Pythagorean Theorem in the Third Dimension

I guess I have found the only possible 3D deformation of the Euclidean plane which - regarding of the 9 vertices of the basic shapes of the Pythagorean Theorem, like the triangle and the squares - is preserving the validity of the theorem. The triangle in the 2D plane is rotating and shrinking, but its proportions and angles don't change. I tried to make some drawings to demonstrate this interesting swirling deformation below.


This picture shows how to make SpHidron deformation from the classical Spidron deformation


The arms of the Sphidron disc are rotating around themselves. The longer distance from the center increases the measure of the rotation. Here the change of the rotation is continuous while in the case of Spidrons it is discrete.


In between every pair of twirled arms must be another arm which like the twirled ones remains in one plane but it isn't not rotating around itself - like an axis - during this deformation.




The spiral arms one by one remain on their own plane, but these planes on which they lie on can be lifted up and pushed down alternately. This way you can make more and more dense surfaces in the same volume.




To make this deformation possible, you have to change your previous imaginations on the plane. It is a little more complicated, as the elements of the plane are not points with no dimension, but they do have dimensions, what make possible their rotation and sliding. While two of the neighboring "spoints" remain neighbors, the rest of them can be changed. Just like in the case of a simple pearl-string where every bounce has only and maximum two permanent neighbors.



Copyright 2009 -2010
Drawings by D Erdely, computer visualization by Janos Erdos

Mon, Mar 15, 2010  Permanent link

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Visualisation of the Hungarian Pavilion at the 12. Venice Biennale of Architecture 2010

Design of the pavilion is a result of Dániel Erdély and The Spidron Team

It is not the winner!, It is the looser application!
Please don't let me misunderstood!




The newest item in the exhibition space is the Caduceus. It is a rotating attribute of Hermes' wand.

Animation can be seen:
http://www.youtube.com/watch?v=NobDdvmq9yI

The bent and twisted surface of the wings can be unfolded into the area between the spiraling tubes.


Entrance


Atrium


Exhibition Hall


Spidronized Archimedean Solids and SpHidron movement


Detail of the wall


Detail of the Main Hall

We are going to exhibit the Spidronized Archimedean Spidroballs,
http://spidron.hu/archispidron/

 http://spidron.hu/archispidron/deforming  and swirling Sphidron disks,
http://www.youtube.com/watch?v=a7hFn1ZDHHA

deforming reliefs,
http://www.youtube.com/watch?v=rvCBO9xChGA&NR=1

toys,
http://www.youtube.com/watch?v=xupWsbc4zhg&NR=1

tiles,
http://spidron.hu/sparchicards/

and 42 brand new space-fillers patented recently.
http://spidron.hu/spidronised_spacefillers/

You can see more animations here:
http://www.youtube.com/watch?v=lwrTVhtn0Pg
http://www.youtube.com/watch?v=RbNcXdnXrkc


Thank you for your kind attention.

Copyright 2010 Daniel Erdely and The Spidron Team

Sun, Feb 21, 2010  Permanent link

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Possible Unfolding of the Sphere

There is no point, no straight line, no circle, no sphere and no space in a manner how Euclide described them. There must be tiny elements, of which the plane is composed. These elements must have axes and radius to be able to slide and rotate around each other. If the particles -
i.e. atoms - of the material (what have been called "ideal plane" by Euclid) can rotate like a spiral string, the sphere can be unfold to the plane without any problem preserving its metrical measures along the string and the other distances and angles also can be calculated. The inner arms are the rotating string arms, it will be a half of the final sphere. They are becoming straight lines on the surface of the sphere. The outer parts of these "S"-like figures can be bent and twisted and creating another halfsphere on the top of the other one.



Enjoy this music:
http://www.youtube.com/watch?v=5T3FXFnoTzE&feature=quicklist&playnext=8&playnext_from=QL



The plane what bisects the sphere can be deformed similar way as it is shown on (partly) the figure above. The orange ribbons are showing the tangent planes with simultaneously equivalent Gaussian curvature.

Tue, Feb 16, 2010  Permanent link

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No Point - No Sphere - No Space

Misleading Euclidean Geometry can cause total chaos in the future of humanity. It is quite obvious that somebody or some superpowers in the past hide the secret of the true geometry. Maybe it was the knowledge what sunk with Atlantis. Descartes and Johann Heinrich Lambert had the last attempt to recover the ancient knowledge, but his colleagues killed his theories on swirling "space" and other structures. The Sphidron Geometry renewed his and his ancestors ideas and made clear that the universe is a discrete entity made of finite elements. No point exists, the spheres are tori (plural of torus) with axes and the space is only 2D in an embedding physical context.



you can see here some elements of the New Sphidronized Geometry without points and spheres.

we are looking for the polyhedra which have changing egdes meeting in its vertices and surfaces which can be applied on the surface of the torical spheres. Gauss knew very well, that Kant made a mistake saying that the three dimensional picture of the universe is an a priori knowledge of the mankind. Einstein, Bolyai and Riemann made the situation more difficult with more complex visions of reality, while the real life is a comprehensive, simple two dimensional phenomenon with some thickness. Everything can be unfold. We are looking for the surface which is a deformed disc on the paths of two or three logarithmic golden spirals. On this surface the Pythagorean theory might be surviving. It is the only surface what can be said really 3 dimensional, but this dimension is only a tricky shadow of the imagined one. The rest is only is a birth trauma of the fetus, who was confronted to the gravity after delivery.

How could a point rotate without a radius? How could a sphere rotate without an axis?
But Point with a radius is not a point anymore, and a sphere with only one axis is not a sphere anymore. They are only abstractions of limited human fantasy.

