
Buttons in Geometry, defined Re: what are hyperbolic objects? Re: unification of conic sections with regular polyhedra
Posted:
Oct 7, 2017 2:09 AM


On Friday, October 6, 2017 at 4:47:05 AM UTC5, Archimedes Plutonium wrote: > Alright, making some progress. What I need now is the negative geometry of polyhedra. The negative geometry is Hyperbolic geometry. I have 1 great resolution so far. > > Eucl = Ellipt unioned Hyperb. > > It is summarized in one picture )( >
Alright, I am making progress on this, really great progress. And I scheduled myself to have this done by January 2018. Looks like I could do this by November 2017 at the rate I am going.
> Where the elliptic ) is counterbalanced by the hyperbolic ( yielding the Euclidean straightline  > > So, what is the Hyperbolic of circle, oval, ellipse, 3gon, 4gon, etc? Here we have some clue as the parabola and hyperbola appear to be "holes in space" so that the elliptical geometry figures can fit inside the hole. > > So, what type of figure emerges when we insert a dodecahedron inside a cube, fill in the empty space. Then remove the dodecahedron and examine the figure remaining. We can call it a "hole". Is it a fancy type of parabola and hyperbola in 3rd D. >
Alright the idea of a Euclidean straightline is the result of a Ellipt geometry curve canceling a Hyperbolic geometry curve and resulting in a Euclidean straight line is the model to go by. So, we ask how can we cancel in 2nd D plane figures and how in 3rd D solid figures? How can we possibly get to something like
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> To answer some of these questions, I have fallen back to 2nd D. And trying to describe the hole inside a square where a circle fits. Now there are 4 points of contact of circle with square, and how do we describe those 4 crescent shaped figures, are they parabolas or hyperbolas triangle shaped with a curved hypotenuse. > > Now, if we take all the ngons, 3gon, 4gon, etc etc and placed inside circle we get many of these crescent shaped figures, 3 for the 3gon, 4 for the 4gon, interrupted only by vertices of the polygon. Now I think these are called Digons, crescent shaped figures. >
So, here, let us do an EXPERIMENT. We draw a circle and a inscribed square and we cut out the circle. Then we cut off those "button" portions leaving us with 4 buttons and the square.
Now I call them buttons for there are 4 of them, each with a straightline and a circle arc.
Now, I invert the buttons and lay tip to tip the four and recreate the square.
Now, I inscribe a circle inside the square the cut the buttons from, a smaller circle than the original circle, and cut out that circle, leaving me with 4 triangle shaped pieces. They have 3 sides, 2 are straightlines and the hypotenuse is a arcedcurve. Now, I assemble the 4 arced triangles forming a star like figure.
> So, now, in the Plane, in 2nd D the goal is to match up a positive curvature figure with a negative curvature and end up with a straightline > > )( > > The curvature figures canceling one another yielding the straight line  > > The goal in 3rd D, is for a positive curvature figure regular polyhedra, to combine with a negative and end up as a cube. > > Getting tougher and tougher by the day, but, I have until January to achieve. >
Alright, the above is a great advance, for what I aim to do now is consider all the regular polyhedra of 3rd dimension enclosed inside a sphere. And what we intend on doing is cut at a perpendicular every side of the entrapped hollow regular polyhedra, cut at a perpendicular so that we can collect BUTTONS of the faces. Each face has a rounded capped form as we cut inside the polyhedra, due to the sphere. For example the icosahedron has 20 buttons, the dodecahedron has 12 buttons. The octahedron has 8 buttons. Now once collected and inverted, can they go back into building say a inverted octahedron where the outside is a plane face but inside is the arc curve of the sphere cut away.
So, here, what I am looking for is a elemental Elliptic geometry unit and a elemental Hyperbolic geometry unit. I believe the units are buttons. If so, then the unification of conics with polyhedra, is one of straightlines versus arcedcurves. And it is conics that shows us two of the conics are hyperbolic arced curves parabola and ellipse.
As for polyhedra arced curves those have never before been discovered, totally new on the scene to mathematics. Where we take buttons and craft a object. Just as the star shaped figure of the circle cutout of the square its 4 pieces form a star of arcedcurves.
AP

