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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 far-reaching 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.