Shape plays a fundamental role across all organisms at all observable levels. Molecules and proteins constantly fold and wrap into intricate designs inside cells. Cells arrange into elaborate motifs to form sophisticated tissues. Layers of different tissues come together to form delicate vascular systems that sustain leaves. Each of these tissues evolved as part of a distinct branch of the ever-growing tree of life. From micro-biology to macro-evolutionary scales, shape and its patterns are foundational to biology. Measuring and understanding the shape is key to extracting valuable information from data, and push further our insights.Shape is too complex to be comprehensively tackled with traditional methods. Landmark-based morphometrics fail if there are not enough homologous points shared across all sampled individuals. Elliptical Fourier Descriptors are not suitable for 3D data. These limitations are especially pressing when we combine our plant visualizations with X-ray Computed Tomography (CT) technology to also record the sophisticated internal structure of stems, seeds, and fruits.℗ Here, I study the potential of Topological Data Analysis (TDA) for plant shape quantification. TDA is a combination of different mathematical and computational disciplines that seeks to describe concisely and comprehensively the shape of data in a general setting. In very succinct terms, TDA consists of two basic ingredients. First we think the data as a collection of points. Second, we define a notion of distance between every pair of points. The points could be atoms, biomolecultes, cells' nuclei, image pixels, or an organism itself. Distances between points could be the Euclidean, geodesic, genetic, or correlation-based. Once we have data points and distances, we merge systematically the points, starting with those that are closer to each other. The key idea is to keep track of distinct blobs, loops, and voids that form and disappear as we merge several points. This versatile idea is not constrained to a particular dimension or set of landmarks, which makes it ideal to compare a vast array of possible different shapes.In this work, we will explore new techniques for mathematical plant phenotyping by studying three concrete cases. For the first case, we digitally extract the totality of shape information from X-ray CT scans of tens of thousands of barley seeds. With the Euler Characteristic Transform, topological and traditional morphological descriptors of the seeds are then used to successfully characterize different barley varieties based solely on the grain shape. This result later enables us to deduce potential genes that contribute to distinct morphology, bridging the phenotype with its genotype. A future goal is to link these genes to climate adaptability to breed better crops for an ever changing weather. For the second case, we use directional statistics and persistence homology to model the shape and distribution of citrus and their oil glands. This leads us to a novel path to explore developmental constraints that govern novel relationships between fruit dimensions from both evolutionary and breeding perspectives. For the third case, we comprehensively measure the shape of walnut shells and kernel. Combining novel size- and shape-specific descriptors, we explore the relationship between shell morphology and traits of commercial interest such as the easiness to remove the kernel intact or the integrity of the shell after being cracked. From the perspective that all data, whether phenotypic or genotypic, has shape, TDA can extract the totality of morphological information. We have interest applying this approach to more crops, to more plant biology inspired datasets, and to large-scale gene expression and population genetic data.
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