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Publishing 152nd book of AP by this weekend//6th Regular Polyhedron composed of hexagons. Archimedes Plutonium<plutonium.archimedes@gmail.com> Oct 20, 2021, 2:26:57 PM to Plutonium Atom Universe AP's 152nd book of science-- The 6th Regular Polyhedra
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Archimedes Plutonium
Oct 21, 2021, 5:31:02 PM10/21/21
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Publishing 152nd book of AP by this weekend//6th Regular Polyhedron composed of hexagons.
Archimedes Plutonium<plutonium....@gmail.com>
Oct 20, 2021, 2:26:57 PM
to Plutonium Atom Universe
AP's 152nd book of science-- The 6th Regular Polyhedron has regular hexagon faces.
So, let me start the proof that the 6th regular polyhedron exists and that its faces are regular hexagons. Why hexagons? Why not 60-60-60 triangles or regular pentagons or even squares? Why regular hexagons? And the reason it is the 6-gon is because it is dual to the 3-gon, for you divide the 6-gon internally into 6 triangles and the dual of the tetrahedron is this CurvedStraightlineSphere CSS of hexagons.
One would think the 4-gon is better at being the infinity regular polyhedron since the Decimal Grid Number System is built of 4-gons in graphing. And square are far far more easy to negotiate a graphing system built on squares with empty space in between one number and the next number. It is far more difficult to tile a graph system with 6-gons than with 4-gons. But the 4-gon has a dual in the octahedron.
Science, Nature, wants symmetry and duality obeyed and so the face of the 6th Regular Polyhedron is a regular hexagon, only tiny tiny hexagons.
In this proof we recognize that we need a gap or hole and cannot have exactly 360 degrees. The Discrete true numbers of mathematics gives us enough of a gap and hole to allow for the existence of a 6th Regular Polyhedron.
So the gap and hole in 10 Decimal Grid is a gap of 0.1 distance from one number to the next number. And if we multiply 360 degrees by 0.1 we would have an angle deficit of 36 degrees to allow for a "go around rotation" and fill the entire sphere surface in 10 Decimal Grid System with tiling of flat planar regular hexagons.
Now in 100 Decimal Grid System our holes and discrete numbers have a gap of 0.01 empty space from one point to the next point so in 100 Grid, our deficit angle to make hexagons go around on a sphere is 360 x 0.01 = 3.6 degrees of going around to fill the sphere surface with flat planar regular hexagons.
Likewise for 1000 Decimal Grid system in that the gaps and holes to allow "go around" is 0.36 degrees with tiny regular hexagons.
So this is the pattern for the proof of the existence of the 6th Regular Polyhedron of faces of regular hexagons.
QED
But alongside the formal proof above, is the hands on, eye witness, I haul in or bring into the lecture hall or symposium, I bring in a actual sample of a 6th Regular Polyhedron. Meaning, I hold in my hands Calcite Crystals of geology mineralogy.
Seeing is believing and far far better of a proof than any actual write-up, such as the above write-up. For keep in mind, for over 2 milleniums, it was thought the Ancient Greek proof that only 5 regular polyhedron can ever exist, yet here in 2021, AP is showing the world the 6th regular polyhedron, holding calcite crystals as a proof.
2nd QED
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Oct 20, 2021, 9:23:07 PM (19 hours ago)
to Plutonium Atom Universe
AP's 152nd book of science-- The 6th Regular Polyhedron has regular hexagon faces.
It is a construction proof.
Construction in the idea that Space is point then empty space gap hole, then another point.
So a sphere surface is not a continuum, nor is a box surface. But rather a network of points with empty space in between one point and the next point.
It is this empty space that allows us to construct a Regular Polyhedron from plastering regular hexagons onto a point of a sphere where we ask the question, how large or small is a point in Space and where the point itself is the hexagon allowing for the empty space between points to be a angle short of 360 subtract sum of internal angles.
