Wallpaper Maps

A wallpaper map is a conformal projection of a spherical earth onto regular polygons with which the plane can be tiled continuously. A complete set of distinct wallpaper maps that satisfy certain natural symmetry conditions is derived and illustrated. Though all of the projections have been published before, the family had not been characterized as a whole. Some wallpaper maps generalize to one-parameter subfamilies in which the sphere is pre-transformed by a conformal automorphism

Real-space recipes for general topological crystalline states

Topological crystalline states are short-range entangled states jointly protected by onsite and crystalline symmetries. While the non-interacting limit of these states, e.g., the topological crystalline insulators, have been intensively studied in band theory and have been experimentally discovered, the classification and diagnosis of their strongly interacting counterparts are relatively less well understood. Here we present a unified scheme for constructing all topological crystalline states, bosonic and fermionic, free and interacting, from real-space "building blocks" and "connectors". Building blocks are finite-size pieces of lower dimensional topological states protected by onsite symmetries alone, and connectors are "glue" that complete the open edges shared by two or multiple pieces of building blocks. The resulted assemblies are selected against two physical criteria we call the "no-open-edge condition" and the "bubble equivalence", which, respectively, ensure that each selected assembly is gapped in the bulk and cannot be deformed to a product state. The scheme is then applied to obtaining the full classification of bosonic topological crystalline states protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our real-space recipes give the complete set of topological crystalline states for bosons and fermions, and prove the boson case analytically using a spectral sequence expansion of group cohomology.Comment: 17+44 pages, 7+1 figures, 0+2 tables. The content is the same as the published version, but arranged differentl

Nullification functors and the homotopy type of the classifying space for proper bundles

Let G be a discrete group for which the classifying space for proper G-actions is finite-dimensional. We find a space W such that for any such G, the classifying space PBG for proper G-bundles has the homotopy type of the W-nullification of BG. We use this to deduce some results concerning PBG and in some cases where there is a good model for PBG we obtain information about the BZ/p-nullification of BG.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-46.abs.htm

A study of user perceptions of the relationship between bump-mapped and non-bump-mapped materials, and lighting intensity in a real-time virtual environment

The video and computer games industry has taken full advantage of the human sense of vision by producing games that utilize complex high-resolution textures and materials, and lighting technique. This results to the creation of an almost life-like real-time 3D virtual environment that can immerse the end-users. One of the visual techniques used is real-time display of bump-mapped materials. However, this sense of visual phenomenon has yet to be fully utilized for 3D design visualization in the architecture and construction domain. Virtual environments developed in the architecture and construction domain are often basic and use low-resolution images, which under represent the real physical environment. Such virtual environment is seen as being non-realistic to the user resulting in a misconception of the actual potential of it as a tool for 3D design visualization. A study was conducted to evaluate whether subjects can see the difference between bump-mapped and nonbump-mapped materials in different lighting conditions. The study utilized a real-time 3D virtual environment that was created using a custom-developed software application tool called BuildITC4. BuildITC4 was developed based upon the C4Engine which is classified as a next-generation 3D Game Engine. A total of thirty-five subjects were exposed to the virtual environment and were asked to compare the various types of material in different lighting conditions. The number of lights activated, the lighting intensity, and the materials used in the virtual environment were all interactive and changeable in real-time. The goal is to study how subjects perceived bump-mapped and non-bump mapped materials, and how different lighting conditions affect realistic representation. Results from this study indicate that subjects could tell the difference between the bump-mapped and non-bump mapped materials, and how different material reacts to different lighting condition