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Jodo and the Architecture of Matter
Plan of
workshop :
1. Introduction
. Two important concepts, the integrality of the universe, everything
that is has come into being , and that it has a history, structure and
function are closely related, understanding structure is important.
Universality and structure. Building blocks.
2. This also has importance for the teaching and learning of mathematics. Space filling and mathematics learning. Geometry
the two important theorems. Tesselations. Why
we cant fill space with regular pentagons. Pythagorus
theorem.
3. What is the generalisation to three dimensions ? The Jodo kit and
three dimensions.
4. Plato and the structure of matter. The regular solids and the five
elements..
5. The chemistry of polygons. What happens when we join triangles and squares ? The Archimedian semiregular solids.
6. The dual shapes, the rhombic dodecahedron.
7. Eulers theorem.
8. Descartes Theorem.
9. Space filling in three dimensions, and the structure of matter. What
are the polyhedra that close pack ?
10. The shapes in the fruit shop. Pyramids. Tetrahedra don’t close
pack.
11. Chemistry and the structure of matter. Diamond. Carborundum.
12. C 60.
13. Viruses and geodesic domes
14. Radiolaria, Pollen.
15. Bees hives.
16. The compression experiment and the rhombic dodecahedron.
17. The Pharaohs Pyramid.
18. The pineapple
19. The four colour problem and the tetrahedron.
Plan of work shop
1. In this workshop
we will play with some toys and also discuss some simple but fundamental
things about science, mathematics , learning and teaching.
2. In the last century
advances in science have led to an increasing acceptance of a number
of very farreaching propositions. These are
a. The integrality
of the real world : i.e. reality doesn’t divide into separate
subjects like physics, chemistry, biology etc. All these subjects are
various manifestations of the structure of matter and the functions
exhibited by material structures which have come to be organised in
different ways.. The concept of the integrality of the real world also
underlies our observation that matter behaves in the same way in distant
stars and galaxies as it does here on earth and its vicinity. This has
led to our getting to know what is happening in places where we never
imagined we could reach. The integrality of the universe/real world
has also led to fundamental and deep advances in theoretical physics
leading to the discovery of electromagnetic theory, special and general
relativity, quantum electrodynamics. Making this assumption we have
been able to plumb the structures of molecules, biomolecules and begin
to decipher the genetic code.
b. Structure and function are not only closely related but in many ways,
determine each other.
c. Every thing has a history. Everything that exists has come into being
and is going through a process of change. Understanding how it has come
into being and how it is changing is what science is all about.
3. Putting the above ideas together in the last century we have accumulated
impressive evidence that there is only one story , and that is the story
of the real world. All the science that we study is different pieces
of this single story. Science is everywhere we care to look. All the
world’s a laboratory.
4. Science education
however is yet to come to terms with the above understanding. Many students,
myself included, have arrived at the above understanding only at the
end of their graduate studies. They spend their lives learning different
subjects. They miss the powerful learning synergies that are contained
in the above perspective of an integral real world. In this workshop
we will play with building blocks, and try to understand the integrality
of structures.
5. Structures are
things that fill space. Children like to play with space filling in
different ways.
6. We begin with
space filling in two dimensions. We can fill space with different kinds
of tiles. Equilateral triangles, squares, hexagons,. But we can’t
fill space with regular pentagons. We all know why. The angles at the
vertex don’t divide 360 degrees exactly.
7. Interestingly,
triangles of arbitrary shape can be arranged to fill space. So we discover
that this is because the three angles of a triangle always add up to
180 degrees. Space filling is a nice way to introduce kids to this fundamental
theorem of school geometry.
8. That a rectangle with sides a and b+c can be filled with two rectangles,
one with sides a and b, and another with sides a and c, is a nice way
to teach students what is probably the most important formula in school
mathematics : a.(b+c) = a.b +a.c.
9. Similarly the formula for (a+b) squared emerges from how a square
of side a+b can be filled with squares and rectangles.
10. Even Pythagorus theorem, that Mount Everest of school mathematics
becomes quite obvious, when we observe that a square of side a+b can
be filled in two different ways.
11. These approaches go over quite readily and naturally into three
dimensions. Children begin their play in three dimensions. The jodo
blocks can be used to show that a cube of side a+b can be filled with
two
cubes of sides a and b respectively plus other pieces which represent
boxes of sides a,a and b and a,b and b. We get the well known formula
for
(a+b) cubed.
1. Is there a similar three dimensional generalisation of the theorem
that the sum of the angles of a trianle add up to 180 degrees ? The
Jodo kit enables us to build three dimensional structures with great
ease. We believe that Jodo is the world’s most versatile kit to
build polyhedra and geodesic domes.
