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Here we investigate a variety of ways to represent polytopes and related objects. We define a class of posets, which includes all abstract polytopes, giving a unique representative among posets having a particular labeled flag graph and characterize the labeled flag graphs of abstract polytopes. We show that determining the realizability of an abstract polytope is equivalent to solving a low rank matrix completion problem. For any given polytope, we provide a new construction for the known result that there is a combinatorial polytope with a specified ridge that is always projectively equivalent to the given polytope, and we show how this makes a naturally arising subclass of intractable problems tractable. We give necessary and sufficient conditions for realizing a polytope's interval poset, which is the polytopal analog of a poset's Hasse diagram. We then provide a counter example to the general realizablity of a polytope's interval poset. / Mathematics

This qualitative research seeks to identify and understand the beliefs of 10 high school geometry teachers that help resolve apparent inconsistencies between their espoused and enacted beliefs. Data was collected using an initial interview, classroom observations, and a follow-up interview to gather evidence of teacher beliefs based on what they say, do, and intend respectively. Open coding, analytical coding, cluster identification, coding memos, and analytical memos were used to analyze the data and write summaries of the teachers' explanatory beliefs with beliefs as the unit of analysis. It was identified that teachers consistently and inconsistently enact their espoused beliefs, but there are also instances when teachers both consistently and inconsistently enact particular espoused beliefs. This endeavor necessitates a shared understanding of terms, and it was found what it means to "understand" needs to be clarified with a definition and examples from teachers. When teachers appear to not enact their espoused beliefs, explanatory beliefs were pinpointed that resolve the conflict and found the explanatory beliefs exist in at least seven macro clusters. These explanatory beliefs interact with espoused beliefs by overriding, limiting, or preventing the espoused beliefs to resolve the apparent inconsistency in teachers espoused and enacted beliefs. The explanatory beliefs with limiting and overriding interactions were found to coexist for some teachers around a teaching practice as overriding interactions are connected to constraints on the classroom whereas limiting interactions are not. It was also found that belief clusters are nested within clusters of beliefs, and these clusters allow for beliefs to cluster in isolation in different ways. This work also shows empirically that some geometry teacher beliefs are socially constructed due to the presence of common cultural artifacts and influence from mathematics teacher educators. This work has implications and future research directions in the areas of using beliefs as the unit of analysis, mapping teacher's belief systems, considering the social construction of beliefs and role of community, connecting beliefs to specific teaching practices, and educating teachers. / Doctor of Philosophy / This research seeks to understand and interpret the beliefs of 10 high school geometry teachers that resolve apparent inconsistencies between what teachers say they believe and what they do in the classroom. Data was collected using an initial interview, classroom observations, and a follow-up interview to gather evidence of teacher beliefs based on what they say, do, and intend respectively. It was identified that teachers consistently and inconsistently enact their stated beliefs, but there are also instances when teachers both consistently and inconsistently enact their stated beliefs. When teachers appear to inconsistently enact their stated beliefs, it was found that teachers have logical reasons why they do so, and these reasons relate to specific teaching practices. It was also found that teacher beliefs interact with each other in different ways. Teachers' beliefs can limit or prevent the enaction of their other beliefs. In addition, school level constraints can override the enaction of some teacher beliefs. This research shows that some beliefs are held by different teachers from vastly different schools which suggests that some geometry teacher beliefs are held socially. The findings from this research have implications for teacher education.

The design of this architecture academy for one hundred students is an exploration of educational function, geometric construction with digital drawing tools, perspectival manipulation, and minimalist architectural aesthetics. Within the school are studio spaces, a shop, a library, a lecture room, faculty studies, and outdoor building yards. Itself technologically derived and constructed, this project fits into the framework of the modernist architectural movement. But like a wayward atom in an otherwise rigidly determined crystalline matrix, the building's form inserts itself into this stanchion at a point intended to commence a generative rift of new possibility. The goal for the institution is to foster an environment of exploration and progressive innovation through exchange and collaboration. This charge would fall squarely on the faculty of course, but hopefully the building itself would act as a catalyst for the search. Through its programmatic organization as well as its perspective-warping angular form, the building is designed to encourage or at least accommodate a participatory, dialectic way of studying and making architecture. Open studio spaces and the proximity and availability of the faculty studies are intended to foster ease of communication and collaborative interchange within the academy. Exchange of knowledge across multiple levels and points of view would occur both intellectually and spatially. The specific outcome of such a dialectically educational environment cannot be accurately predicted, and therein lays the vibrant potential of the proposal. The rigid predictability of enlightenment rationalism has outlived its modality and a new model is needed in order to move forward. The goal of the contemporary architecture academy should not be to establish the primacy of one mode of operation but to invite a plurality of approaches to the table. / Master of Architecture

It is in human nature to preserve things and objects from the past, study, enjoy and cherish our history. This need to learn from and cherish the objects from the past has resulted in the development and evolution of spaces such as museums where people can come and see these objects, either to know or learn something or out of personal interest and curiosity. Somewhere in all this, is architecture from the past taken for granted? A lot of the prominent historic buildings have been well preserved and are known to people. But at the same time there are numerous historic structures, story tellers from the past, being ignored and even trampled upon. Should we not look at these also as valuable objects that have to say so much about our social, cultural and technological past? Do they need a museum space as well? Can architecture be housed and preserved in a museum? Or maybe become a museum, displaying itself, allowing people to experience it from outside and within. Watson's Hotel is one such historical building that lies today unnoticed, uncared for, decaying and falling apart. My thesis is an intervention into this urban situation. The goal of the design has not been just preservation but rather an elevation or celebration of the structure, bringing forth its true nature that lies in its structural framework, a cast iron grid of columns and beams. It aims to highlight this essential core of the building by revealing the grid in different spatial conditions. There is also a constant wish to tie the structure back to its surroundings, to bring back the dialogue that the building shared with its surroundings in the past. The structural framework is revealed and experienced in different spatial conditions achieved with the help of geometry, light and material, surfaces added in and around it, and the grid runs through all these elements bold, undisturbed and uninterrupted. / Master of Architecture

Thesis (Ed.M.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2999-01-01

Thesis (Ed.M.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2999-01-01

Thesis (Ed.M.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2999-01-01