Runcinated tesseracts

In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract. There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations. The runcinated tesseract or (small) disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron. The runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra (vertex figures), cubes (face prisms), and triangular prisms (edge figure prisms). The same process applied to a 16-cell also yields the same figure. The Cartesian coordinates of the vertices of the runcinated tesseract with edge length 2 are all permutations of: Eight of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces. The runcinated tesseract can be dissected into 2 cubic cupolae and a rhombicuboctahedral prism between them. This dissection can be seen analogous to the 3D rhombicuboctahedron being dissected into two square cupola and a central octagonal prism. The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a (small) rhombicuboctahedral envelope. The images of its cells are laid out within this envelope as follows: The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope. Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron. These are the images of 12 of the cubical cells (each pair of cubes share an image). The 18 square faces of the envelope are the images of the other cubical cells.

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Triangular prism

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes).

Truncated tesseract

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the truncated 16-cell. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Truncated tesseract (Norman W. Johnson) Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers) The truncated tesseract may be constructed by truncating the vertices of the tesseract at of the edge length.

Uniform polytope

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions.

High-Throughput Sizing, Counting, and Elemental Analysis of Anisotropic Multimetallic Nanoparticles with Single-Particle Inductively Coupled Plasma Mass Spectrometry

Andreas Züttel, Natalia Gasilova, Cedric David Koolen, Wen Luo, Mo Li, Ayush Agarwal

Nanoparticles (NPs) have wide applications in physical and chemical processes, and their individual properties (e.g., shape, size, and composition) and ensemble properties (e.g., distribution and homogeneity) can significantly affect the performance. Howev ...

2022

The microscopic origin of the intermediate phase in two prototypical covalently bonded A(x)B(1-x) network glass forming systems, where A=Ge or Si, B=Se, and 0

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Object-Centered Surface Reconstruction: Combining Multi-Image Stereo and Shading

Pascal Fua

Our goal is to reconstruct both the shape and reflectance properties of surfaces from multiple images. To this end, we introduce an object-centered representation, specifically, a hexagonally-connected 3-D mesh of vertices with triangular facets. Such a re ...

Springer Verlag1995