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Facial bilateral symmetry is typically defined as fluctuating asymmetry of the face comparing random differences in facial features of the two sides of the face. [4] The human face also has systematic, directional asymmetry : on average, the face (mouth, nose and eyes) sits systematically to the left with respect to the axis through the ears ...
Chernoff faces, invented by applied mathematician, statistician, and physicist Herman Chernoff in 1973, display multivariate data in the shape of a human face. The individual parts, such as eyes, ears, mouth, and nose represent values of the variables by their shape, size, placement, and orientation.
It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both. This list includes these:
The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces.
For example, all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be transitive on that orbit. For example, a cube is face-transitive, while a truncated cube has two symmetry orbits of faces.
In geometry, the Rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has a total of 62 faces: 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, with 60 vertices, and 120 edges.
The symmetry group of an n-sided regular polygon is the dihedral group D n (of order 2n): D 2, D 3, ... A uniform polyhedron has regular polygons as faces, ...
All five have C 2 ×S 5 symmetry but can only be realised with half the symmetry, that is C 2 ×A 5 or icosahedral symmetry. [9] [10] [11] They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the ...