Sat, Nov 28, 2009  Permanent link

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42 New Spidronized Spacefillers are Patented in Hungary

Please visit our new site.

http://spidron.hu/spidronised_spacefillers/

I promise, you will be satisfied.
One of the most beautiful Spacefillers - multiplied.



FS-41 - This complex spacial model - just like the cube or any kind of brick, fills the space without any ovelappings and gaps.

Fri, Nov 20, 2009  Permanent link

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Rational Sphidron Ruler

Make it simple!
Make the irrational rational!

All Rational Shidron Rulers are made of two similar spiral arms.



The irrational distances, like sqrt2 or Pi can be transformed to rational ones as requested.
All you need to perform this magic is a Golden Logarithmic Sphidron Ruler for this operation.
As by Fibonacci numbers - by increasing the numbers of subsections - you can finely approach the Golden Mean, you can divide any sequence of the spiral to as many parts as you like to approach it. If you have a Golden Logarithmic spiral, you can slide any sequence of this spiral to any "point" of it from which you can approach the A and/or B as precisely as you like by smaller and smaller golden sequences.


With the Rational Sphidron Ruler you can make any distance rational.
Special thanx to Kathy Pedro, Lisboa

Sun, Nov 8, 2009  Permanent link

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New Spidron Geometry - ELEMENTS

Expected results

I made Euclide's elements with golden log spirals and it works. I think you know the most about it.
Please help to communicate and solve the final parts of the work.
It is a good feeling, especially the parallels postulate 5 are proved very elegantly, but I like all of them.
You can try if you give me the credits: use metric golden log spirals and it's segments (it is the nicest!) instead of the concepts of fixed points and a "straight line" between them!

The points A and B can be fixed, while the sequence of the golden log spiral can be moved rotated or chosen in different ways) keeping its measure. postulate 1 For example it is 1 inch.
And the 1 inch section of the golden log spiral is (GLS) always similar to a part of an other GLS. postulate 2
The point A can be the source of a finite long spiral or a segment of it.

In case of postulate 3 where Euclide determined point and "distance", we could use "sequence" or "half-spiral" which can have different shape.
Half spiral is where A is a source, and whole spiral is a spiral where both A and B are sources and the spiral arms are meeting exactly in halfway of them. This theory was based on my spidrons and the newest spHidrons!



All the lines are the same length. the last two figures are: half logarithmic golden spiral and the whole one.




All consequences derived from the new postulates are solving infinite more dimensional questions, paradoxes and anomalies. Including optical illusions and incommesurability.
More philosophical results are on work on the last years. Fig by Jano Erdos

About postulate 4 it is true that A does not have to be a source point, because the normal cross section of the two spiral is remaining normal while the whole shape (for example looking through a circular whole) is looking as a rotating or coming closer and closer to us, while the normal cross section does not change under any circumstances. And the parts to what the spiral arms are cutting the plane (or a circle) are always equal.

If the spirals are whole spirals (A and B are sources) the parallel sections of the third SPline will never meet (only!!!) in the infinity while they are crossing each other several times. In case of an equilateral triangle's A, B and C points (where C is the center of the space [circle, universe, horizon] they do meet and exactly ONLY one time and only in the infinity! postulate 5 any spidronised regular polygon can be made this way with uncrossing arms.

So it seems to me that it the total merge of the Euclidean, spherical and Bolyai (and who knows which else? May be Emil Molnar knows it better!) geometries in one system.












Nautilus snail is not a golden log spiral. John Sharp's remark.

Details of the SpELEMENTS are on note papers. I'll publish it soon.


www.spidron.hu
www.spidron.hu/spidronised_spacefillers
www.spacecollective.org/edanet


Erdély Dániel
+ 36 70 514 8885
www.spidron.hu

Sun, Nov 1, 2009  Permanent link

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When the Deformation is Spherically Symmetrical

You draw a triangle on a surface of a sheet of circular shaped paper.
You swirl and twist it by two perpendicular logarithmic spiral
one of them is twisted around the axis which is in the center of the circle and the other one is around an axis which is a radius of the circle.



As the deformation is spherically symmetrical, the vertices of the triangle are preserving the ratio of the original triangle.
The original triangle is laying on a 2D paper, but the third triangles is "laying" in the embedding physical 3D space.

Sun, Oct 4, 2009  Permanent link

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Celebrate the New Geometry - It was born TODAY

Dear friends, and loved ones!

Listen to the new geometry based on
fluid physics vortex-, swirl dynamics,
and (only!) 2D Eucledian geometry!



No more Eucledian dimension exists!
All the rest is only a trick, "space" of manipulation or a result of a birth trauma.
When we had to experience the existence of gravity instead of the equilibrium and unorienatbility of our reception in the womb of our mother.

The rest of the dimensions are the values and measures
of the embedding physical world's properties.

Break the mirror now!

You can link the so called THIRD dimension's parameters
to the points - defined by x and y parameters - of the Eucledian ones
in infinite ways like: gravity, time, colour, temperature, popularity, smell or fear.
Each of these parameters are only randomly choosen properties,
just like the measure of the illusion what you are insisiting on,
the 3rd dimension. It does not exist at all! All right?

• There is no Spheric Geometry.
  I deny

• There is no Hyperbolic Geometry.
  I deny

• There are no 8 Geometries, at all
  only some, maybe only one

Can you release it? Keep on trying! It takes some time, for sure...
... but at last ...
... the life will be happier for all of us - suddenly.

Hugs

d erdély
cell: +36 70 514 8885
edan@spidron.hu

Sat, Sep 12, 2009  Permanent link

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New Sphidron Geometry: Sphaper - Interplanetary Medium?

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

Thu, Jul 30, 2009  Permanent link

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