So the pentagon is 108+108+108 = 324 shy of 360 by 36 degrees to be a planar tiling but be a dodecahedron.
So now in 10 Grid the empty space is quite large at being 0.1 distance apart to the next neighboring point and it is this network of points that go to compose a sphere, a box a figure in 3D and even figures in 2D. But in something like 10^604 Grid the points are so close together separated by at minimum 10^-604 that one would think it is a continuum. So when we have hexagons on the size of 10^-60 tiling each point of a sphere in 10^40 Grid that the hexagons would be the points themselves.
And so we can easily imagine that if we had a Sphere in 10^604 Grid that the empty space separation is 10^-604 allowing for hexagons to cover the surface and be a Regular Polyhedron.
AP
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Archimedes Plutonium<plutonium....@gmail.com>
Oct 20, 2021, 11:50:30 PM (17 hours ago)
to Plutonium Atom Universe
The Construction Proof of 6th Regular Polyhedron
Few if any mathematicians alive today know of Discrete Geometry and a geometry without the obnoxious fake continuum and its Reals. This is why a 6th Regular Polyhedron was never taken serious and why so many hid behind the fakery of the proof of only 5 regular polyhedron.
Construction is long and we have to get to it piece by piece.
So all spheres and boxs and figures in 2D or 3D are not continuums but discrete points with empty space in between.
So this is one edge line segment of a pentagon in a dodecahedron.
..................
And this is the second pentagon with a edge that intersects the first
..................
The points of the first pentagon edge have to line up exactly with the points and empty space with the second pentagon edge as thus.
..................
..................
becoming intersected as thus
..................
So, my message in this post is that if the regular hexagons are made so small, and the sphere surface so so large, that it is similar to a tiny planar hexagon on Earth Surface tiling the entire planet Earth sphere.
So say the Earth was a perfect sphere and we had tiny hexagons on the size of a microscope bacteria. We then can tile the entire Earth to be a Regular Polyhedron.
What the empty space in between true points does, is allow for the tiling to have no gaps nor to have buckling or a bad fit. The empty space allows for this CSS CurvedStraightlineSphere tiled by tiny hexagons be a Regular Polyhedron.
Now some would say, why not tile with squares instead of regular hexagons? And my answer to that is that squares do not rotate around, but hexagons naturally rotate around to tile the sphere surface where each layer is naturally formed, while a square layer if fabricated taking a decision after each layer is built. There is a second objection to a square, in that a square cannot cover the sphere surface and leaving holes behind but the hexagon can as seen in this depiction.
. . . . .
. . . . .
. . . . .
. . . . .
Take a point in the center of that network of points on a sphere surface. Keep in mind that we are focused in on the microscopic scale of a large huge sphere. The 6-gon edge can coincide with the center point and then 6 more 6-gons can occupy the 6 points and empty space of that center, which a square cannot do.
. . .
. . .
Of course, no-one in Old Math could ever contemplate a 6th Regular Polyhedron for they had continuum, they had Reals with always more points between any two given points, and in a geometry like that, based on a liaring, that the Old Math so called proof of 5 and only 5 regular polyhedron made sense in that fake world geometry.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
2:18 AM (14 hours ago)
to Plutonium Atom Universe
Now there is a Uber Engineering Blog H3: Uber's Hexagonal Hierarchical Spatial Index showing a sphere almost tiled by hexagons on the 1/2 sphere with only 4 pentagons visible.
Another website shows this sphere with only 1 pentagon visible but with a variation in size of hexagons on the 1/2 sphere.
So here I am wondering if both websites reduced the size of the hexagons, reduced them to the size of the few pentagons they have, whether they can eliminate the pentagons altogether on the 1/2 sphere.
Size does matter, and so I am thinking that if you get the 1/2 sphere reduced to 2 pentagons, then make all the hexagons the size of those pentagons, that in the next round of tiling you need just 1 pentagon. Now reduce the hexagons further to that size of the lonely pentagon and then you no longer need a pentagon in the 1/2 sphere.
