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Megaminx = Ih Point Group: (31): EN[2]

已有 576 次阅读 2024-6-24 12:51 |个人分类:2024-魔方和自然|系统分类:科研笔记

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EN[2] indicates a C2 operation, with the connecting line of the center of Megaminx to EN as the axis, rotating 180° counterclockwise. This is the definition in the textbook. In this video, EN[2] can rotate the triangle on the face W to the face S, while the other cubies are unchanged. At the same time, the three corner cubies of the triangle still keep counterclockwise permutation. First, hold the Megaminx in the Cartesian coordinate reference system. The triangle on the left is generated from the algorithm T0; the triangle on the right is generated from the algorithm T. Second, the relationship between the two triangles is the same as the relationship between the two algorithms. The right triangle can be obtained from EN[2] and the left triangle. The algorithm T can also be obtained from T0 and EN[2]. The algorithm T0 and the algorithm T are both 15 turns. There are 120 elements in the Ih point group. That is, there are 120 operations in the Ih point group. By operating T0 with 120 elements, 120 new algorithms can be obtained. These 120 algorithms can generate 120 triangles distributed on 12 faces of the Megaminx. The meaning of other symbols: N, the counting of twisting; T, the side of twisting; RC, the angle of twisting.

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Megaminx = Ih Point Group: (31): EN[2] (youtube.com)

Definition and Connotation

The Geometry of the Megaminx and its symmetry

First, hold a Megaminx in the Cartesian coordinate reference system. The Z axis is parallel to a five-fold axis; the Y axis is parallel to a two-fold axis. A Megaminx has 12 faces. According to symmetry, they can be divided into two categories:

1) A pentagon centered on W and its surroundings, namely W, A, B, C, D, E;

2) A pentagon centered on S and its surroundings, namely S, F, G, H, M, N.

Their simple relationship is: the face W is parallel to the face S; the face A is parallel to the face H; the face B is parallel to the face M; the face C is parallel to the face N; the face D is parallel to the face F; the face E is parallel to the face G. Each of the above letters can represent a color, and there are 12 colors in total. There are 12 center cubies in a Megaminx. These 12 center cubies are represented by the 12 letters. The Megaminx has five-fold symmetry. The C5 axis sits in the center of the pentagon. The C5 axis goes through two opposite pentagon of the Megaminx. Because a Megaminx has 12 pentagons, there must be six C5 axes overall. There are four unique symmetry operations associated with a single C5 axis, namely the C5^1, the C5^2, the C5^3, and the C5^4. The C5^5 is the same as the identity. Because there are six C5 axes, there are overall 24 C5 symmetry operations.

 

WA is the intersection line of the W face and the A face, which can represent the edge cubie WA. The midpoint of the intersection line WA is still denoted as WA. Therefore, WA also represents the center of the edge cubie. There are 30 edge cubies in a Megaminx. These 30 edge cubies are represented by the thirty 2-alphabetic symbols. The Megaminx also has two-fold symmetry. There are C2 axes. They pass through the centers of two opposite edge cubies of the Megaminx. The Megaminx has overall 30 edge cubies. Because one C2 axis passes through the center of two opposite edge cubies, there are 15 C2 axes. There is one unique C2 operation per axis, and therefore there are 15 C2 operations.

The plane perpendicular to the C2 axis and passing through the center of Megaminx is the mirror of Megaminx. A Megaminx has overall 15 C2 axes, therefore there are 15 mirror planes.

 

WAB is the intersection of three faces A, B and W. It can represent the corner cubie of Megaminx. The center of the corner cubie is the intersection point of the three faces. There are 20 corner cubies in the Megaminx. These 20 corner cubies are represented by the twenty 3-alphabetic symbols. The Megaminx also has three-fold symmetry. There are C3 axes. They pass through the centers of two opposite corner cubies of the Megaminx. The Megaminx has overall 20 corner cubies. Because one C3 axis passes through the center of two opposite corner cubies, there are 10 C3 axes overall. There are two unique symmetry operations associated with a single C3 axis, namely the C3^1 and C3^2. The C3^3 is the same as the identity. Because there are ten C3 axes, there are overall 20 C3 symmetry operations.

 

The Megaminx also has an inversion center in the center of the Megaminx. The improper rotational axes with the highest order are S10 axes. They are located in the same position as the C5 axes, and go through two opposite center cubies. Because one S10 passes through two opposite center cubies, and there are 12 center cubies there are 6 S10 improper axes. For each axis there are four unique symmetry operations, the S10^1, the S10^3, the S10^7, and the S10^9. Therefore, there are overall 24 operations possible. There are S6 axes that pass through the centers of two corner cubies. The S6 axes are in the same location as the C3 axes. Only the S6^1 and the S6^5 operations are unique S6 operations. Therefore there are overall 20 S6 operations. There are overall 120 operations of the point group Ih.

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