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Particle Geometry Space: A Circumdiameter Interpretation

This paper enhances the Particle Geometry Space (PGS) framework by demonstrating that the particle circumdiameter, a key maximum dimension, can be accurately inferred from the y-axis intercept of the PGS plot, thereby enabling the direct estimation of maximum particle size from specific surface and volume without explicit length measurements.

Original authors: Dipak Dahal, Seung Jae Lee

Published 2026-09-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Dipak Dahal, Seung Jae Lee

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Every grain of sand, every pebble in a riverbed, and every crushed stone in a road base is a three-dimensional object with its own unique personality. These particles are not just tiny rocks; they have a specific size, a total amount of surface area, a volume, and a distinct shape that ranges from nearly perfect spheres to jagged, irregular shards. For engineers and scientists who study how these materials behave—how they pack together, how they flow, or how they support heavy loads—understanding these individual geometric traits is essential. The way a pile of gravel settles or a road holds up under traffic depends heavily on whether the stones inside are round and smooth or sharp and elongated. For a long time, researchers have tried to measure these traits, but they often looked at size and shape as separate problems, missing the bigger picture of how they work together as a single, unified system.

To solve this, scientists developed a way to map all these traits into a single analytical space, a kind of coordinate system where every particle can be plotted based on its volume and its surface area relative to that volume. This framework, known as the Particle Geometry Space, allows researchers to see how shape, size, and surface area relate to one another mathematically. However, this map had a gap. While it could describe the volume of a particle and its surface area, it could not explicitly account for the maximum length of the particle—the distance from one end to the other, or the size of the smallest sphere that could completely enclose it. This missing piece was crucial because the maximum length of a particle often dictates how it fits into a mixture, yet the existing map could only estimate size based on volume, which fails to capture how long or stretched out a particle might be.

In a recent study, researchers at Florida International University set out to fill this gap and complete the map. They asked a simple but profound question: what does the missing dimension in their geometric space actually represent? By analyzing a massive collection of 2,700 different particles, they discovered that the missing coordinate corresponds directly to the particle's maximum length. To reach this conclusion, the team examined a diverse array of materials. Their dataset included real-world samples such as limestone and granite gravel from the United States, crushed quartz sand from Wisconsin, and railway ballast from South Africa and Portugal. They also looked at industrial by-products like steel slag and even created thousands of perfect, theoretical shapes using computer software, ranging from classic geometric solids to highly elongated, needle-like forms.

The researchers plotted the specific surface area and volume of each of these 2,700 particles on their map. They found that when they looked for the mathematical point that represented a particle's maximum length, it aligned almost perfectly with the particle's circumdiameter. The circumdiameter is simply the diameter of the smallest sphere that can wrap around a particle, touching its farthest points. The study revealed a strikingly strong connection between the theoretical length derived from the map and the actual physical maximum length of the particles. The data showed that for every type of particle they tested, from a perfect cube to a jagged piece of coal, the value calculated from the map was a nearly exact match for the physical maximum length. This means that scientists can now determine how long a particle is, even without physically measuring the distance between its two farthest points, simply by knowing its volume and surface area.

This discovery does more than just add a new number to a chart; it fundamentally changes how we can describe the geometry of granular materials. Before this work, if a researcher wanted to know the maximum length of a particle, they often had to perform a direct measurement, which can be difficult for irregular, jagged stones. Now, the study shows that this length can be inferred directly from the particle's volume and surface area. The researchers also explored how this new understanding connects to older ways of describing particle shape. They found that a classic measure of how "sphere-like" a particle is, known as sphericity, is actually a combination of two things: the particle's overall form and a smaller, residual factor related to its roundness. The study suggests that while the overall shape of a particle is the dominant factor in how it behaves, the specific roundness of its edges plays a secondary, though still important, role.

The implications of this work are practical and immediate for anyone working with granular materials. By incorporating the maximum particle length into the unified geometric framework, engineers can now predict how a mixture of stones or sand will behave with greater accuracy. They can estimate the maximum dimension of a particle without needing to measure it directly, which simplifies the process of characterizing complex mixtures. The study confirms that the relationship between a particle's volume, its surface area, and its maximum length is consistent across a vast range of shapes and sizes. Whether the material is a natural river stone or a manufactured industrial product, the geometric rules hold true. This unified approach allows for a more holistic view of particle behavior, treating size, shape, and surface characteristics not as isolated facts, but as interconnected features of a single, coherent system.

Ultimately, this research provides a clearer, more complete picture of the microscopic world that builds our macroscopic infrastructure. It bridges the gap between abstract mathematical models and the physical reality of the stones and grains that make up our roads, foundations, and natural landscapes. By identifying the missing piece of the geometric puzzle, the researchers have given the scientific community a powerful new tool to understand and predict the behavior of the materials that hold our world together. The study does not claim to have solved every problem in granular mechanics, but it has successfully defined a previously unknown relationship, turning a theoretical gap into a practical method for measuring the unseen dimensions of the particles that surround us.

