By studying self-avoiding polygons (SAPs) in (L, M )-tubes (a tubular sublattice of the simple cubic lattice) as a sequence of 2-spans, transfer matrices can be used to obtain theoretical and numerical results for these SAPs. As a result, asymptotic properties of these SAPs, such as pattern densities in a random SAP and the expected span of a random SAP, can be calculated directly from these transfer matrices. These same results can also be obtained for compact polygons, as well as SAPs under the influence of an external force (called compressed or stretched polygons). These results can act as tools for examining the entanglement complexity of SAPs in (L, M )-tubes. In this thesis, it is examined how transfer matrices can be used to develop these tools. The transfer matrix method is reviewed, and previous transfer matrix results for SAPs in (L, M )-tubes, as well as SAPs subjected to an external force, are presented. The transfer matrix method is then similarly applied to compact polygons, where new results regarding compact polygons are obtained, including proofs for a compact concatenation theorem and for a compact pattern theorem. Also in this thesis, transfer matrices are actually generated (via the computer) for relatively small tube sizes. This is done for the general case of SAPs in (L, M )-tubes, as well as for the compact and external force cases. New numerical results are obtained directly from these transfer matrices, and a new algorithm for generating polygons is also developed from these transfer matrices. Compact polygons are actually generated (via the computer) for relatively small tube sizes and spans by using the developed polygon generation algorithm, and new numerical results for pattern densities and limiting free energies are obtained for stretched and compressed polygons.

This thesis explores the procedural generation of 3D buildings from the floor plan all the way to the fa¸cade and building’s details such as doors, windows, and roof. Through this, the study explores several techniques and approaches to create different layers of the building generation pipeline. The focus is on implementing a set of algorithms that, when running sequentially, are able to create complete 3D buildings in a short time (so they can be used in online generation), could be used in any open-world game, and with a limited count of triangles per building. Furthermore, the tool provides a clear and easy-to-use interface for designers in Unity, where they can interact with the multiple parameters of each building layer, giving designers a high degree of controllability. The tool is evaluated using the resulting buildings based on different metrics and how individual changes to different parameter starting from a template affect the output of the generator in terms of the metrics and the resulting building. The result from the analysis show that the polygon mathematics is well suited for generating 3D buildings for games.