There probably is a math formula to use that inputs the size of the sphere, then calculates the size of the hexagon needed to tile the surface without any pentagon needed.
Of course some of these web pictures are computer simulations in which there is immense distortion. But in this website they include the continent landmasses. But still, much distortion.
Now another idea worth pursuing is from the soccer ball with its 12 pentagons and 20 hexagons, and to find out the size of the hexagon that can tile the interior of the pentagon. Then that size of hexagon should be able to tile the entire sphere surface. This uses the size of the 12 pentagons and 20 hexagons and factors in the size of the soccer ball to determine the size of what the hexagon should be to tile the entire surface in hexagons alone.
Now in theory, the soccer ball is not flat to accommodate hexagons, but if the hexagons are so tiny, it makes that patch of the ball so big that the hexagon thinks it is laying down on a flat surface.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
11:25 AM (5 hours ago)
to Plutonium Atom Universe
On Wednesday, October 20, 2021 at 11:50:30 PM UTC-5 Archimedes Plutonium wrote:
Now some would say, why not tile with squares instead of regular hexagons? And my answer to that is that squares do not rotate around, but hexagons naturally rotate around to tile the sphere surface where each layer is naturally formed, while a square layer if fabricated taking a decision after each layer is built. There is a second objection to a square, in that a square cannot cover the sphere surface and leaving holes behind but the hexagon can as seen in this depiction.
This started out as a conjecture that there exists a 6th Regular Polyhedron, and in the proof of its existence I am able to prove more new theorems such as the above suggests.
I got out a mosquito suit that was a netting of square blocks and wrapped it around my glass sphere to see the difference between that tiling and the hexagon tiling netting. And the trouble with squares is that they form lines and lines on a sphere inevitably are destroyed, so a square cannot be a Grid system for the sphere surface. However, the hexagon + pentagon in ratio of 20 to 12 for a total of 32, naturally tiles the sphere surface and the reason for the 32 in total, as I magic marker my soccer ball an rotate from north pole of pentagon and sure enough the 32nd is the south pole as pentagon, as I rotate and magic marker the tiling. I come to realize that 32 is just past the 3.14159... of the pi number for in 10 Grid System pi is approximated by 0.1 X 32 = 3.2.
Squares cannot be a Regular Polyhedron at infinity borderline of 1*10^604 and its inverse 1*10^-604.
However, the Hexagon faces at infinity constitute the worlds 6th Regular Polyhedron.
And in this proof, I had to prove that Hexagons Naturally Tile the Sphere Surface and no other n-gon can do that tiling of naturally going around and not intersecting, for the hexagon is a rotating polygon while the square creates long lines in tiling, the hexagon creates long rotation circles in tiling.
So this is one great benefit of the proof of 6th regular polyhedron, a Proof that Sphere Surface Grid System is built from regular polygons of the hexagon.
Now one can say the hexagon is composed of 6 equilateral triangles 60-60-60 and then say the 6th regular polyhedron is connected to the tetrahedron. But keep in mind that a hexagon face is a unit.
I should have this 152nd book of mine published by the end of this week in October 2021.
This book is a proof of the existence of the 6th Regular Polyhedron.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
12:12 PM (4 hours ago)
to Plutonium Atom Universe
The proof is a construction proof, the best type of proof possible because you have the object in your hands at the end of the existence proof.
STATEMENT: There exists a 6th Regular Polyhedron composed of microscopic hexagon faces which is formed at Infinity borderline of 1^10^604 and its inverse 1*10^-604 where pi digits have three zero digits in a row for the first time in pi digits.
