Do spheres exist?

[Joseph Austin]

Do spheres exist?

I was thinking about infinity and the Riemann sphere,

and then considered how once could create a uniform coordinate system for the surface of a sphere.

Though at first thought I would expect a sphere to appear "the same" and isotropic from any point on it,

I'm not even sure it is possible to cover the surface of a sphere in points at a uniform density.

That is, beyond the 20 apexes of the inscribed regular icosahedron, is it even possible to place additional points on the surface equidistant from all neighbors?  Or equivalently, divide the surface into smaller congruent areas?  (Of course geodesics generate congruent triangles, but not "regular" and not uniform.)

Is a sphere the limit of some sequence of regular polyhedra the way a circle is the limit of regular polygons?

If so, it would seem the surface of the sphere would be "rough" or bumpy, with the apexes of the polyhedron not all at the same distance from a "center".

 

Perhaps this is an issue Buckminster Fuller addressed?

And speaking of circles, do "degrees" exist?

Given the tools of Euclidean Geometry (compass, straight-edge), or for that matter some other device, 

how is it possible to subdivide a circle into degrees, that is, 360 equal arcs?

[kirby urner]

On Mon, Jul 11, 2016 at 1:57 PM, Joseph Austin <drtec...@gmail.com> wrote:

Do spheres exist?

I was thinking about infinity and the Riemann sphere,

and then considered how once could create a uniform coordinate system for the surface of a sphere.

Though at first thought I would expect a sphere to appear "the same" and isotropic from any point on it,

I'm not even sure it is possible to cover the surface of a sphere in points at a uniform density.

That is, beyond the 20 apexes of the inscribed regular icosahedron, is it even possible to place additional points on the surface equidistant from all neighbors?  Or equivalently, divide the surface into smaller congruent areas?  (Of course geodesics generate congruent triangles, but not "regular" and not uniform.)

Is a sphere the limit of some sequence of regular polyhedra the way a circle is the limit of regular polygons?

If so, it would seem the surface of the sphere would be "rough" or bumpy, with the apexes of the polyhedron not all at the same distance from a "center".

 

Perhaps this is an issue Buckminster Fuller addressed?

He does, but remember Synergetics is a discrete math and geometry with nothing touching anything else, no continuity or continua, just really high frequency energy events close together, as in granite and quartz.

Clearly such metaphysics has its problems but then so does the pure continuum analog flavor.  If drawing a spectrum, I'd say Synergetics is on the Democritus end of the Democritus <--> Euclid spectrum, Fuller says so himself.

So spheres as commonly imagined, as either completely solid (like bowling balls) or having a completely smooth and continuous shell (like... actually I can't think of any real examples, knowing about atoms as I do) have been defined out of existence.

Let me dig up a random passage to get the flavor:

224.07 Sphere: The Greeks defined the sphere as a surface outwardly equidistant in all directions from a point. As defined, the Greeks’ sphere's surface was an absolute continuum, subdividing all the Universe outside it from all the Universe inside it; wherefore, the Universe outside could be dispensed with and the interior eternally conserved. We find local spherical systems of Universe are definite rather than infinite as presupposed by the calculus's erroneous assumption of 360-degreeness of surface plane azimuth around every point on a sphere. All spheres consist of a high-frequency constellation of event points, all of which are approximately equidistant from one central event point. All the points in the surface of a sphere may be interconnected. Most economically interconnected, they will subdivide the surface of the sphere into an omnitriangulated spherical web matrix. As the frequency of triangular subdivisions of a spherical constellation of omnitriangulated points approaches subvisibility, the difference between the sums of the angles around all the vertex points and the numbers of vertexes, multiplied by 360 degrees, remains constantly 720 degrees, which is the sum of the angles of two times unity (of 360 degrees), which equals one tetrahedron. Q.E.D.

http://www.rwgrayprojects.com/synergetics/s02/p2400.html#224.07

You'll see he segues into Descartes' Deficit, which I've been harping on, along with V + F == E + 2 and 1, 12, 42, 92... as elementary level topics.

Am I saying this synergetics definition provides "the only right way" to think about spheres, and that Euclideans are "wrong" to play their preferred board game instead of ours? 

Nope.  Many sandcastles on this beach.  Some geometries are non-Euclidean, big deal.

