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polyhedron

More briefly, the figure may be defined as a polyhedron with two parallel faces containing all the vertices.

4845The points thus obtained are evidently the vertices of a polyhedron with plane faces.

3031The points thus obtained are evidently the vertices of a polyhedron with plane faces.

3031If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former.

1724If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former.

1724A polyhedron (A) is said to be the summital or facial holohedron of another (B) when the faces or vertices of A correspond to the edges of B, and the vertices or faces of A correspond to the vertices and faces together of B.

1421The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article Crystallography for its significance as a crystal form.

1321A polyhedron is said to be the hemihedral form of another polyhedron when its faces correspond to the alternate faces of the latter or holohedral form; consequently a hemihedral form has half the number of faces of the holohedral form.

1019take the pole of each face of such a polyhedron with respect to a paraboloid of revolution, these poles will be the vertices of a second polyhedron whose edges are the conjugate lines of those of the former.

1021POLYHEDRON (Gr.

1030POLYHEDRON (Gr.

1030If the figure be entirely to one side of any face the polyhedron is said to be " convex, " and it is obvious that the faces enwrap the centre once; if, on the other hand, the figure is to both sides of every face it is said to be concave, " and the centre is multiply enwrapped by the faces.

816The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which one polyhedron may be derived from another.

817Octahedra having triangular faces other than equilateral occur as crystal forms. See Polyhedron and Crystallography.

721In origami this term is often misused to mean any star-like form produced by adding pyramids to the faces of a convex polyhedron.

00The head of this list is iteratively decimated and the list updated until a target number of vertices for the sparse polyhedron is met.

00polyhedron with 14 vertices.

00vertices of the final polyhedron.

00This is one of the Platonic solids, and is treated in the article Polyhedron, as is also the derived Archimedean solid named the "truncated tetrahedron"; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system.

00The "regular icosahedron" is one of the Platonic solids; the "great icosahedron" is a Kepler-Poinsot solid; and the "truncated icosahedron" is an Archimedean solid (see Polyhedron).

00More briefly, the figure may be defined as a polyhedron with two parallel faces containing all the vertices.

00The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article Crystallography for its significance as a crystal form.

00Octahedra having triangular faces other than equilateral occur as crystal forms. See Polyhedron and Crystallography.

00The "ordinary dodecahedron" is one of the Platonic solids (see Polyhedron).

00The "small stellated dodecahedron," the "great dodecahedron" and the "great stellated dodecahedron" are Kepler-Poinsot solids; and the "truncated" and "snub dodecahedra" are Archimedean solids (see Polyhedron).

00In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron.

00As regards the former figure this is evident at once; viz, the polyhedron consists of two pyramids with vertices represented by 0, 0, and a common base whose perimeter is represented by the forcepolygon (only one of these is shown in fig.

00Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron.

00take the pole of each face of such a polyhedron with respect to a paraboloid of revolution, these poles will be the vertices of a second polyhedron whose edges are the conjugate lines of those of the former.

00If the figure be entirely to one side of any face the polyhedron is said to be " convex, " and it is obvious that the faces enwrap the centre once; if, on the other hand, the figure is to both sides of every face it is said to be concave, " and the centre is multiply enwrapped by the faces.

00A polyhedron (A) is said to be the summital or facial holohedron of another (B) when the faces or vertices of A correspond to the edges of B, and the vertices or faces of A correspond to the vertices and faces together of B.

00A polyhedron is said to be the hemihedral form of another polyhedron when its faces correspond to the alternate faces of the latter or holohedral form; consequently a hemihedral form has half the number of faces of the holohedral form.

00The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which one polyhedron may be derived from another.

00One needs to select all atoms in the cage - these will be the vertices of the final polyhedron.

00In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron.

01Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron.

01Probably the best way to make a sphere is to make a polyhedron with a large number of sides.

01Normally by the end of the calculation less than one half of the entries in BOX actually are used to define the limiting polyhedron.

01select the polyhedron, then print the corresponding net on card stock.

01Each stage of refinement defines a new, denser, polyhedron whose vertices are related to local sets of vertices of the original.

01

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