Preliminary Construction Proof: Get a soccer ball whose cover surface is a pattern of 12 pentagons with 20 hexagons. With a magic marker pen I started a north pole of a pentagon and numbered each polygon as I rotated around the ball and at the 32nd polygon which is the south pole, it also was a pentagon. The reason for the pentagons being the poles is that in hexagon or mixed hexagon tiling, hexagons have a natural rotation, unlike squares if attempted to tile the sphere surface forms straight lines and those straight lines start to intersect as layed on a sphere surface. The hexagon tiling of rotation does not form straight lines to intersect with one another as tiled. This makes the hexagon a unique polygon in forming a Sphere Grid System. And the proof of the 6th Regular Polyhedron easily proves the conjecture that hexagons form a unique Sphere Grid System.
So we have our soccer ball. And we now also have a glass sphere as in my cover picture of my 152nd book of science shows with me pressing a hexagon netting fabric to form the Sphere Grid System. I need to press only to show 1/2 of the sphere surface is a hexagon, pure hexagon grid system. As without loss of generality in proof, if we an show 1/2 of the surface as covering by hexagons proves the whole sphere responds in the same way.
Now, in Old Math they had a fake proof that a 6th regular polyhedron was impossible due to if adding up the "internal angles" equals or exceeds 360 degrees means no regular polyhedron possible. The trouble with Old Math proof is that they were based in a fictitious world of geometry of a continuum and Reals as numbers when in fact the world of geometry is Discrete geometry where there are holes and gaps of empty space in between one number and the next successor number. In logic, you quickly learn if your math proof has a liaring lie in amongst your assumptions -- continuum and Reals --- then your proof based on that liaring lie will end up being a fake proof, a invalid proof.
The hexagons of a soccer ball are far far too large. If those hexagons were microscopic and tiling the soccer ball, we need no pentagons to tile that soccer ball and can tile it on pure hexagons. And this idea is in agreement with the idea that Sphere Surface Grid is a grid system of Pure Hexagons.
We prove the grid system of Sphere is a pure hexagon pattern.
So say we have a Unit Sphere, a sphere of radius 1 and our radius of hexagon is 1*10^-604. That is very microscopic for the smallest number in Old Physics is 10^-120 in Planck measure.
First we prove the Sphere has a Pure Hexagon Grid system of points, and as a corollary of that proof, we thence prove that the sphere possesses a 6th Regular Polyhedron of tiny faces of hexagons, no pentagons needed.
So if we had a 1 radius unit sphere and given 1*10^604 hexagons of radius 1*10^-604 (radius or diameter), Then as we tile this unit sphere, we have the 6th Regular Polyhedron.
This 6th Regular Polyhedron of vertices, edges and faces when reduced in lowest terms has faces of 1 in a formula of V-E+F where the tetrahedron is 6,4,6 and reduced to lowest terms is 3,2,3. The 6th Regular Polyhedron is of reduced terms as a,b,1 compared to tetrahedron 3,2,3.
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Archimedes Plutonium<plutonium....@gmail.com>
4:21 PM (now)
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Alright, so my proof of there exists a 6th Regular Polyhedron rests on a proof that a Sphere exists as a NetWork of evenly spaced points that constitute a Sphere Grid System and those points be the vertices of regular hexagons. Hexagons that start at a north pole as a tiny hexagon and keeps going round and round until the last hexagon is placed as a south pole. We can see the mechanics of this already in a soccer ball of 12 pentagons and 20 hexagons played out. Only in the 6th Regular Polygon we have 1*10^604 hexagons on the size radius or diameter of 1*10^-604 radius or diameter.
And as we tile the what can be called the Universal Sphere, the sphere becomes those network of vertice points with empty space in between the network, as seen in my cover picture of me tiling a glass ball sphere with a nylon netting of hexagons.
So as I prove that the Sphere in reality is a network of tiny straightline segments of 1*10^-604 in distance, these vertices is the sphere itself. As I prove that the sphere has this Grid System, or else a sphere does not exist at all.
So the Proof of Sphere = this network of points on the surface of at least 1*10^604 points on the surface. As I prove the Sphere grid system is constructed from regular hexagons twirling around from a north pole that creates a circumnavigation of the entire sphere ending up at a south pole, directly opposite north pole in construction of hexagon rotation. And only the hexagon can do this Sphere construction. Then the proof of Sphere Grid System also proves the existence of 6th Regular Polyhedron built of 1*10^604 hexagons as faces.
