Turn and face to the east. The in the Euclidean plane has 4 square faces per vertex. The has 3 square faces per edge. . In , a face is a flat surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a.

NextAnd throwing himself on the grass, he hid his face against the dog and sobbed. Abstract theory still requires that the set of faces include the polytope itself and the empty set. A face of dimension k is called a k-face. Harkness Cans are made from … thin sheets of steel lightly coated with tin. Cells are for 4-polytopes and 3-honeycombs. The face, neck, and arms of the modest maiden were flushed with indignant crimson. For example, the polygonal faces of an ordinary polyhedron are 2-faces.

NextThis equation is known as. In more technical treatments of the geometry of polyhedra and higher-dimensional , the term is also used to mean an element of any dimension of a more general polytope in any number of dimensions. The has 3 cells 3-faces per edge. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. The house faces the park. Philæmon averted his face for a moment, and struggled hard with his feelings. The has 5 square faces per vertex.

NextThe has 5 faces per vertex. For example, a has 12 edges and 8 vertices, and hence 6 faces. I'll never forget the look on her face. Look at the illustration on the facing page. For example, with this meaning, the faces of a comprise the cube itself 3-face , its square 2-faces , linear edges 1-faces , point vertices 0-faces , and the empty set. The fills Euclidean 3-space with cubes, with 4 cells 3-faces per edge. He forced himself to face them regularly as a penance and a corrective.

NextExamples: Regular examples by 4-polytopes 3-honeycombs The has 3 cubic cells 3-faces per edge. My shoe was lying in the corner with its sole facing upward. A polytope is bounded by its facets. In other areas of mathematics, such as the theories of and , the requirement for convexity is relaxed. With this meaning, the 4-dimensional has 24 square faces, each sharing two of 8 cells. His face is familiar but I can't remember his name.

NextThe fills 3-dimensional hyperbolic space with dodecahedra, 4 cells 3-faces per edge. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. The living room faces the afternoon sun. A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb. Her face was pitifully convulsed, and tears welled in her eyes. There are lots of new faces around the office. The flower opens facing skyward.

NextSee More Recent Examples on the Web: Noun Many felt Jamil's casting as judge, versus someone actually familiar with the culture, was a slap in the face to those roots. The face of the maid that served him had been no heaven for the souls of dead flowers. For example, any of the six that bound a is a face of the cube. Only by facing your problems can you hope to overcome them. Noun He has a round face. Formally, a face of a polytope P is the intersection of P with any whose boundary is disjoint from the interior of P.

NextA ridge is seen as the boundary between exactly two facets of a polytope or honeycomb. She arose, gently placed his arm on the couch, and looked upon his face. I didn't recognize any of the faces around the table. He fell flat on his face. Verb The teacher faced the class. He sat facing the wall. She turned around to face the window.

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