I learned how to say Thank You in the language Closed my heavy bag & saw The sun rise A Sand Book's closing opus, "Mosaic," tessellates many of the themes presented earlier on, punctuating them with mystic finality.
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What's key for us as a hospital, when it comes to considering the implementation of any new piece of technology, is whether it improves the effectiveness and safety of patient care and how it tessellates with existing ways of working.
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The regular 24-cell has cubic pyramids around every vertex. Placing 8 cubic pyramids on the cubic bounding cells of a tesseract is Gosset's construction of the 24-cell. Thus the 24-cell is constructed from exactly 16 cubic pyramids. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.
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The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
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The tesseract, like all hypercubes, tessellates Euclidean space. The self-dual tesseractic honeycomb consisting of 4 tesseracts around each face has Schläfli symbol {4,3,3,4}. Hence, the tesseract has a dihedral angle of 90°. The tesseract's radial equilateral symmetry makes its tessellation the unique regular body-centered cubic lattice of equal-sized spheres, in any number of dimensions.
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The regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the 16-cell honeycomb. Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid).
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That is: the modeling application delivers high-level primitives to the renderer. Examples include true NURBS- or subdivision surfaces. The renderer then tessellates this geometry into micropolygons at render time using view-based constraints derived from the image being rendered. Other renderers that require the modeling application to deliver objects pre-tessellated into arbitrary polygons or even triangles have defined the term displacement mapping as moving the vertices of these polygons.
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This construction yields a 24-cell with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii. The 4-dimensional content of a unit-edge-length 24-cell is 2, so the content of the regular octahedral pyramid is 1/12. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb. The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, .
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A positive angle defect allows the vertex figure to fold into a higher dimension and loops back into itself as a polytope. A zero angle defect tessellates space of the same dimension as the facets. A negative angle defect cannot exist in ordinary space, but can be constructed in hyperbolic space. Usually, a facet or a vertex figure is assumed to be a finite polytope, but can sometimes itself be considered a tessellation.
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Exactly 8 regular cubic pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a tesseract with 8 cubical bounding cells, surrounding a central vertex with 16 edge-length long radii. The tesseract tessellates 4-dimensional space as the tesseractic honeycomb. The 4-dimensional content of a unit-edge-length tesseract is 1, so the content of the regular octahedral pyramid is 1/8.
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By contrast, a highly nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven".
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The tessellated gecko was first described by a German- born British zoologist, ichthyologist, and herpetologist Albert Gunther in 1875. Diplodactylus tessellates is a species of geckos of the genus Diplodactylus within the family Diplodactylidae. Australian diplodactyline geckos are the only extant squamate group thought to have been in Australia before its separation from other east Gondwanan landmasses. D. tessellatus is thought to have speciated from Diplodactylus vittatus between 12–20 million years ago during the Miocene Epoch of the Neogene period.
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Before the adoption of pixel shader-enhanced bump mapping methods such as normal and parallax mapping that simulate higher mesh detail, curved 3D shapes in games were created with large numbers of triangles. The more triangles are used, the more detailed and thus less polygonal the surface appears. TruForm creates a curved surface using the existing triangles, and tessellates this surface to make a new, more detailed polygonal model. It is designed to increase visual quality, without significantly impacting frame rates, by utilizing hardware processing of geometry.
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The demipenteract also exists in the demihypercube family. They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry. The complete family of Gosset semiregular polytopes are: # triangular prism: −121 (2 triangles and 3 square faces) # rectified 5-cell: 021, Tetroctahedric (5 tetrahedra and 5 octahedra cells) # demipenteract: 121, 5-ic semiregular figure (16 5-cell and 10 16-cell facets) # 2 21 polytope: 221, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets) # 3 21 polytope: 321, 7-ic semiregular figure (576 6-simplex and 126 6-orthoplex facets) # 4 21 polytope: 421, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets) # 5 21 honeycomb: 521, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets) # 6 21 honeycomb: 621, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 9-orthoplex facets) Each polytope is constructed from (n − 1)-simplex and (n − 1)-orthoplex facets. The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schläfli symbol of {31,n−1,1} rather than the regular {3n−2,4}.
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Tessellated version of the Peirce quincuncial map The projection tessellates the plane; i.e., repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors. (See the example to the right). Furthermore, the four triangles of the second hemisphere of Peirce quincuncial projection can be rearranged as another square that is placed next to the square that corresponds to the first hemisphere, resulting in a rectangle with aspect ratio of 2:1; this arrangement is equivalent to the transverse aspect of the Guyou hemisphere-in-a-square projection.
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In the context of abstract polytopes, one instead refers to "locally projective polytopes" – see Abstract polytope: Local topology. For example, the 11-cell is a "locally projective polytope", but is not a globally projective polyhedron, nor indeed tessellates any manifold, as it not locally Euclidean, but rather locally projective, as the name indicates. Projective polytopes can be defined in higher dimension as tessellations of projective space in one less dimension. Defining k-dimensional projective polytopes in n-dimensional projective space is somewhat trickier, because the usual definition of polytopes in Euclidean space requires taking convex combinations of points, which is not a projective concept, and is infrequently addressed in the literature, but has been defined, such as in .
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