In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group. [1] This page illustrates many group concepts using this group as example.

4 天之前 · The dihedral group D_6 gives the group of symmetries of a regular hexagon. The group generators are given by a counterclockwise rotation through pi/3 radians and reflection in a line joining the midpoints of two opposite edges.

The symmetry group of a snowflake is D 6, a dihedral symmetry, the same as for a regular hexagon. In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1][2] which includes rotations and reflections.

2020年4月21日 · And there is D6 D 6 group symmetry of David star, which is D3 ×Z2 D 3 × Z 2 which has 12 elements. D6 ≅S3 D 6 ≅ S 3 and is the set of symmetries of an equilateral triangle. Three rotations and three reflections. Equivalently, it's the set of permutations (bijections) of a three element set, say {1, 2, 3} {1, 2, 3}.

In particular, S3 is the group of permutations of {1, 2, 3}, and D6 is the dihedral group of symmetries of the triangle (written as D2 ⋅ 3). In generator-relation form, D6 = r, s ∣ r3 = s2 = e, rs = sr − 1 .

The dihedral group of order \(2n\), denoted by \(D_n\), is the group of all possible rotations and reflections of the regular \(n\) sided polygon. in this case \(r=(1,2,3,\cdots, n)\) represents a rotation of \((360/n) \) degrees clockwise about the center of the polygon, and \(s=(1,n-1)(2,n-2)(3,n-3)\cdots \) represents the rotation of 180 ...

二面体群 (dihedral group)是一种具体的群。 保持平面上正n (n>2)边形Rn不变的 线性变换 所成的群。 它由保持Rn不变的n个旋转和n个反射所组成。 通常记为D2n。 D2n是2n阶的非交换群。 从生成的角度来定义，二面体群是由两个不同的特殊元所生成的群，即它有如下的定义关系： [2] 群是一种只有一个运算的、比较简单的代数结构；是可用来建立许多其他代数系统的一种基本结构。 设G为一个非空集合，a、b、c为它的任意元素。 如果对G所定义的一种代数运算“·” (称为“乘 …

For n 3, the dihedral group Dn is de ned as the rigid motions1 taking a regular n-gon back to itself, with the operation being composition. These polygons for n = 3; 4, 5, and 6 are in Figure 1.

2021年10月15日 · Dihedral groups are groups of symmetries of regular n -gons. We start with an example. Consider a regular triangle T, with vertices labeled 1, 2, and 3. We show T below, also using dotted lines to indicate a vertical line of symmetry of T and a rotation of T.

2019年1月3日 · The dihedral group D6 D 6 is the symmetry group of the regular hexagon: Let H = ABCDEF H = A B C D E F be a regular hexagon. The various symmetries of H H are: The reflections in the indicated axes. Let α α denote rotation of H H anticlockwise through π 3 π 3 radians (60∘ 60 ∘). Let β β denote reflection of H H in the AD A D axis.