Some say space is 3D, others 4D.  Live and let live.

[ We also allow others besides Euclideans to use the ruler and compass of course.  It's not like Euclidean geometers don't "own" these signature tools as their intellectual property, computers either ]

Kirby

[kirby urner]

On Mon, Jul 11, 2016 at 4:50 PM, kirby urner <kirby...@gmail.com> wrote:

 

[ We also allow others besides Euclideans to use the ruler and compass of course.  It's not like Euclidean geometers don't "own" these signature tools as their intellectual property, computers either ]

Kirby

Garbled...

"Own" or "don't own"?  -- that's not the question, I could have said.

The ruler and compass, and the digital computer, are all just sitting there, available to Euclideans and non-Euclideans alike, a level playing field.  Plenty of room in cyberspace (Cyberia) for all players.

Kirby

[Joseph Austin]

On Mon, Jul 11, 2016 at 1:57 PM, Joseph Austin <drtec...@gmail.com> wrote:

Do spheres exist?

On Jul 11, 2016, at 7:50 PM, kirby urner <kirby...@gmail.com> wrote:

http://www.rwgrayprojects.com/synergetics/s02/p2400.html#224.07

From your ref
"The demonstration thus far made discloses that the sum of the angles around all the vertexes of a sphere will always be 720 degrees or one tetrahedron^__less than the sum of the vertexes times 360 degrees^__ergo, one basic assumption of the calculus and spherical trigonometry is invalid.

So is this what the Lie Group people mean by saying a Sphere is not a Manifold?

I've been wondering whether curvature can be defined differentially.

That is, can I define a circle by the amount of "turn" instead of by the distnace from a center?

My concern with the sphere was that, 

unlike a circle which can be defined as the limit of a sequence of regular polygons, i.e. straight lines,

it seems the sphere cannot be defined as the limit of a sequence of regular planar figures (faces),

and hence a "sphere" would either not have uniform curvature or not have uniform surface density,

more like a golf ball that a ball bearing.

If I understand what you are saying, you agree with my understanding.

Joe

[kirby urner]

On Tue, Jul 12, 2016 at 10:03 AM, Joseph Austin <drtec...@gmail.com> wrote:

 

My concern with the sphere was that, 

unlike a circle which can be defined as the limit of a sequence of regular polygons, i.e. straight lines,

it seems the sphere cannot be defined as the limit of a sequence of regular planar figures (faces),

and hence a "sphere" would either not have uniform curvature or not have uniform surface density,

more like a golf ball that a ball bearing.

If I understand what you are saying, you agree with my understanding.

Joe

I think so Joe.

With a hexapent, you can get as many hexagons as you want to tile a sphere (no upper limit). 

Think of bathroom floor tiles defining a ball the size of the Earth.  It'd be a smooth as a ball bearing in terms of curvature.

The math is well summarized by these six slides:

http://s783.photobucket.com/user/danshep8/media/Civ_5_Hex_World/p016.png.html

​

Yes, each hexagon may in turn be triangulated with a local apex, a hub, giving six triangles. 

Omni-triangulating in this way gives us 1:2:3 in terms of N:F:E where N=V-2 (the total number of vertexes minus 2). 

V + F == E + 2 as usual

But wait: six equilateral triangles around a dot is perfectly flat, so are we saying these triangles are not actually equiangular?  Exactly right, they're not.

Every point in this graph is surrounded by at most |360 - epsilon| degrees where epsilon > 0. 

There's a "grain of sand", a tiny bit of "missing flatness" (a wedge, a sinus), that each vertex contributes to overall curvature.

If you add up all the slivers, wedges, the differences from 360 (perfect flatness), that total is precisely 720 degrees, the number of degrees in the four triangles of a tetrahedron.

In subtracting 720 degrees from a flat sheet (of like graphene) we make it curve around to connect with itself as a ball (like a fullerene).

The hexapent also has 12 pentagons.

Today July 12 is Bucky Fuller's birthday by the way, so apropos to be having this discussion. 

In recognition of the occasion, I was just given the gift of a new bibliography, a catalog to a nearby (Portland-based) collection.  My thanks to synchronfile.com

Kirby

[Joseph Austin]

Kirby,

You six math sided didn't show up in my mail reader.