AP, King of Science, especially Physics
P.S. I should have this book published in the next few days as my 152nd science book.
Archimedes Plutonium<plutonium....@gmail.com>
Oct 20, 2021, 2:26:57 PM
to Plutonium Atom Universe
AP's 152nd book of science-- The 6th Regular Polyhedron has regular hexagon faces.
So, let me start the proof that the 6th regular polyhedron exists and that its faces are regular hexagons. Why hexagons? Why not 60-60-60 triangles or regular pentagons or even squares? Why regular hexagons? And the reason it is the 6-gon is because it is dual to the 3-gon, for you divide the 6-gon internally into 6 triangles and the dual of the tetrahedron is this CurvedStraightlineSphere CSS of hexagons.
One would think the 4-gon is better at being the infinity regular polyhedron since the Decimal Grid Number System is built of 4-gons in graphing. And square are far far more easy to negotiate a graphing system built on squares with empty space in between one number and the next number. It is far more difficult to tile a graph system with 6-gons than with 4-gons. But the 4-gon has a dual in the octahedron.
Science, Nature, wants symmetry and duality obeyed and so the face of the 6th Regular Polyhedron is a regular hexagon, only tiny tiny hexagons.
In this proof we recognize that we need a gap or hole and cannot have exactly 360 degrees. The Discrete true numbers of mathematics gives us enough of a gap and hole to allow for the existence of a 6th Regular Polyhedron.
So the gap and hole in 10 Decimal Grid is a gap of 0.1 distance from one number to the next number. And if we multiply 360 degrees by 0.1 we would have an angle deficit of 36 degrees to allow for a "go around rotation" and fill the entire sphere surface in 10 Decimal Grid System with tiling of flat planar regular hexagons.
Now in 100 Decimal Grid System our holes and discrete numbers have a gap of 0.01 empty space from one point to the next point so in 100 Grid, our deficit angle to make hexagons go around on a sphere is 360 x 0.01 = 3.6 degrees of going around to fill the sphere surface with flat planar regular hexagons.
Likewise for 1000 Decimal Grid system in that the gaps and holes to allow "go around" is 0.36 degrees with tiny regular hexagons.
So this is the pattern for the proof of the existence of the 6th Regular Polyhedron of faces of regular hexagons.
QED
But alongside the formal proof above, is the hands on, eye witness, I haul in or bring into the lecture hall or symposium, I bring in a actual sample of a 6th Regular Polyhedron. Meaning, I hold in my hands Calcite Crystals of geology mineralogy.
Seeing is believing and far far better of a proof than any actual write-up, such as the above write-up. For keep in mind, for over 2 milleniums, it was thought the Ancient Greek proof that only 5 regular polyhedron can ever exist, yet here in 2021, AP is showing the world the 6th regular polyhedron, holding calcite crystals as a proof.
2nd QED
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Oct 20, 2021, 9:23:07 PM (19 hours ago)
to Plutonium Atom Universe
AP's 152nd book of science-- The 6th Regular Polyhedron has regular hexagon faces.
It is a construction proof.
Construction in the idea that Space is point then empty space gap hole, then another point.
So a sphere surface is not a continuum, nor is a box surface. But rather a network of points with empty space in between one point and the next point.
It is this empty space that allows us to construct a Regular Polyhedron from plastering regular hexagons onto a point of a sphere where we ask the question, how large or small is a point in Space and where the point itself is the hexagon allowing for the empty space between points to be a angle short of 360 subtract sum of internal angles.
So the pentagon is 108+108+108 = 324 shy of 360 by 36 degrees to be a planar tiling but be a dodecahedron.
