On Friday, October 6, 2017 at 4:47:05 AM UTC-5, 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, 3-gon, 4-gon, 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
> 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 n-gons, 3-gon, 4-gon, etc etc and placed inside circle we get many of these crescent shaped figures, 3 for the 3-gon, 4 for the 4-gon, 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 arced-curve. 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 arced-curves. 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 cut-out of the square-- its 4 pieces form a star of arced-curves.