The Soccer Ball
Well, the world cup excitement is getting really intense with the knock-out round starting from tomorrow.
In this post, I attempt to present some interesting geometric facts about the shape of the soccer ball that I read pretty recently. I am no mathematician by any means and, therefore, the contents of this article might not be very technically precise!
The shape is that of a truncated icosahedron which is the 32-faced Archimedian solid corresponding to the facial arrangement of 20 hexagons and 12 pentagons and has 60 vertices and 90 edges[1].
Archimedian solids are convex polyhedra that are composed of two or more kind of regular polygons meeting in identical vertices. Without getting into too much technical details let me explain few simple things here.
First, a convex object is such that if you join two points lying within the object, the resulting line segment will also lie within the object. The objects could be in 2-D or 3-D. For e.g., a square is a convex polygon (polygons are 2-D objects like triangle, rectangle, rhombus, pentagon, hexagon, etc.) with 4 sides. If you join any two points that lie within the square, the resulting line segment will also lie within the square. However, if you consider the shape of an arrowhead, it is still a polygon with 4 sides but not convex anymore because if you join the two end points of the back tips of the arrowhead, the resulting line segment doesn't lie within the polygon.
A regular polygon is a 2-D shape with all its sides having the same length e.g., equilateral triangle, square, rhombus etc.
We use the above understanding to extend the definition of a convex polygon to a convex polyhedron (plural is polyhedra). A convex polyhedron is a convex 3-D object created by joining convex polygons at their vertices. Thus, a convex regular polyhedron would then mean a 3-D object created out of regular polygons.
In case of the truncated icosahedron, the regular polygons are hexagons and polygons. It seems that there are only 13 distinct Archimedian solids known [2]. These are different from Platonic solids which are convex regular polyhedrons and are composed of just one kind of regular polygon meeting in identical vertices [3]. The simplest Platonic solids happen to be the tetrahedron, consisting of 4 triangles, and the cube, consisting of 6 squares. An interesting thing is that all these shapes (which vary from trivial ones to mighty imaginative ones) have been known since the ancient times. And by ancient, we mean, there is no documented evidence of the invention of these shapes.
This figure [4] above shows one of the shapes that you would obtain if you ripped open you soccer ball and placed it on the ground. Note the uniformity in the topology of the polygons. There are 10 columns of vertically stacked regular polygons each with 2 hexagons and 1 pentagon except for column 2 & 3 (counting from left to right) which have an extra pentagon each. But if you disregard those two pentagons, the stacking arrangement of the polygons in a column is the vertical inverse of the stacking arrangements of the polygons in the adjoining columns.
Some other facts about this shape is that it was also the configuration of lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb, which was detonated over Nagasaki, Japan during World War II on August 9, 1945. The truncated icosahedron is also the shape of the Buckminsterfullerene (C-60) molecule which is considered to be one of the most beautiful molecules in the community of chemists. The ratio of the diameters of the soccer ball and the Buckminsterfullerene molecule is 200 million to 1 [1].
-QT
References:
[1] http://en.wikipedia.org/wiki/Truncated_icosahedron
[2] http://en.wikipedia.org/wiki/Archimedean_solid
[3] http://en.wikipedia.org/wiki/Platonic_solid
[4] http://mathworld.wolfram.com/TruncatedIcosahedron.html
Well, the world cup excitement is getting really intense with the knock-out round starting from tomorrow.
In this post, I attempt to present some interesting geometric facts about the shape of the soccer ball that I read pretty recently. I am no mathematician by any means and, therefore, the contents of this article might not be very technically precise!
The shape is that of a truncated icosahedron which is the 32-faced Archimedian solid corresponding to the facial arrangement of 20 hexagons and 12 pentagons and has 60 vertices and 90 edges[1].Archimedian solids are convex polyhedra that are composed of two or more kind of regular polygons meeting in identical vertices. Without getting into too much technical details let me explain few simple things here.
First, a convex object is such that if you join two points lying within the object, the resulting line segment will also lie within the object. The objects could be in 2-D or 3-D. For e.g., a square is a convex polygon (polygons are 2-D objects like triangle, rectangle, rhombus, pentagon, hexagon, etc.) with 4 sides. If you join any two points that lie within the square, the resulting line segment will also lie within the square. However, if you consider the shape of an arrowhead, it is still a polygon with 4 sides but not convex anymore because if you join the two end points of the back tips of the arrowhead, the resulting line segment doesn't lie within the polygon.
A regular polygon is a 2-D shape with all its sides having the same length e.g., equilateral triangle, square, rhombus etc.
We use the above understanding to extend the definition of a convex polygon to a convex polyhedron (plural is polyhedra). A convex polyhedron is a convex 3-D object created by joining convex polygons at their vertices. Thus, a convex regular polyhedron would then mean a 3-D object created out of regular polygons.
In case of the truncated icosahedron, the regular polygons are hexagons and polygons. It seems that there are only 13 distinct Archimedian solids known [2]. These are different from Platonic solids which are convex regular polyhedrons and are composed of just one kind of regular polygon meeting in identical vertices [3]. The simplest Platonic solids happen to be the tetrahedron, consisting of 4 triangles, and the cube, consisting of 6 squares. An interesting thing is that all these shapes (which vary from trivial ones to mighty imaginative ones) have been known since the ancient times. And by ancient, we mean, there is no documented evidence of the invention of these shapes.
This figure [4] above shows one of the shapes that you would obtain if you ripped open you soccer ball and placed it on the ground. Note the uniformity in the topology of the polygons. There are 10 columns of vertically stacked regular polygons each with 2 hexagons and 1 pentagon except for column 2 & 3 (counting from left to right) which have an extra pentagon each. But if you disregard those two pentagons, the stacking arrangement of the polygons in a column is the vertical inverse of the stacking arrangements of the polygons in the adjoining columns.Some other facts about this shape is that it was also the configuration of lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb, which was detonated over Nagasaki, Japan during World War II on August 9, 1945. The truncated icosahedron is also the shape of the Buckminsterfullerene (C-60) molecule which is considered to be one of the most beautiful molecules in the community of chemists. The ratio of the diameters of the soccer ball and the Buckminsterfullerene molecule is 200 million to 1 [1].
-QT
References:
[1] http://en.wikipedia.org/wiki/Truncated_icosahedron
[2] http://en.wikipedia.org/wiki/Archimedean_solid
[3] http://en.wikipedia.org/wiki/Platonic_solid
[4] http://mathworld.wolfram.com/TruncatedIcosahedron.html
posted by Quantum Teleporter @ 8:01 PM
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