Handwritten digit images lie in a high-dimensional pixel space but exhibit strong geometric and statistical structure. This paper investigates the latent organization of handwritten digits in the MNIST dataset using three complementary dimensionality reduction techniques: Principal Component Analysis (PCA), Factor Analysis (FA), and Uniform Manifold Approximation and Projection (UMAP). Rather than focusing on classification accuracy, we study how each method characterizes intrinsic dimensionality, shared variation, and nonlinear geometry. PCA reveals dominant global variance directions and enables high-fidelity reconstructions using a small number of components. FA decomposes digits into interpretable latent handwriting primitives corresponding to strokes, loops, and symmetry. UMAP uncovers nonlinear manifolds that reflect smooth stylistic transitions between digit classes. Together, these results demonstrate that handwritten digits occupy a structured low-dimensional manifold and that different statistical frameworks expose complementary aspects of this structure.