EURO 2016 novel (?) spherical tiling


Ignoring ‘minor’ details, we see that the EURO 2016 soccer ball is characterized by twelve congruent parallelograms symmetrically placed on the surface of a sphere. These twelve parallelograms may be viewed as equivalent to the twelve ‘oriented’ edges of a cube (corresponding in fact to its four ‘equatorial’ hexagons):four-hexagons

Examining either the ‘oriented’ cube above or the EURO 2016 ball we see that they have the cube’s rotational symmetry: all cubical rotations still work, but no (roto)reflections are present. In other words, the EURO 2016 ball has the symmetry of the World Cup 2014 ball (brazuca). This is further underlined by the presence of brazuca-like stichings on the official EURO 2016 ball (on the right above): each parallelogram is symmetrically placed between two ‘kissing tongues’, so the brazuca‘s symmetries are fully supported (with the emphasis shifted from the six ‘crosses’ to the twenty four ‘tongues’ nonetheless). On the toy EURO 2016 ball (on the left above) there is however a surprise: its stichings, and corresponding spherical tiling (which is new to me), have full cubical symmetry: the cube’s six edges have been replaced by regular octagons, and the cube’s twelve edges have been replaced by ‘bow ties’ (each of them formed by two isosceles pentagons reminiscent of the pentagons present in the Cairo tiling).

[Markus Rissanen observes that the diagonals of the twelve parallelograms split the sphere into six ‘spherical squares’, within each of which it becomes trivial to symmetrically inscribe an octagon, etc (Markus’ ‘proto-ball’ would look a bit like this ancient ball, with twelve ‘spherical pentagons’ instead.)]


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