About spidrondeformation:
1. it is not a folding in a traditional sense, as there is no stage when I folded the paper all across the paper surface.
With my friend the perpetuum mobile Walt van Ballegooijen
2. I folded the traditional honeycomb accordion spidron net in 1979 edge by edge, vertex by vertex, that is why it took so much time, and needed so big trust to go on. Only in the last stage it became clear that the paper can be folded this way.
Hard work of the Spidron Team
3. In case of spidrons, it has an extra unique property: I did not want to fold the triangles by their edges but I tried to rotate them along axes which were pointed from the middlepoints of their base (which remained always on the baseplane) toward the center of the nests. The level of freedom of the whole system allowed to get rigid triangles at the end.
One of the reliefs
4. I know, it is strange for most of the mathematicians to hear, that the center of the spidronnest are closed, But I can prove it. I am sorry but for it I need personal meeting or I can write it only in Hungarian, because it is not so easy.
Special transformation of the nest
5. There is an extraordinary property of the spidron deformation: at a certain stage the monotone rotation of edges results two different, but equivalently possible rotation at the neighbouring - coresponding - edge. This property leads me to very strange idea about the 2D plane. The reversibility of the 3D movement during the monoton change - rotation - of one of the edges means that this movement is not a real 3D movement, only similar to it. To be extreme: I think the Spidron deformation proves that the 3D Euclidean space does not exist at all. All what we experience is only an embedding physical "space" in which the deformations of a 2D object takes place.
Dodecaspidron
6. The stage where this extraordinary property appears is a very unique stage. The equations are divided into two equivalently valid equations, so that the last undivided value of the "f", what is the angle between the edge - of the second ring's from outside of a nest's periphery - and the baseplane - is 35,26˘. It does not look a specific number, but if we measure the angle between the adjacent edges on the same (second) ring we find that this angle is exactly 90°! After this stage the angle is decreasing to 30°, inspite of continuing the rotation of the outer edge monotonously from 0° to 60°.
Sphidronised spacefillers (42 new shapes)
7. I described more (very many) consequences of this interesting deformation, but - I think - it is enough for now.
Sphidron rings
8. More Archimedean - and other different - reliefs can be fold from a flat 2D paper.
Timaean Serie - Sphidrondisc
9. I started to work with sphidrons (spidron with a "h" in between "p" and "i" - what means spherical or bent, twisted spidrons), which has more freedom and this investigation leads me to the description of a new type of geometry, where the points of the plane are little spheres, which can rotate, roll and slide on each other making possible infinite incredible deformations. To preserve the measure of the sphidrons I introduced logarithmic and perpendicularly swirling deformations. We can say that the spidron and its movement is the "sceleton movement" of the smoother and slider (does this word exist?) sphidron movements. I work on this project with a young Hungarian student, called Janos Erdos.
Conclusion:
Spidron deformation is not a traditional way of folding, and its unexpected properties allows us to think over the "facts" what we have learned about the 3D Euclidean space.
I found references of these ideas in lower dimensional topology, quantum gravity and other fields of mathematics and physics as crystallography and quasy-crystal theory. Most of them were mentioned by Hungarian scientists like
Mr. Peter Hamburger, professor of Indiana, Purdue University, author of the book:
Set theory - with A. Hajnal
Janos Hajdu from Koln, author of the book:
Introduction to the Theory of Integer Quantum Hall Effect, he was the student and personal colleague of Mr. Werner Heisenberg, and
Janos Polonyi, from Theoretical Physics Laboratory, Louis Pasteur University, Strasbourg, France
Emil Molnar - the professor of Budapest University of Technology and Economics
and
Andras Nemethi from Renyi Mathematical Institute, Hungary
... but as the main actor of the Spidron Story I have to mention
Mr. Lajos Szilassi who was the first to describe this very complex and new kind of deformation. I attach the link of his terrific paper which made the spidron movement a mathematical fact:
you can dowload it from here:
http://www.zentralblatt-math.org/matheduc/en/search/?q=au:Szilassi%2C%20L*
I have to thank the results of the spidron project to a lot of persons, you can find their names on our websites.
www.spidron.hu
www.spidron.hu/spidronised-spacefillers
www.spacecollective.org/edanet
...etc.
best regards,
daniel e