Thesis advisor: Keerthi Madapusi Pera / We study the necessary and sufficient conditions for a Newton stratum on a Shimura variety of Hodge type to admit a hypersymmetric point, more explicitly by showing that hypersymmetric points arise as the image of points in the basic stratum of a smaller Shimura variety. We compare this condition with the combinatorial conditions proved by Chai-Oort and Zong in certain PEL-type cases. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.