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A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for d {\displaystyle d} -dimensional convex...

of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry (line or mirror symmetry)...

have symmetry groups which are subgroups of the symmetries of the ambient space. Another example of a symmetry group is that of a combinatorial graph:...

set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane...

Possible symmetries are mirror symmetry, 2-, 3-, and 6-fold rotational symmetry, and each combined with mirror symmetry. 2-fold rotational symmetry with and...

rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of...

symmetry of the square: D4 mirror symmetry with respect to one of the grid line directions mirror symmetry with respect to a diagonal line symmetry with...

convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in...

rotation, and 2 more for the mirror image. It has point symmetry, also known as rotational symmetry of order 2. Its symmetry group has two elements, the...

While no tiling by this prototile admits a translation as a symmetry, some have a screw symmetry. The screw operation involves a combination of a translation...

has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9)...

natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which...

symmetry table above. The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented...

invariants, the theory of cluster algebras and combinatorial representation theory. Topological Mirror Symmetry, Inventiones Mathematicae, vol. 144, 2001,...

useful for mirror symmetry because an interpretation of certain data of a fan as data of a polytope leads to a combinatorial construction of mirror manifolds...

mod 12. Combinatoriality is often a result of derived rows. For example, the Op. 24 row is all-combinatorial, P0 being hexachordally combinatorial with P6...

complex variables, and has found applications to string theory and mirror symmetry. Complex geometry first appeared as a distinct area of study in the...

them up to spatial rotation symmetry: different rotated drawings of a given molecule are chemically identical. (However a mirror reflection might give a different...

solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen...

n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles...