missed an 8

4 times pi divided by 43,200 is 2.908882086657215961539615394846141477e-4

 

Also I would like to note that the angles around the outside of the diagram in #162 are for figuring purposes only and are as if the diamond is extended.

On the actual sphere the borders would involve being next to diamonds whose division lines would wind up coming in in a mirror image angle manner.

That is the middle intersection would not read 105\75\105\75 it would show as 105\75\75\105. And the 60 degree angles at the 120 corner are there as well for figuring purposes, as the 120 corner is actually a three point, as in 120/120/120, not the 120/60/120/60 shown for smooth progression from 60 to 90 degrees in 7.5 degree increments.

Edited April 24, 20178 yr by tar

Thread,

 

Like DrP's orange slices, each of the four color wheels division lines is part of a great circle, where they all intersect at two opposite points which happen to be the three points each at the intersection of the three diamonds on either side of the color wheel you are looking at that are not part of that color wheel. For instance, considering the red color wheel, which consists of diamonds 1,2,3,4,5, and 6, you can spin that wheel around an axis that goes through the point where diamonds 9,10 and 11 touch and the opposite point where diamonds 7,8, and 12 touch.

 

The 16 divisions of lets say diamond 1 are created by 5 red slice lines at 15 degree intervals, 5 yellow slice lines at 15 degree intervals. In the case of diamond #1 the degree lines would be 330R, 345R, 0(360)R, 15R, and 30R, for the red wheel division lines, and 330Y, 345Y, 0(360)Y, 15Y, and 30Y for the yellow wheel division lines.

 

The convention I suggest is to consider the "bottom" of each diamond, the direction from which the increasing degrees are coming from, suggesting that the bottom corner be the one consistent with the base of both arrows depicting the direction of increasing degrees running around the sphere from 0(360) around to that same point in the center of diamond 1 for the red and yellow wheels, and the center of diamond 11 for the blue and green wheels.

 

As a picture is worth a thousand words, I drew a silver balloon out, with the 15 degree divisions and the arrows showing the direction of increasing degrees.

 

 

Also, by convention, I am suggesting that a particular division size be stated for the consideration of the entire sphere, and that the midpoint of a designated division be considered the "direction" of that designated division described by the coordinates of the bottom corner of that division.

 

Regards, TAR

When I use the term "three points" I just realized it could be interpreted as three points. I mean it to mean two opposite points where three diamonds touch. Here I am making a distinction between the 6 points where 4 diamonds touch, which I call four points, and the 8 points where three diamonds touch, which I call three points. Perhaps I should say threepoints, or three-points. But in any case, these 8 three points, analogous to the corners of a cube, in spacing and in the intersection of three faces of the cube in a "corner" can be thought of as 4, opposite corner pairs. Each of these pairs, if an axis is drawn through them is the axis of a color wheel.

raw garnet shape

 

https://www.bing.com/images/search?view=detailV2&ccid=nowVWIX1&id=C007BF16FABC5E16D6B36DCC9973301243F428EF&thid=OIP.nowVWIX1VsYoEfOxVWtUVgEsEs&q=garnet+shape&simid=608004509805773192&selectedIndex=0&ajaxhist=0

almadine Garnet

https://www.bing.com/images/search?q=almandine+garnet&id=0C7E8E6902765A577EC8D82E8674F7CCD2F4DE60&FORM=IARRTH

chemical composition, crystal category, etc

 

https://en.wikipedia.org/wiki/Almandine

TAR spherical coordinate system consists of designating the four three points adjacent to the South pole four point as Red, Yellow, Green and Blue, looking at the South Pole, moving counter clockwise around the pole.

Each of the four threepoints becomes the center of rotation of an axis going through the center of the sphere.

These are analogous to the four axis of a tetrahedron. 

Looking at them from the south, each axis can be imagined as putting out an infinite number of great circles that intersect at the other end of the axis on opposite side of the sphere,

The line going through the South pole is the 0/360 line and the other lines are designated in degrees in a clockwise direction around each of the four axis.

