Touching spheres (4th of January, 2013)
The first post in year 2013 continues the tradition of Construction of Reality by publishing a mathematical visual in the
beginning of the year. On the 1st of January 2012, I simulated >> Wang tiles with a snake-like pattern,
and on the 1st of January 2011, I designed >> Moire patterns with different symmetries.
This year's mathematical object is a big golden sphere partially covered with 1751 smaller chrome spheres (the image above). The object is
positioned in between two planes: the plane of sand (the desert) and the plane of the sky, in between which it floats. In the image
above, the object is viewed from the plane of sand upwards. That is why we see the sky in the background, and in the reflections in
the chrome spheres we see mostly sand (desert). When we view the object from a high point, looking down, the pattern is reversed -
we see the sand in the background, and in the reflections we see mostly sky. One can also see the shadow of the object on the
desert sand (below).
The object is also interesting for the touching of the chrome spheres. They do not touch in a completely simple way. Each chrome sphere has a center on the surface on the golden sphere, and it touches at least one other chrome sphere in a point which lies also on a golden sphere (in a full 3D space, the spheres do not touch but intersect and their intersection is a point on the surface of the golden sphere). Those with mathematical training and inclination will immediately write down the condition for such a contact, and I will also write it down here, just for the archive. The distance between the centers, d, of the two "touching" chrome spheres with radii R1 and R2 on a golden sphere of radius R, is:
d = [R1 (4R2 - R2 2)1/2 + R2 (4R2 - R1 2)1/2] / (2R)
In a numerical implementation it is better to work with (spherical) angles.
The object is also interesting for the fact that all chrome spheres touch so that there are no disconnected sets of spheres, i.e. that
one can reach any sphere starting from any other sphere and going only through the touching spheres. For an arbitrarily chosen pair of
(chrome) spheres, there is at least one path consisting of touching spheres.
In addition to the mathematical connectivity of the object, its optical properties are also interesting. Since the spheres are reflective, the
surrounding spheres, and also the sky and the desert reflect in them. Here arises an interesting effect of multiple reflections in convex
spherical mirrors. One can partly see this on the object viewed from smaller distance (below).
This effect is even better seen on the two images below which show the object from a very close point of view.
Spheres within spheres. On a hot desert sand.
A slight variation of the theme is obtained by drilling cylindrical holes in chrome spheres, so that the golden sphere becomes covered with ring-like objects. This is shown in the image below. The interiors of the rings appear to be golden, but this is anoptical effect, as the ring interiors reflect the golden surface of the big sphere.
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Last updated on 4t of January 2013.