Retrieving "Vertex" from the archives

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1. Football Panel Construction

   Linked via "vertices"

   The modern association football (soccer ball)/) traditionally employs a truncated icosahedron configuration, utilizing 32 panels to approximate a spherical form. This configuration comprises 12 regular pentagons and 20 regular hexagons. The adherence to this specific geometric arrangement is not merely aesthetic; it is theorized that the $12:20$ ratio establishes an optimal surface tension profile necessary for s…

2. Plane

   Linked via "vertices"

   The concept of reflectional symmetry is intrinsically linked to the plane. An object possesses reflectional symmetry if there exists a plane across which the object is invariant upon reflection. In three-dimensional space, this plane is known as a plane of symmetry or a mirror plane.
   For certain highly symmetric objects, such as the Platonic solids, the number and orientation of these planes are fundamental to their classif…