So now in 10 Grid the empty space is quite large at being 0.1 distance apart to the next neighboring point and it is this network of points that go to compose a sphere, a box a figure in 3D and even figures in 2D. But in something like 10^604 Grid the points are so close together separated by at minimum 10^-604 that one would think it is a continuum. So when we have hexagons on the size of 10^-60 tiling each point of a sphere in 10^40 Grid that the hexagons would be the points themselves.
And so we can easily imagine that if we had a Sphere in 10^604 Grid that the empty space separation is 10^-604 allowing for hexagons to cover the surface and be a Regular Polyhedron.
AP
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Archimedes Plutonium<plutonium....@gmail.com>
Oct 20, 2021, 11:50:30 PM (17 hours ago)
to Plutonium Atom Universe
The Construction Proof of 6th Regular Polyhedron
Few if any mathematicians alive today know of Discrete Geometry and a geometry without the obnoxious fake continuum and its Reals. This is why a 6th Regular Polyhedron was never taken serious and why so many hid behind the fakery of the proof of only 5 regular polyhedron.
Construction is long and we have to get to it piece by piece.
So all spheres and boxs and figures in 2D or 3D are not continuums but discrete points with empty space in between.
So this is one edge line segment of a pentagon in a dodecahedron.
..................
And this is the second pentagon with a edge that intersects the first
..................
The points of the first pentagon edge have to line up exactly with the points and empty space with the second pentagon edge as thus.
..................
..................
becoming intersected as thus
..................
So, my message in this post is that if the regular hexagons are made so small, and the sphere surface so so large, that it is similar to a tiny planar hexagon on Earth Surface tiling the entire planet Earth sphere.
So say the Earth was a perfect sphere and we had tiny hexagons on the size of a microscope bacteria. We then can tile the entire Earth to be a Regular Polyhedron.
What the empty space in between true points does, is allow for the tiling to have no gaps nor to have buckling or a bad fit. The empty space allows for this CSS CurvedStraightlineSphere tiled by tiny hexagons be a Regular Polyhedron.
Now some would say, why not tile with squares instead of regular hexagons? And my answer to that is that squares do not rotate around, but hexagons naturally rotate around to tile the sphere surface where each layer is naturally formed, while a square layer if fabricated taking a decision after each layer is built. There is a second objection to a square, in that a square cannot cover the sphere surface and leaving holes behind but the hexagon can as seen in this depiction.
. . . . .
. . . . .
. . . . .
. . . . .
Take a point in the center of that network of points on a sphere surface. Keep in mind that we are focused in on the microscopic scale of a large huge sphere. The 6-gon edge can coincide with the center point and then 6 more 6-gons can occupy the 6 points and empty space of that center, which a square cannot do.
. . .
. . .
Of course, no-one in Old Math could ever contemplate a 6th Regular Polyhedron for they had continuum, they had Reals with always more points between any two given points, and in a geometry like that, based on a liaring, that the Old Math so called proof of 5 and only 5 regular polyhedron made sense in that fake world geometry.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
2:18 AM (14 hours ago)
to Plutonium Atom Universe
Now there is a Uber Engineering Blog H3: Uber's Hexagonal Hierarchical Spatial Index showing a sphere almost tiled by hexagons on the 1/2 sphere with only 4 pentagons visible.
Another website shows this sphere with only 1 pentagon visible but with a variation in size of hexagons on the 1/2 sphere.
So here I am wondering if both websites reduced the size of the hexagons, reduced them to the size of the few pentagons they have, whether they can eliminate the pentagons altogether on the 1/2 sphere.
Size does matter, and so I am thinking that if you get the 1/2 sphere reduced to 2 pentagons, then make all the hexagons the size of those pentagons, that in the next round of tiling you need just 1 pentagon. Now reduce the hexagons further to that size of the lonely pentagon and then you no longer need a pentagon in the 1/2 sphere.
There probably is a math formula to use that inputs the size of the sphere, then calculates the size of the hexagon needed to tile the surface without any pentagon needed.
Of course some of these web pictures are computer simulations in which there is immense distortion. But in this website they include the continent landmasses. But still, much distortion.