At least, I didn't see any math.

Can you send a link?

I can imagine simply close-packing balls around a central ball in successive layers

1, 12, ...

But isn't that begging the question?  If balls don't exist, I can't "pack" them to generate

successively larger and "smoother" balls.  Can you do such packing with regular polyhedra? Of course, the points of contact of the imaginary balls could form apexes or perhaps tangents to faces or midpoints of edges of polyhedra.  I haven't studied geodesics enough know off the top of my head.

But if we start with polyhedra, the surface would never become "smooth", but only relatively so relative to the radius, a "quantum thickness" so to speak.

I guess where I'm coming from is that "circles" (Euclid's 3rd postulate) play a critical role in Euclid's geometry, yet from physics we are led to suspect that the "infinitesimal continuum" might not exist.  So then the question is, can we create a geometry without the third postulate--a "raster" or "quantum" geometry?

Joe

On Jul 12, 2016, at 4:39 PM, kirby urner <kirby...@gmail.com> wrote:

The math is well summarized by these six slides:

http://s783.photobucket.com/user/danshep8/media/Civ_5_Hex_World/p016.png.html

​

Yes, each hexagon may in turn be triangulated with a local apex, a hub, giving six triangles. 

Even so, are you not left with 12 anomalous regions at the pentagons?

In other words, are the triangles forming pentagons congruent to triangles forming hexagons, and if so, how do they maintain the same curvature?  if not,

how do they maintain the same point area density?

And finally, it would seem in such a planet you would have three independent choices of quadrays, or alternatively, dozenal rays.

An alternative question would be, given

[kirby urner]

On Tue, Jul 12, 2016 at 5:23 PM, Joseph Austin <drtec...@gmail.com> wrote:

Kirby,

You six math sided didn't show up in my mail reader.

At least, I didn't see any math.

Can you send a link?

This was the link I shared:

http://s783.photobucket.com/user/danshep8/media/Civ_5_Hex_World/p016.png.html

I also embedded one of the six slides.  

The slides hint at how the icosahedron is the basis for hexapent subdivision algorithm.  Maybe not news. 

 

I can imagine simply close-packing balls around a central ball in successive layers

1, 12, ...

But isn't that begging the question?  If balls don't exist, I can't "pack" them to generate

successively larger and "smoother" balls.  

Anything that exists in reality will be fine to pack with e.g. bowling balls, ball bearings... ping pong balls.  

They're not continuua either.  Made of atoms.  Democritus.  Neither are atoms "perfect solids" -- more like knotted energy.

1, 12, 42, 92... is the Icosahedral Number Series, so yes, you can use those as a guide to create more and more V, to whatever frequency limit.

http://www.4dsolutions.net/ocn/xtals101.html

https://oeis.org/A005901

 

Can you do such packing with regular polyhedra? Of course, the points of contact of the imaginary balls could form apexes or perhaps tangents to faces or midpoints of edges of polyhedra.  I haven't studied geodesics enough know off the top of my head.

They don't have to be "imaginary balls".  Use the ping pong balls of ordinary experience.  Or just imagine them (ping pong balls).

Whether balls appear continuous or not has to do with frequency i.e. the length of the intervals between so-called nodes.  

Microscopic or macroscopic?  Resolution matters.

 

But if we start with polyhedra, the surface would never become "smooth", but only relatively so relative to the radius, a "quantum thickness" so to speak.

Or define polyhedra to be wireframes.  They're skeletons.  Empty windows.  No "surfaces" need apply.

I'm happy to pinball around among these various definitions.  

The neural nets involved are smart enough to recognize the patterns.  Polyhedra are also graphs.

 

I guess where I'm coming from is that "circles" (Euclid's 3rd postulate) play a critical role in Euclid's geometry, yet from physics we are led to suspect that the "infinitesimal continuum" might not exist.  So then the question is, can we create a geometry without the third postulate--a "raster" or "quantum" geometry?

Joe

Yes, but it would be Non-Euclidean.  

I'd say Synergetics embeds one such geometry.  

Karl Menger had a "geometry of lumps" proposal, which I merged with the Bucky stuff as they were on the same page.

I also call it a "discrete geometry".  More atomic.  Like Democritus.  Different maxims & definitions, live and let live.

Kirby