The research summarized in this thesis consists essentially of two parts. In the first, we generalize a coloring theorem of Baxter about triangulations of the plane (originally used to prove combinatorially Brouwer's fixed point theorem in two dimensions) to arbitrary dimensions and to oriented simplicial and cubical pseudomanifolds. We show that in a certain sense no other generalizations may ...

Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of 'noncommutative topology', agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity stri...

Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of ‘noncommutative topology’, agreeing with some results of noncommutative geometry but lacking the metric aspects of C∗-algebras. Here, we make such similarity stri...

We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). Thus we obtain the first instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms the existence conjecture of Hetyei [17, Conj. 2, p. 325]. More systematically, we prove that every normal crossing...

This paper proposes a new cubical space model for the representation of continuous objects and surfaces in the n-dimensional Euclidean space by discrete sets of points. The cubical space model concerns the process of converting a continuous object in its digital counterpart, which is a graph, enabling us to apply notions and operations used in digital imaging to cubical spaces. We formulate a d...

This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topological ...

It is shown that the cubical nerve of a strict omega-category is a sequence of sets with cubical face operations and distinguished subclasses of thin elements satisfying certain thin filler conditions. It is also shown that a sequence of this type is the cubical nerve of a strict omega-category unique up to isomorphism; the cubical nerve functor is therefore an equivalence of categories. The se...

Having addressed the issue ofgenerating octrees from raster scan efficiently using parallel computation, it is worthwhile to ensure that the encoding is accomplished without much loss of information. Knowing that each cube usually stores an average ofthe intensity value in the cubical subspace, the encoding could lead to a "blocky" (as against ''patchy'') solid during regeneration. Retaining th...

Cell shape in vitro can be directed by geometrically defined micropatterned adhesion substrates. However conventional methods are limited by the fixed micropattern design, which cannot recapitulate the dynamic changes of the cell microenvironment. Here, we manipulate the shape of living cells in real time by using a tightly focused pulsed laser to introduce additional geometrically defined adhe...

This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra. In this framework it is easy to compute with osculating circles for the description of local curvature. Also methods for the fitting of spheres as well as bounding spheres are presented. In a nutshell, this paper provides a starting point for ...