Now another idea worth pursuing is from the soccer ball with its 12 pentagons and 20 hexagons, and to find out the size of the hexagon that can tile the interior of the pentagon. Then that size of hexagon should be able to tile the entire sphere surface. This uses the size of the 12 pentagons and 20 hexagons and factors in the size of the soccer ball to determine the size of what the hexagon should be to tile the entire surface in hexagons alone.
Now in theory, the soccer ball is not flat to accommodate hexagons, but if the hexagons are so tiny, it makes that patch of the ball so big that the hexagon thinks it is laying down on a flat surface.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
11:25 AM (5 hours ago)
to Plutonium Atom Universe
On Wednesday, October 20, 2021 at 11:50:30 PM UTC-5 Archimedes Plutonium wrote:
Now some would say, why not tile with squares instead of regular hexagons? And my answer to that is that squares do not rotate around, but hexagons naturally rotate around to tile the sphere surface where each layer is naturally formed, while a square layer if fabricated taking a decision after each layer is built. There is a second objection to a square, in that a square cannot cover the sphere surface and leaving holes behind but the hexagon can as seen in this depiction.
This started out as a conjecture that there exists a 6th Regular Polyhedron, and in the proof of its existence I am able to prove more new theorems such as the above suggests.
I got out a mosquito suit that was a netting of square blocks and wrapped it around my glass sphere to see the difference between that tiling and the hexagon tiling netting. And the trouble with squares is that they form lines and lines on a sphere inevitably are destroyed, so a square cannot be a Grid system for the sphere surface. However, the hexagon + pentagon in ratio of 20 to 12 for a total of 32, naturally tiles the sphere surface and the reason for the 32 in total, as I magic marker my soccer ball an rotate from north pole of pentagon and sure enough the 32nd is the south pole as pentagon, as I rotate and magic marker the tiling. I come to realize that 32 is just past the 3.14159... of the pi number for in 10 Grid System pi is approximated by 0.1 X 32 = 3.2.
Squares cannot be a Regular Polyhedron at infinity borderline of 1*10^604 and its inverse 1*10^-604.
However, the Hexagon faces at infinity constitute the worlds 6th Regular Polyhedron.
And in this proof, I had to prove that Hexagons Naturally Tile the Sphere Surface and no other n-gon can do that tiling of naturally going around and not intersecting, for the hexagon is a rotating polygon while the square creates long lines in tiling, the hexagon creates long rotation circles in tiling.
So this is one great benefit of the proof of 6th regular polyhedron, a Proof that Sphere Surface Grid System is built from regular polygons of the hexagon.
Now one can say the hexagon is composed of 6 equilateral triangles 60-60-60 and then say the 6th regular polyhedron is connected to the tetrahedron. But keep in mind that a hexagon face is a unit.
I should have this 152nd book of mine published by the end of this week in October 2021.
This book is a proof of the existence of the 6th Regular Polyhedron.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
12:12 PM (4 hours ago)
to Plutonium Atom Universe
The proof is a construction proof, the best type of proof possible because you have the object in your hands at the end of the existence proof.
STATEMENT: There exists a 6th Regular Polyhedron composed of microscopic hexagon faces which is formed at Infinity borderline of 1^10^604 and its inverse 1*10^-604 where pi digits have three zero digits in a row for the first time in pi digits.
Preliminary Construction Proof: Get a soccer ball whose cover surface is a pattern of 12 pentagons with 20 hexagons. With a magic marker pen I started a north pole of a pentagon and numbered each polygon as I rotated around the ball and at the 32nd polygon which is the south pole, it also was a pentagon. The reason for the pentagons being the poles is that in hexagon or mixed hexagon tiling, hexagons have a natural rotation, unlike squares if attempted to tile the sphere surface forms straight lines and those straight lines start to intersect as layed on a sphere surface. The hexagon tiling of rotation does not form straight lines to intersect with one another as tiled. This makes the hexagon a unique polygon in forming a Sphere Grid System. And the proof of the 6th Regular Polyhedron easily proves the conjecture that hexagons form a unique Sphere Grid System.
