphericity differences between environments exist but are not distinctive enough to allow confident environmental discrimination. The measurement is therefore most useful in combination with other textural properties, particularly roundness and grain-size data.

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