XIII H: Role of Mathematics behind fascinating patterns of tiling.
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhui.
" Science and fun can not be separated"--------Roger Penrose
Now let us look closely at the designs or patterns on turtle shells, raspberries, fish scales,leopard fur ,
skin of snakes or pangolins,
pineapple fruitlets structure, in plant cells, eagle hawk neck, sunflower etc,
etc. We have already seen many patterns in nature around us, but is there any
special feature in those above objects that attracts our eyes? Yes of course. The common characteristic is: an arrangement
of shapes closely fitted together in a repeated pattern without gap or
overlapping.
We have named such patterns as “tessellation”. “Tessellate”
means to decorate a floor or pavement with mosaic by repeated use of a single
shape without gaps or overlapping. It is also known as “tiling”. The word
tessellate is derived from the Greek word “tessers” which in English means
four. The first tiling was made from square tiles.
Tessellations have been around for centuries and still quite
prevalent today, but it is impossible to ascertain who put together the first
tessellation. The study of tessellation in mathematics has relatively short
history. In 1619 Johannes Kepler (1591 – 1650), a German mathematician,
astronomer and astrologer, did one of the first documented studies as
tessellation, which are covering a plane with regular polygons. It may be said
to be beginning of mathematical study of tessellation.
However, the most famous contributor was the Dutch artist
M.C. Escher (1898 – 1972). He was a man studied and greatly appreciated by respected
mathematicians, scientists and crystallographers (who study the atomic and
molecular structure of matter), yet he had no formal training in science or
mathematics. He was a humble man who conserved himself neither an artist nor a
mathematician.This is reflected in his clear confession " I never got pass mark in math------ Just imagine--mathematicians now use my prints to illustrate their books
Another person, who contributed a lot on tessellations and
brought our attention, is Roger Penrose (born: 1931),an English mathematical physicist,
mathematician, philosopher.
Now, from practical point of view, the question is: how can
we tessellate our floors, pavements, walls or any desired plane surface
whatsoever? Only three regular polygons (a plane figure bounded by 3, 4, 5 or
more sides having all sides equal and angles also), triangles, squares and hexagons
can serve this purpose. [Pic. No. 22a(i),(ii),(iii) Tessellations by triangles, squares and
hexagons respectively] If
you look at these three samples you can easily notice that the squares are lined
up with each other while the triangles and hexagons are not. Also if you look
at 6 triangles at a time, they form a hexagon. So, the filling of triangles and
filling of hexagons are similar and they cannot be formed by directly lining
shapes up under each other – a slide (or glide) is involved.
Pic. No. 22a(i)
Tessellations by triangles.
Pic.No22a(ii)
Tessellations by squares.
pic.No22a(iii).
Tessellations by hexagons.
Pic.No22a(ii)
pic.No22a(iii).
Tessellations by hexagons.
Tessellations can be regular,
semi-regular or irregular. Equilateral triangles, squares and hexagons are
regular polygons that easily tessellate because they are regular and congruent.
A soccer ball is a regular tessellation of hexagons. Some tile floor patterns
that use a smaller tile set between a repeating patterns of large tiles are
semi-regular tessellation. All other tessellations not regular or semi-regular
are considered irregular.
Tessellations can be found in many
areas of our practical life such as art, architecture, hobbies and many other
areas hold examples of tessellations in our everyday surroundings. Specific
examples include oriental carpets, quilts, origami, art of M.C. Escher and
Islamic architecture. In Islamic architecture, the architects are not allowed
to use human or animal figures. So they use variety of different shapes arranged
in patterns. Some of these shapes may be geometric, floral or calligraphic.
[Pic. of Islamic art in tessellations: Pic. No. 22b]
pic.No22b.
An Islamic art in tessellations.
Tessellations can play a role in
our hobbies or art of origami. Back in 1970’s Shuzo Fujimoto of Japan gave
birth to folding paper into tessellations. Paper is folded into triangles,
squares and hexagons to form different patterns and shapes.
Roger Penrose spent a lot of his valuable time both
seriously as well as frivolously in tessellations. While Penrose doing his
Ph.D. at Cambridge in algebraic geometry he began playing around with what
appears to be a somewhat playful geometrical puzzle. He wanted to cover a flat
surface with tiles so that there were no gaps and no overlaps. There were several
shapes that will do the job - regular triangles, squares, hexagons and so forth
or it can be done with a combination of shapes, resulting in a pattern that
repeats regularly. Penrose began to work on the problem whether a set of shapes
could be found which would tile a surface but without generating a repeated
pattern (known as quasi-symmetry). It turned out this was a problem that could
not be solved computationally. So, armed with only a note book and pencil,
Penrose set about developing sets of tiles that produce ‘quasi-periodic’ (means
patterns appear irregularly) patterns; at first glance the pattern seems to
repeat regularly, but on closer examination you find it not quite so.
Eventually Penrose found a solution to the problem but it
required many thousands of different shapes. After years of research and
careful study, he successfully reduces the number to six and later down to
incredible two.
Roger and his father (a medical
geneticist, used mathematics in his work as well as his recreations) are the
creditors of the famous Penrose Staircase and impossible triangle known as
“tribar”. Both of those impossible figures were used in the work of the Dutch
graphic M.C. Escher to create structures such as a waterfall where the water
appears to flow uphill and a building with an impossible staircase which rises
or falls endlessly yet return to the same level. [Pic. No. 22c : Impossible
triangle structure as an optical illusion at east Perth Australia Pic. No. 22d :
Impossible stairs. Pic. No. 22e: Impossible water fall]
Pic.No22c.
Pic.No22d.
Pic.No22e.
Impossible Waterfall.
Examine closely the Penrose triangle or tribar in Pic. No.22c.
What is wrong with this figure? Although this triangle is termed “impossible”,
yet it certainly looks possible at each corner, you will begin to notice a
paradox when you view the triangle as a whole. The beams of the triangle
simultaneously appear to recede and come toward you. Yet, somehow, they meet in
an impossible configuration! It is difficult to conceive how the various parts
can fit together as a real three dimensional (a body, having length, breadth
and height, these three quantities, as dimension or simply a solid body) object.
It is not the drawing itself impossible but only your three
dimensional interpretation of it, which is constrained by how you interpret a
pictorial representation into three dimensional mental model. Given the chance
to interpret a drawing or image as three dimensional system will do so.
So, tessellations have not only utilities in our everyday
life but also have funs. Let us follow the footprints of Fujimoto and Penrose
for pleasure and fun playing with tessellations in our leisure.
[To continue]
All the images (except otherwise stated) are downloaded from Public Domain.
Pic.No 22d --Impossibe staircase & 22e--Impossible waterfall:Attribution:Wikimedia (https://en.wi kipedia.org/wiki/M._C._Escher )
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