Technical Summary: Particle Geometry Space: A Circumdiameter Interpretation

Problem Statement
The Particle Geometry Space (PGS) is an analytical framework designed to unify the representation of three-dimensional particle attributes, including shape ( $\beta$ ), surface area ( $A$ ), volume ( $V$ ), specific surface ( $A/V$ ), and size ( $D$ ). While the original PGS framework successfully integrated these attributes into a single log-log space, it contained a significant limitation: it did not explicitly incorporate a maximum particle dimension measurable as a length, such as the circumdiameter ( $L$ ). The existing size parameter ( $D$ ) within the PGS is a volume-equivalent diameter, which differs fundamentally from the maximum linear dimension of a particle. Consequently, the geometric attribute corresponding to the slope of -2 in the PGS (where $k=2$ in the generalized expression) remained undefined, leaving a gap in the framework's ability to holistically characterize particle geometry, particularly regarding particle length and form.

Methodology
To address this gap, the authors investigated the physical and geometric meaning of the undefined $k=2$ attribute. The study utilized a comprehensive dataset of 2,700 three-dimensional particles sourced from ten distinct origins to ensure broad morphological coverage. The dataset included:

Natural Particles: Scanned mineral particles (limestone, granite, quartz, coal, railway ballast) from various global locations.
Theoretical Solids: Classical polyhedra (Platonic, Archimedean, Catalan, and Johnson solids) and author-created synthetic solids (including elongated cylinders, capsules, and prisms).
Data Sources: The analysis combined previously published 3D scanning data, publicly available digital repositories, and theoretical models generated via Mathematica.

The methodology involved analyzing the relationship between the newly defined particle length parameter ( $\mathcal{L}$ ), derived from the PGS formulation for $k=2$ , and the particle circumdiameter ( $L$ ). The authors also examined the relationship between Wadell's true sphericity ( $S$ ), the alternative sphericity measure ( $\psi = D/L$ ), and the shape index ( $M$ ) introduced by Su et al., utilizing regression analysis to determine correlations and geometric interpretations.

Key Contributions and Results
The study yields several key findings regarding the geometry of particles and the structure of the PGS:

Definition of the $k=2$ Attribute: The geometric attribute corresponding to the slope of -2 in the PGS is identified as a new particle length parameter, $\mathcal{L}$ . Mathematically, $\mathcal{L}$ is defined as $(A/V)^2 V$ , which simplifies to a "shape-factored" particle size.
Approximation of Circumdiameter: Through the analysis of 2,700 particles, the study demonstrates a strong correlation ( $R^2 = 0.99$ ) between the derived parameter $\mathcal{L}$ and the particle circumdiameter ( $L$ ). Specifically, the y-axis intercept at $A/V = 1$ associated with a slope of -2 corresponds to $6\pi L$ . This allows for the direct estimation of the maximum particle dimension ( $L$ ) using only specific surface ( $A/V$ ) and volume ( $V$ ) via the approximation $L \approx A^2 / (6\pi V)$ , without explicitly measuring the distance between the two farthest points of a particle.
Decomposition of Sphericity: The study provides a new interpretation of Wadell's true sphericity ( $S$ ). It is decomposed into two multiplicative components: a particle form component ( $MS$ ) and a residual shape component ( $MS^3$ ). The analysis reveals that $MS^3$ remains statistically close to unity (mean value of 1.05), indicating that variations in $S$ are governed primarily by particle form ( $D/L$ ), with only a modest contribution from residual shape characteristics (such as roundness).
Superiority of Shape Index $M$ : The relationship between the shape index $M$ and the form parameter $\psi$ ( $D/L$ ) is found to be significantly stronger ( $R^2 = 0.97$ ) than the relationship between true sphericity $S$ and $\psi$ ( $R^2 = 0.76$ ). This suggests that $M$ is a more direct measure of particle form than $S$ .

Significance and Claims
The paper claims that incorporating the circumdiameter into the Particle Geometry Space resolves a critical limitation of the original framework. By defining the $k=2$ intercept as a proxy for the maximum particle dimension, the extended PGS now comprehensively represents all key 3D particle attributes: shape, surface area, volume, specific surface, size, and length.

The authors assert that this unified representation eliminates the need to treat particle size, length, and shape as separate, isolated variables. Instead, it offers a mathematically consistent framework for investigating the combined effects of particle geometry on macroscopic granular material behavior. Furthermore, the ability to infer the maximum particle dimension ( $L$ ) directly from surface area and volume measurements provides a practical alternative to direct length measurement, which can be difficult for irregular particles. The study modestly concludes that while the residual shape component contributes to sphericity, the primary driver of variation in Wadell's sphericity is particle form, a finding clarified by the geometric mean decomposition of $S$ .

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