Alice C. Niemeyer

1. Building in a resource-efficient manner is becoming increasingly important as sustainable construction practices gain prominence. One well-known approach in this context is the use of modular an...

Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He established that such an embedding exists if and only if the dual graph contains a particular subgraph. Here, strong embeddings are investigated motivated by the cycle double cover conjecture and the relation to triangulated surfaces. We provide a complete characterization of strong embeddings on the projective plane, the torus, and the Klein bottle in terms of a distinguished subset of Enami's subgraphs. This characterization not only deepens the structural understanding of graph embeddings on non-spherical surfaces, but also establishes a robust foundation for computing cycle double covers. As a direct consequence, we derive explicit criteria that determine when a graph does not admit a strong embedding on these surfaces-offering new tools for both theoretical analysis and algorithmic applications.

Topological interlocking assemblies (TIA) are arrangements of blocks such that rigid‐body motions of the blocks are fully constrained by their neighbours and a fixed frame. In this work, we investigate tubular interlocking structures derived from the sine curve and parametrised by several geometric design parameters. We analyse the behaviour of these parametrised tubular interlockings under various boundary conditions and examine how our proposed parameters influence the mechanical response. For this purpose, we first develop a simplified multibody dynamics formulation that enables an efficient exploration of how the design parameters of the block influence the load transfer within the assembly. To further corroborate these results, we perform several finite element simulations, which give insights into the mechanical behaviour of our proposed TIA. Our results show that the block geometry plays a decisive role in the mechanical performance of the corresponding TIA. We additionally discuss the problem of exploding TIAs and demonstrate that the TIA resulting from our Sine Block does not exhibit this behaviour. Lastly, we provide evidence that non-exploding TIAs possess better mechanical properties than exploding ones.

A topological interlocking assembly (TIA) consists of rigid blocks together with a fixed frame, such that all blocks are kinematically constrained and can therefore not be removed from the assembly. In recent years, these assemblies have gained a lot of interest due to their suitability for the construction of sustainable modular systems. We pursue a modular approach to construct interlocking blocks by exploiting the tetroctahedrille. This leads to the discovery of two interlocking blocks, which we call kitten and UFO, whose corresponding TIAs realise spiral forms. We show that given TIAs can be deformed continuously in order to obtain new TIAs. We also present a geometric application of the tetroctahedrille for approximating given geometric objects. Additionally, we discuss how physical models with incorporated magnets can be employed to motivate the construction of such geometries with special application in civil engineering.

A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. This conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a 3-connected cubic planar graph G has exactly one orientable circuit double cover if and only if G is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic.

In 1957, Steinhaus conjectured that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop, does not exist; this was later proven by Świerczkowski. We show that modifying the statement by requiring the tetrahedra of a chain to be isosceles results in the first examples of closed chains with all tetrahedra being congruent. As a result, we provide a census of toroidal polyhedra arising from closed chains consisting of up to 20 isosceles tetrahedra. Moreover, we establish the existence of an infinite family of toroidal polyhedra emerging from chains of isosceles tetrahedra. Additionally, we generalise the notion of chains of isosceles tetrahedra and therefore introduce clusters of isosceles tetrahedra along with a combinatorial tool called tetrahedral symbol to describe them. Finally, we exploit our methods to construct clusters of isosceles tetrahedra that yield polyhedra of higher genera.

Woven fabrics have a long history of study across the fields of art, mathematics, and mechanics. While weaves with symmetries in R2 have been extensively formalized, classified, and characterized, a systematic framework for representing and designing weaves in R3 remains absent. Despite their relevance to engineering applications—particularly in composite materials—volumetric weaves are often designed in an ad hoc manner, typically by stacking planar weaves and introducing trivial thread connections along the stacking axis. In this article, we establish a formal framework for volumetric weaves by defining them through the isometries of Bravais lattices and their corresponding Voronoi cells in R3. This approach provides a structured description of the design space for a specific family of volumetric weaves, which we call volumetric Bravais weaves. As an example of volumetric Bravais weaves, we analyze, cubic primitive weaves (cP-weaves) in detail as the simplest example within the volumetric Bravais weave framework. This example demonstrates how volumetric Bravais weave structures naturally emerge from 3D lattice isometries. Furthermore, we show that all possible cP-weaves can be systematically generated using a set of cP lattice isometries and cube isometries. Our findings reveal that even just the space of cP-weaves is at least one order of magnitude larger than that of conventional two-way 2-fold weaves, highlighting the potential of our approach for expanding the design space of volumetric weaves.