Related papers: Measuring Shape with Topology
We extend a formal framework that previously derived time from the multifractal structure of biological lineages (Hudnall \& D'Souza, 2025). That work showed that time itself is multifractal -- not a universal background dimension, but an…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
For many materials, a precise knowledge of their dispersion spectra is insufficient to predict their ordered phases and physical responses. Instead, these materials are classified by the geometrical and topological properties of their…
The root is an important organ of a plant since it is responsible for water and nutrient uptake. Analyzing and modelling variabilities in the geometry and topology of roots can help in assessing the plant's health, understanding its growth…
Mathematical morphology is a theory and technique to collect features like geometric and topological structures in digital images. Given a target image, determining suitable morphological operations and structuring elements is a cumbersome…
Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been…
In this paper we present a mixed projection- and density-based topology optimization approach. The aim is to combine the benefits of both parametrizations: the explicit geometric representation provides specific controls on certain design…
The precise theoretical characterization of a fractionalized phase in spatial dimensions higher than one is through the concept of ``topological order''. We describe a physical effect that is a robust and direct consequence of this hidden…
3D shape creation and modeling remains a challenging task especially for novice users. Many methods in the field of computer graphics have been proposed to automate the often repetitive and precise operations needed during the modeling of…
Algorithmic fractal dimensions -- constructs of computability theory -- have recently been used to answer open questions in classical geometric measure theory, questions of mathematical analysis whose statements do not involve computability…
We present a robust method to find region-level correspondences between shapes, which are invariant to changes in geometry and applicable across multiple shape representations. We generate simplified shape graphs by jointly decomposing the…
Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for…
Measurement plays a quintessential role in the control of quantum systems. Beyond initialization and readout which pertain to projective measurements, weak measurements in particular, through their back-action on the system, may enable…
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…
The morphometric approach is a powerful ansatz for decomposing the chemical potential for a complex solute into purely geometrical terms. This method has proven accuracy in hard spheres, presenting an alternative to comparatively expensive…
Context. A porous and/or fractal description can generally be applied where particles have undergone coagulation into aggregates. Aims. To characterise finite-sized, porous and fractal particles and to understand the possible limitations of…
Here we propose a new method for the classification of texture images combining fractal measures (fractal dimension, multifractal spectrum and lacunarity) with local binary patterns. More specifically we compute the box counting dimension…
The science of fractography revolves around the correlation between topographic characteristics of the fracture surface and the mechanisms and external conditions leading to their creation. While being a topic of investigation for…
The local structure of a fractal set is described by its dimension $D$, which is the exponent of a power-law relating the mass ${\cal N}$ in a ball to its radius $\epsilon$: ${\cal N}\sim \epsilon^D$. It is desirable to characterise the…
We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties…