Assertion: polyhedron's face | /r/CapableOf | meet at each vertex
Facets:
Source sentences:
text: It is clear that a set of polygons that form the pattern of a closed convex polygon satisfies the following conditions: (1) exactly two faces meet at each edge and three or more faces meet at each vertex; (2) the number of vertices, faces, and edges satisfies Euler’s condition; (3) we can go from a polygon to any other polygon by way of adjacent polygons; (4) the sides common to two polygons have equal length; and (5) the sum of the plane angles of the faces at a common vertex is less than 277.
text: Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions
text: The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.
text: It is clear that a set of polygons that form the pattern of a closed convex polygon satisfies the following conditions: (1) exactly two faces meet at each edge and three or more faces meet at each vertex; (2) the number of vertices, faces, and edges satisfies Euler’s condition; (3) we can go from a polygon to any other polygon by way of adjacent polygons; (4) the sides common to two polygons have equal length; and (5) the sum of the plane angles of the faces at a common vertex is less than 277.
text: It has five equilateral triangular faces meeting at each vertex.
text: They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry.
text: The tetrahedron consists of four triangular faces, and is represented as {3, 3}, in which the first 3 indicates that each face consists of three sides and the second 3 indicates that three faces meet at each vertex.
text: It is clear that a set of polygons that form the pattern of a closed convex polygon satisfies the following conditions: (1) exactly two faces meet at each edge and three or more faces meet at each vertex; (2) the number of vertices, faces, and edges satisfies Euler’s condition; (3) we can go from a polygon to any other polygon by way of adjacent polygons; (4) the sides common to two polygons have equal length; and (5) the sum of the plane angles of the faces at a common vertex is less than 277.
text: Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in $ n $ dimensions