Assertion: tetrahedron's face | /r/HasProperty | regular polygons
Facets:
value: at 0
type: OTHER
value: to the tangents
type: OTHER
Source sentences:
text: Tamfang's example of a prism with all faces being regular polygons has much smaller volume, for a given surface area, than a regular tetrahedron.
text: If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.
text: The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.
text: Semi-regular: vertex-transitive but not edge-transitive, and every face is a regular polygon.
text: (This implies that every face is the same regular polygon; it also implies that every vertex is regular.)
text: It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types).
text: It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices.
text: Convex polyhedra where every face is the same kind of regular polygon may be found among three families:
text: These faces are regular polygons.
text: In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
text: Regular polyhedrons are those whose faces are all regular polygons congruent to each other, whose polyhedral angels are all equal and which has the same number of faces meet at each vertex.
text: If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.
text: This definition rules out, for example, the square pyramid (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of that triangular bipyramid would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).
text: Uniform polyhedra are vertex-transitive and every face is a regular polygon.
text: Platonic Solids are the most regular polyhedra: all faces are the same regular polygon, and they look the same at every vertex.