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The ability to engineer sophisticated two-dimensional tessellation organic nanoarchitectures based on triangular molecules and on-surface-synthesized covalent multimers is investigated using scanning tunneling microscopy. 1,3,5-Tris(3,5-dibromophenyl)benzene molecules are deposited on high-temperature Au(111) surfaces to trigger Ullmann coupling. The self-assembly into a semi-regular rhombitrihexagonal tiling superstructure not only depends on the synthesis of the required covalent building blocks but also depends on their ratio. The organic tessellation nanoarchitecture is achieved when the molecules are deposited on a Au(111) surface at 145 ° C . This halogen-bonded structure is composed of triangular domains of intact molecules separated by rectangular rows of covalent dimers. The nearly hexagonal vertices are composed of covalent multimers. The experimental observations reveal that the perfect semi-regular rhombitrihexagonal tiling cannot be engineered because it requires, in addition to the dimers and intact molecules, the synthesis of covalent hexagons. This building block is only observed above 165 ° C and does not coexist with the other required organic buildings blocks. [ABSTRACT FROM AUTHOR]

The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (Td) fullerene cages. Cages in the first series have 28n² vertices (n ≥ 1). Cages in the second (leapfrog) series have 3 x 28n². We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (Td) symmetry, these new polyhedra constitute a new class of "convex equilateral polyhedra with polyhedral symmetry". We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host's polyhedral symmetry. [ABSTRACT FROM AUTHOR]

In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure. [ABSTRACT FROM AUTHOR]

Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions. [ABSTRACT FROM AUTHOR]

There is increasing interest in two-dimensional and quasi-two-dimensional materials and metamaterials for applications in chemistry, physics and engineering. Some of these applications are driven by the possible auxetic properties of such materials. Auxetic frameworks expand along one direction when subjected to a perpendicular stretching force. An equiauxetic framework has a unique mechanism of expansion (an equiauxetic mode) where the symmetry forces a Poisson's ratio of − 1 . Hinged tilings offer opportunities for the design of auxetic and equiauxetic frameworks in 2D, and generic auxetic behaviour can often be detected using a symmetry extension of the scalar counting rule for mobility of periodic body-bar systems. Hinged frameworks based on Archimedean tilings of the plane are considered here. It is known that the regular hexagonal tiling, { 6 3 } , leads to an equiauxetic framework for both single-link and double-link connections between the tiles. For single-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found here to be equiauxetic: these are { 3. 12 2 } , { 4.6. 12 } , and { 4. 8 2 } . For double-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found to be equiauxetic: these are { 3 4. 6 } , { 3 2. 4.3. 4 } , and { 3.6. 3.6 } . [ABSTRACT FROM AUTHOR]

Euclidean tilings are constantly applied to many fields of engineering (mechanical, civil, chemical, etc.). These tessellations are usually named after Cundy & Rollett's notation. However, this notation has two main problems related to ambiguous conformation and uniqueness. This communication explains the GomJau-Hogg's notation for generating all the regular, semi-regular (uniform) and demi-regular (k-uniform, up to at least k = 3) in a consistent, unique and unequivocal manner. Moreover, it presents Antwerp v3.0, a free online application, which is publicly shared to prove that all the basic tilings can be obtained directly from the GomJau-Hogg's notation. [ABSTRACT FROM AUTHOR]

Beauty judgments for regular polygon tessellations were examined in two experiments. In experiment 1 we tested the three regular and eight semi-regular tilings characterized by a single vertex. In experiment 2 we tested the 20 demi-regular tilings containing two vertices. Observers viewed the tessellations at different random orientations inside a circular aperture and rated them using a numeric 1–7 scale. The data from the first experiment show a peak in preference for tiles with two types of polygons and for five polygons around a vertex. Triangles were liked more than other geometric shapes. The results from the second experiment demonstrate a preference for tessellations with a greater number of different kinds of polygons in the overall pattern and for tiles with the greatest difference in the number of polygons between the two vertices. Ratings were higher for tiles with circular arrangements of elements and lower for those with linear arrangements. Symmetry group p6m was liked the most and groups cmm and pmm were liked the least. Taken as a whole the results suggest a preference for complexity and variety in terms of both vertex qualities and symmetric transformations. Observers were sensitive to both the underlying mathematical properties of the patterns as well as their emergent organization. [ABSTRACT FROM AUTHOR]