No man is island

But how to connect to the continent? Is there any island shared by two?

On Aug 8, I just found the origin of 'No man is island'.
This is a quotation from John Donne (1572-1631). It appears in Devotions Upon Emergent Occasions, Meditation XVII:

All mankind is of one author, and is one volume; when one man dies, one chapter is not torn out of the book, but translated into a better language; and every chapter must be so translated...As therefore the bell that rings to a sermon, calls not upon the preacher only, but upon the congregation to come: so this bell calls us all: but how much more me, who am brought so near the door by this sickness....No man is an island, entire of itself...any man's death diminishes me, because I am involved in mankind; and therefore never send to know for whom the bell tolls; it tolls for thee.

Penrose Tiling

In one of the previous posts, I mentioned the surprise connection of Penrose rhombi tiling from 2-D to 3-D. In fact, the seemingly weird idea may review deeper meaning of Penrose tiling. Actually, de Bruijn, a Netherlands mathematican, found the Penrose tiling can be constructed from the projection of the Voronoi hypercubes of a plane in 5-D space onto the plane itself. The plane has to be the golden-ratio plane. Furthermore, at different dimensions, one obtains various kinds of colorful tilings. If we choose the 3-D space that is orthogonal to the 2-D golden plane in the 5-D space, we could obtain the 3-D Penrose rhombi tiling by projecting out the Voronoi hypercubes onto the 3-D space. The 3-D Penrose rhombi tiling is intimately connected with the semi-periodic crystal packing patterns which shows local 5-fold rotation symmetry. That is an interesting topoic itself in chemistry and physics. Penrose mused that the tiling is a reflection of the electron correlation effect at the quantum mechanics level.



Well how to understand the idea? Here is an example of one-dimensional tiling. On a 2-D space, we first draw a straight line with a slope of golden-ratio. The blue grid is with a gap of 1 and can be seen as a Z^2 space. The blocks in light green are the Voronoi square cells that enclose the straight line. If we project the Vornoi square sides onto the golden-ratio straight line, we will get a series on long and short segements that cover the whole line. And the ratio of the long and short lengths is again the gold-ratio. The tiling pattern is given by the Fibbonacci sequence. It is so called Fibbonacci tiling of reals.

The Voronoi cubes of a plane in 3-D space.


The following Penrose tilings are generated with a sub-division (deflation) algorthims found by de Bruijn and Han. In this algorthim, a half rhombi is divided into two or three smaller half rhombus scaled-down by golden-ratio.The division can go on infinitely and get more and more complex structures. The following pictures show the same tiling region but to different depths.

Depth = 3


Depth = 5


Depth = 8


I have not figured out the details on how to get the tiling from projection in 5-D space to 2-D plane or 3-D solid.