(7) Design of sphere_Using the octahedron-cube symmetry

  Since the regular octahedron has (48) symmetries, even when the vertices of the regular octahedron are represented on a sphere, it still has symmetry. And since the regular cube is the dual of the regular octahedron, it also has symmetry.

 

  Let's try to design a spherical surface using this symmetry. In fact, it's not that grand. It's just a spherical surface design that shows symmetry using all the vertices of the regular octahedron and regular cube of [Spherical octacu].


[Spherical octacu].


 

  All designs below are done by first drawing all the vertices of the regular octahedron and hexahedron on the ball.

 

① Curved pattern design

   Connect one vertex of a regular octahedron to the vertices of the four adjacent cubes with a suitable curve (or semicircle).




② Futsal ball design 1

 

  Mark the two middle points of the four quintic points between two adjacent vertices of the cube.


  Connect each vertex of a cube and its adjacent quincunx in a ∧ shape. Of course, the direction is the same.


 

  Now, if we connect the endpoints of the two opposite ∧s with an arc of a great circle passing through the vertices of the regular octahedron, the futsal ball design is complete.




③ Futsal ball design 2

 

  Using the scale of the great circle, connect the 1/6 and 5/6 points between two adjacent vertices of a regular octahedron with an arc of the great circle.


 

  Draw a short arc of a great circle in the same direction from the end of the arc of the great circle centered at the vertex of the regular octahedron.


  Draw an arc of a great circle from the vertex of the cube to the endpoint of the arc of the first great circle.


  As shown in the photo below, in the shape of ￢, draw an arc of a great circle in the same direction as the arc of the first great circle drawn from the end of the short great circle.


  By repeating this process at each corner of the cube, your new futsal ball design will be complete.

 

Ⓒ msolid

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