2. We first build the regular solids. Building triangles on triangles
using the trinex nexors we get the tetrahedron. Building squares on
a square we get the cube. When we build pentagons on pentagons using
trinexes we get the dodecahedron. What happens when we build hexagons
on hexagons ?
3. Going on to fournexes we build triangles on triangles to get the
octahedron. What happens when we build squares on squares using fournexes?
4. The icosahedron is obtained by building triangles on triangles using
fivenexes. We cant build any other regular shapes because the angles
at the vertex must add up to 360 degrees or less. There is only that
much angle. If we try to pack more angle at a point we get buckling
 i.e. negative curvature.
5. We can use the following notation to describe the polyhedra which
have all vertices identical to each other the so called isogonal polyhedra.
333 means three equilateral triangles at the vertex. The cube is 444.
The dodecahedron is 555. The octahedron is 3333. The icosahedron is
33333. What we are doing is combining faces at vertices  we get the
regular polyhedra by keeping all faces identical. Call this the chemistry
of faces. What happens when we combine faces which are different from
each other ? What kinds of chemical compounds result ?
6. We try different combinations. 344 gives us a triangular prism. 544
is a pentagonal prism. 3434 is the well known cuboctahedron. Experimenting
with different combinations we generate the 14 Archimedian semiregular
polyhedra.
7.There is an interesting property of duality  we interchange faces
and vertices, the cube becomes the octahedron and viceversa. The dodecahedron
becomes the icosahedron and vice versa. The dual shapes to the archimedian
polyhedra are known as the Catalan polyhedra. Interestingly the dual
polyhedra do not appear to have been known to the greeks except for
one or two simple ones like the rhombic dodecahedron. Why the greeks
didn’t discover the dual polyhedra is a real mystery. Why that
discovery had to wait for more than 2000 years until Catalan arrived
in the 19th century is another mystery. This is because the greatest
mathematicians of the world like Descartes, Kepler, Euler and Gauss
all worked on the regular and semiregular polyhedra in their time.
1. Euler is supposed
to have discovered the famous formula which bears his name V + F = E
+ 2. But we feel that this must have been discovered by Descartes who
came many decades earlier.
2. Descartes discovered this lovely formula relating the missing angle
at each vertex to the number of vertices. We check it out for various
Archimedian polyhedra.
3. Having constructed the regular polyhedra, lets look at their spacefilling
properties. Every child knows that cubes fill space. What about the
tetrahedron ? What about the other regular polyhedra ? It turns out
that by themselves these regular solids don’t fill space. However,
the tetrahedron and octahedron combination fills space.
4. We see these shapes in the fruit sellers shop. We also see them in
the structure of diamond crystals and crystals of quartz, flint, and
silicon carbide. It is not surprising that if diamond is the hardest
substance known to us, silicon carbide is the second hardest substance
carborundum. Quartz and flint are also very hard, and were humankinds
first cutting tools.
5. Jodo gives us a hands on elementary understanding of chemistry, moreover
an understanding that does not have to be unlearned later. It is simple
but fundamentally correct. We can conjecture that carbon should also
exist in the shape of the 566 polyhedron, which it does, as C 60. We
can conjecture about other possible polyhedral shapes for Carbon, and
Silicon compounds, based on what can be made with Jodo. Since we are
not professional chemists, we don’t know if all have been discovered
in the real world. In the world of today, the various possibilities
of carbon and silicon are at the forefront of modern technology. Advanced
is not necessarily complicated. We can teach chemistry in a different
way in our schools to equip the children of the world to demystify technology
and master their own futures with self reliant knowledge and understanding.
6. In this brief workshop we can only mention that Jodo also gives us
a hands on understanding of virus structure. Many viruses have the shapes
of Geodesic domes. Many radiolaria have the shapes of the regular and
semiregular polyhedra. Pollen grains also exhibit these shapes.
7. The universal architecture of matter from the crystals of physics,
to the molecules of chemistry and the microworld of biology coincides
with the shapes that can be easily built with Jodo. A simple experiment
with closepacking makes this point in a remarkable and dramatic fashion.
Maybe this is how diamonds got made under the pressure of the early
earth formation. As we play with Jodo, the pages of the single story
slowly start unfolding, we learn to read the story wherever we are.
We understand where the bees learned geometry to make their hives.
8. The close packing properties of tetrahedra and octahedra, which gave
us the structure of quartz also make an interesting toy. The Pharaohs
pyramid toy tells us why the hexagonal numbers add up to give cubes.
(Discussion)
9. We conclude by putting before you some simple but non trivial problems.
Simple, nontrivial, but doable problems are the essence of good science
teaching, of turning on kids to the joys of learning.
10. The pineapple
problem.
11. The fourcolour problem.

