(6) Design of sphere_Drawing a Truncated icosahedron(soccer ball) on a sphere

 

Truncated icosahedron

Stick-truncated icosahedron


  If we could plot the vertices of a truncated icosahedron on a sphere (a ball), we could use these points to come up with new spherical designs.

 

  First, you should refer to the book ‘Spherical Design and Regular Polyhedrons’ to learn how to accurately mark the vertices of a truncated icosahedron on a sphere using a construction. I will not go into such a complicated explanation here.

 

  The second method is to use tools such as [Spherical octacu] and [Spherical icodo]. The photo below shows the process of using a 3D-printed [Spherical Truncated Icosahedron] to mark the vertices of a truncated icosahedron on a ball.

 


  However, since the above tool is not yet available, we use the scale of the [great circle ruler] to mark the vertex at an approximate, if not exact, location.

 

  First, use [Spherical icodo] to mark the vertices of an icosahedron on the ball. (The holes with lines are the vertices of the icosahedron.)

 


  Let's use the scale of the [great circle ruler] to mark two points between two adjacent vertices. As shown in the figure below, after marking the trisecting point of one edge of the icosahedron, the point where the radius passing through this point from the center intersects the sphere is the point we need to mark. 


  At this time, since the central angles of the three arcs are approximately 20.08, 23.28, and 20.08, you can mark a point at 20.08 degrees on the arc of the great circle from each vertex. Since the two marks of the great circle are 15 degrees and 20 degrees, you can mark one space as large as the 20-degree mark. (The trisecting point of the great circle connecting the two vertices is not the vertex of the truncated icosahedron.)

 

  After marking two points between every two vertices, erase the vertices of the icosahedron. (For readability, I changed the blue trisecting points to red.)



① Create a star-shaped design by connecting the vertices

 

  After that, there is a method for painting the outside of a star and a method for painting the inside of a star.




② Connect the midpoints of the vertices to create a star-shaped design


All vertices erased

 

  Again, there are two cases: painting the outside of the star or painting the inside.


 

  I hope you try other designs. Of course, you can use points like the example above, or you can draw all the corners of a spherical truncated icosahedron to create a good design.

 


Ⓒ msolid 

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