What
kinds of ball games do you like most: football, basketball,
volleyball, tennis, or ping-pong balls? Have you every tried
taking a closer look to these balls? Apart from the use of
material and size, are there other differences between them?
Have you counted how many sections are there making up their
surfaces?
A ping-pong ball consists of two sections. A tennis ball
has also two, but the shape of its section is different. For
volleyball, it looks more like a dice because it has six groups
of section.
For all these balls, the most complicated one is the football.
It has sections in the form of pentagon and hexagon, or five-sided
and six-sided shape respectively. Try counting the number
of pentagons and hexagons, how many are there? Well, the answer
should be 12 and 20. Now we have the most difficult count
of the day. How many vertices (places where more than two
sections meet) are there in a football? The answer is 60,
and is the biggest number we will be discussing in this article.
Back to some two thousand years ago, a man named Plato had
made some important observations. Plato was a Greek philosopher.
He noticed that if he had just some regular plane polygons
on hand, such as equilateral triangles, squares or regular
pentagons, he could build different solids using one kind
of polygons only. But he found out that he could make no more
than five different solids.
These solids are tetrahedron with four equilateral triangles,
cube with six squares, octahedron with eight equilateral triangles,
dodecahedron with twelve regular pentagons, and icosahedron
with twenty equilateral triangles. These five solids are now
collectively called Platonic solids to recognize the contribution
from Plato.
The most common of all Platonic solids is the cube. It has
six sides or faces made of squares. Since all its six sides
are identical, it is perfect for making dices for the game
of chance. In fact, although they are less common, you can
find other shape of dices with four, eight, twelve and twenty
faces.
Platonic solids exhibit another interesting property called
symmetry. You can always cut any Platonic solids into two
along any one of its edges in such a way that with the help
of a mirror, half of it provides sufficient information to
recreate a complete Platonic solid.
Extending this property of symmetry along all edges of the
same face, you can use mirrors equal in number to the edges
to recreate a complete Platonic solid from just one face.
Using three properly cut and placed mirrors, you can recreate
from one face a tetrahedron, octahedron, or icosahedron. Using
four mirrors you can get a cube, and five mirrors you can
get a dodecahedron. After completing a mirror well, you can
have funs playing with it.
As you may have noticed, a Platonic solid is an approximation
using plane surfaces to recreate a sphere. The more sides
are employed, the better the approximation will be. Obviously
an icosahedron has more sides than a tetrahedron and so it
looks more “spherical”.
But how about creating a better “sphere” with
more sides? Plato said that you could not use more regular
shaped surfaces. You may have to use shapes that are less
regular (such as isosceles triangle) or shapes of different
size. Of course you may want to have in your warehouse minimum
number of different items for an easier inventory control
if you are a constructor.
An architect called Buckminster Fuller discovered that we
could start from a Platonic solid first. He replaced each
face by a number of smaller surfaces of a few variations to
get a better approximation. Furthermore he found that the
edges of the faces should best be following a geodesic (the
shortest path between two points on a sphere), and so he called
his structure geodesic dome. Geodesic domes have the property
of requiring the smallest amount of material to contain a
biggest and strongest internal volume.
Another way of playing with a Platonic solid is to cut its
corners. If we cut the corners evenly with respect to all
sides joining that corner, you can also get a better approximation
to a sphere. Try cutting the corners of an icosahedron with
all new and remaining edges of the same length, and guess
what you will get? It is just a football. An icosahedron has
twenty faces of triangle and twelve vertices where five faces
meet. A proper cutting generates a football of twenty hexagons
and twelve pentagons. You may notice also that the length
of all edges of the pentagons and hexagons are the same.
Interestingly, for a structure with all its edges having
the same length, it can also be used to represent a stable
chemical molecule. It was only discovered not long ago in
1985 that carbon atoms can naturally bond themselves to create
a family of cage-shaped structure. The discoverers, Curl,
Kroto and Smalley, earned their Nobel Prize in 1996.
The cage with perfect symmetry is structurally the same
as a football and is also most stable. It has 60 carbon atoms.
In order to honour Fuller's work in the study of geodesic
structures, the discovers named this new family of carbon
molecules fullerenes.
Carbon can even form tube-shaped structures. Carbon tubes
can be made very long. Yet it can still be very strong because
they are joining together by their basic chemical bonding.
People soon discovered that industrial processes could be
developed to produce materials consisting of carbon tubes
of precise size. Such carbon tubes, of a size of the order
of nanometre (0.000 000 001 m), help creating a new class
of industrial technique called nanotechnology. Material such
as cloth can now be produced so that the tubes inside allow
only water and oxygen molecules to pass through but not bigger
ones such as bacteria or viruses. It is definitely useful
for fighting against SARS (severe acute respiratory syndrome,
or atypical pneumonia).
Let us get back to the mirror wells. There is a further
property of interest. Because a well helps reproducing a Platonic
solid, and we know that a Platonic solid has all its sides
connected without gap, so we can say that each well is a partition
of a three dimensional space with respect to the centre of
the solid. The number of partitions equal to the number of
sides of the Platonic solid, and all partitions are identical.
Using the mirror well for an icosahedron as an example, we
can pack twenty identical wells surrounding a single point
perfectly.
Now the fun begins. We still use an icosahedron as illustration.
We know that whatsoever you put inside the mirror well, it
will create 19 mirror images of itself. If you make a structure
that has links “passing” through the mirrors connecting
all its neighbouring images, you can always reproduce the
structure with twenty copies (some of them may have to be
laterally inverted) to make a rigid three-dimensional object.
If your structure has some moving parts that can move freely
within the well, your object can always have 20 moving parts
that will never collide.
Imagine that you can modify the structure so that it can
move up and down the well while keeping all its links through
the mirrors. You have now created a solid three-dimensional
object that can expand and contract. As soon as you are sure
that your structure will work in the well, you are also rest
assured that it will also work three dimensionally with twenty
replicas of its (some may have to be laterally inverted) joining
together.
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