So we have our soccer ball. And we now also have a glass sphere as in my cover picture of my 152nd book of science shows with me pressing a hexagon netting fabric to form the Sphere Grid System. I need to press only to show 1/2 of the sphere surface is a hexagon, pure hexagon grid system. As without loss of generality in proof, if we an show 1/2 of the surface as covering by hexagons proves the whole sphere responds in the same way.
Now, in Old Math they had a fake proof that a 6th regular polyhedron was impossible due to if adding up the "internal angles" equals or exceeds 360 degrees means no regular polyhedron possible. The trouble with Old Math proof is that they were based in a fictitious world of geometry of a continuum and Reals as numbers when in fact the world of geometry is Discrete geometry where there are holes and gaps of empty space in between one number and the next successor number. In logic, you quickly learn if your math proof has a liaring lie in amongst your assumptions -- continuum and Reals --- then your proof based on that liaring lie will end up being a fake proof, a invalid proof.
The hexagons of a soccer ball are far far too large. If those hexagons were microscopic and tiling the soccer ball, we need no pentagons to tile that soccer ball and can tile it on pure hexagons. And this idea is in agreement with the idea that Sphere Surface Grid is a grid system of Pure Hexagons.
We prove the grid system of Sphere is a pure hexagon pattern.
So say we have a Unit Sphere, a sphere of radius 1 and our radius of hexagon is 1*10^-604. That is very microscopic for the smallest number in Old Physics is 10^-120 in Planck measure.
First we prove the Sphere has a Pure Hexagon Grid system of points, and as a corollary of that proof, we thence prove that the sphere possesses a 6th Regular Polyhedron of tiny faces of hexagons, no pentagons needed.
So if we had a 1 radius unit sphere and given 1*10^604 hexagons of radius 1*10^-604 (radius or diameter), Then as we tile this unit sphere, we have the 6th Regular Polyhedron.
This 6th Regular Polyhedron of vertices, edges and faces when reduced in lowest terms has faces of 1 in a formula of V-E+F where the tetrahedron is 6,4,6 and reduced to lowest terms is 3,2,3. The 6th Regular Polyhedron is of reduced terms as a,b,1 compared to tetrahedron 3,2,3.
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Archimedes Plutonium<plutonium....@gmail.com>
4:21 PM (now)
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Alright, so my proof of there exists a 6th Regular Polyhedron rests on a proof that a Sphere exists as a NetWork of evenly spaced points that constitute a Sphere Grid System and those points be the vertices of regular hexagons. Hexagons that start at a north pole as a tiny hexagon and keeps going round and round until the last hexagon is placed as a south pole. We can see the mechanics of this already in a soccer ball of 12 pentagons and 20 hexagons played out. Only in the 6th Regular Polygon we have 1*10^604 hexagons on the size radius or diameter of 1*10^-604 radius or diameter.
And as we tile the what can be called the Universal Sphere, the sphere becomes those network of vertice points with empty space in between the network, as seen in my cover picture of me tiling a glass ball sphere with a nylon netting of hexagons.
So as I prove that the Sphere in reality is a network of tiny straightline segments of 1*10^-604 in distance, these vertices is the sphere itself. As I prove that the sphere has this Grid System, or else a sphere does not exist at all.
So the Proof of Sphere = this network of points on the surface of at least 1*10^604 points on the surface. As I prove the Sphere grid system is constructed from regular hexagons twirling around from a north pole that creates a circumnavigation of the entire sphere ending up at a south pole, directly opposite north pole in construction of hexagon rotation. And only the hexagon can do this Sphere construction. Then the proof of Sphere Grid System also proves the existence of 6th Regular Polyhedron built of 1*10^604 hexagons as faces.
AP, King of Science, especially Physics
P.S. I should have this book published in the next few days as my 152nd science book.
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