New Concept:

The Sphaper (Sphidron-paper)

the 2D plane of the

New Sphidron Geometry.

I found some pictures of galaxies, and these made more important to describe some of my thoughts:

IT IS Quite similar to our Sphidron formations:

Interplanetary medium:

http://en.wikipedia.org/wiki/Interplanetary_medium
http://en.wikipedia.org/wiki/Spiral_galaxy#Spiral_arms
Explanations how the real material is moving around the center and how the curves ov different layer circles are SLIDING around each other.

1. If we allow the "points" as to the "atoms" to have a minimal dimension in the sence of measure or volume, which is ROTATING

and

2. If we allow to the line to ROTATE around itself (in the sence of the points which are following each other on a line can ROTATE around the curves as axis) and

3. if we allow for the circular or polyhedral discs to be swirled in a way where the 1. and 2. elements can SLIDE on and around each other, and

4. if we allow to the lines (or curves) of 1. and 2. type to make spiralling movements around different centers, for example around the middlepoints of different regular polygons and circles, and

5. if we allow to the point to create different starfishes in the same vertex (or the same centers of the polygons, this way the quantity of the arms in a vertex or in the center can be changed from 1 to 2 to 3 and so on ... we need to introduce a constant which is a kind of "tolerance or flexibility, what meks possible to the "atoms" to jump over from one arm to another while the deformation and spiralling of the curves (2.) composing the surfaces (3.) - are taking place. (Look in the Proceedings BRIDGES, Banff 2009, Jim Bumgardner, page 304, and visit his website:

http://www.coverpop.com/whitney) and

6. if we extend this idea to the 3D following the two perpendicular logarithmic spiral deformation of the sphidrons, which guaranties the constant area and spherical swirling deformations of the sph(i)idronnest, and

7. if we understand well the curious movement of the spidronnest - and espcially the sph(!)idron nest, which has an extra property of SLIDING points and lines of 1. and 2. kind - where each ring has different deformation around the center and the edge's middle point (which are remaining on the baseplane of the original nest) in a way, and

8. if we understand that the sph(h)idronnests, which are crossing the centerpoint of the sphere are different from the ones which are laying on the surface of the sphere we would like to cover in a platonic and/or in an Archimedean way, as the ones which are crossing the center point of the sphere are swirling surfaces with "S" edges or loxodromes (and with points of Gaussian Curvature=0 in any point, or the the logarithmic curves are laying on a tangent plane - which is related to the Regge Calculus in the theory Quantum gravity and the low-dimensional topology > Peter Hamburger<) on the surface of the sphere, and the ones wich are on the surface of the sphere and have "C" edges after the deformation of 1. 2. and 3. kind, - as they had "S" edges before the deformation in the 2D plane.

If we can accept that this properties and conditions above are corresponding to the empiria and the observations of the "material sciences and theories" which is connecting the mathematical principles with the physical reality, WE CAN UNFOLD THE SPHERE in the 2D PLANE keeping the measures of the planar disks and WE CAN MAKE a (and more) very interesting NEW KIND OF BREATHING AND LIVING POLYHEDRA with constant surface but with changing volume, changing vertices and changing tension using the properties discriben in 1. 2. 3. 4. 5. 6. 7. and 8.

When I say "Living polyhedra" I mean that with universal tolerance the polyhedra can change freely from one shape to another. This change can be take place when the local vertices ar flat (so the starfishes of Jim have straight arms)

All the conclusions what I made are the result of three facts:

1. Sphaper (2D paper) can be bent.

2. Sphaper can be swirled around any choosen inner point

3. Sphaper with central symmetrical "S" sides can be changed in a way fro flat position, where the "S" sides are becoming spherical lines, i.e. mirror symmetrical "C"-s.

I know, there is a lot of work left to finalize this concept, maybe there are some terminological mistake in it, but I thought through a lot of times and I am certain, this will work well.

I think the Spidrons are the dead skeletons of the bending, breathing living Sphidron creatures we shall describe in the near future, with some promising properties in common.

Remark:

"There is no hole in the center of the Sphidronnests."

Thank you for your kind attention, any remark is very welcome

**Daniel Erdely**
www.spidron.hu