Twelve diamonds are described by drawing the great circles at 0 degrees, 60 degrees, 120, 180,, 240, 300 and 360.

The 0, 60, 120 are the same circle as the  180, 240, 300 but retain them all because the intersections of certain of the degrees on the six diamonds that are furthest from the axis ends, around the middle of the sphere, in reference to each axis, allow a description of every possible direction from the center of the sphere with two coordinates.

Diamond 1 through 12 are numbered as follows.

 

1 Red 180-240 Blue 120-180

2  Yellow 240-300 Blue 60-120

3 Green 300-360 Blue 0-60

4 Yellow 180-240 Red 120 180

5 Green 240-300 Red 60-120

6 Blue 300-360 Red 0-60

7. Green 180-240 Yellow 120-180

8 Blue 240-300 Yellow 60=120

9 Red 300-360 Yellow 0-60

10 Blue 180-240 Green 120-180

11 Red 240-300 Green 60-120

12 Yellow 300-360 Green 0-60

Notice each color appears 6 times, the six 60 sections.

So if you have a point in space at the  center of a sphere, and designate a South pole surrounded by four tetrahedral axis you designate one as Red and the whole system is determined and every direction in space can be designated with color degree, color degree and every point in space describable by adding a distance to the direction.

Copyright Thomas A. Roth Aug 8 2021

 

On 8/8/2021 at 2:48 PM, tar said:

TAR spherical coordinate system consists of designating the four three points adjacent to the South pole four point as Red, Yellow, Green and Blue, looking at the South Pole, moving counter clockwise around the pole.

Each of the four threepoints becomes the center of rotation of an axis going through the center of the sphere.

These are analogous to the four axis of a tetrahedron. 

Looking at them from the south, each axis can be imagined as putting out an infinite number of great circles that intersect at the other end of the axis on opposite side of the sphere,

The line going through the South pole is the 0/360 line and the other lines are designated in degrees in a clockwise direction around each of the four axis.

Twelve diamonds are described by drawing the great circles at 0 degrees, 60 degrees, 120, 180,, 240, 300 and 360.

The 0, 60, 120 are the same circle as the  180, 240, 300 but retain them all because the intersections of certain of the degrees on the six diamonds that are furthest from the axis ends, around the middle of the sphere, in reference to each axis, allow a description of every possible direction from the center of the sphere with two coordinates.

Diamond 1 through 12 are numbered as follows.

 

1 Red 180-240 Blue 120-180

2  Yellow 240-300 Blue 60-120

3 Green 300-360 Blue 0-60

4 Yellow 180-240 Red 120 180

5 Green 240-300 Red 60-120

6 Blue 300-360 Red 0-60

7. Green 180-240 Yellow 120-180

8 Blue 240-300 Yellow 60=120

9 Red 300-360 Yellow 0-60

10 Blue 180-240 Green 120-180

11 Red 240-300 Green 60-120

12 Yellow 300-360 Green 0-60

Notice each color appears 6 times, the six 60 sections.

So if you have a point in space at the  center of a sphere, and designate a South pole surrounded by four tetrahedral axis you designate one as Red and the whole system is determined and every direction in space can be designated with color degree, color degree and every point in space describable by adding a distance to the direction.

Copyright Thomas A. Roth Aug 8 2021

 

 

The four images posted are the 4 equatorial diamonds

look for the small pink numbers in the middle of each diamond

the number system is simple and elegant, unlike the arbitrary numbering I used earlier in the thread

to imagine it , 1 is to the upper right of 2, 2 is the equatorial diamond where the date line on the Earth passes through the center, 3 is to the lower left of 2

then rotate as the Earth rotates to diamond 5, again an equatorial diamond. 4 is to the upper left, 6 is to the lower right.

rotate to equatorial diamond 8.  7 is to the upper left 9 to the lower right

rotate to number 11  10 is to the upper left 12 is to the lower righ

Unlike my earlier numbering system this works out perfectly.  The order is easy to follow AND you will notice a pick dot in the left corner of each diamond.  This signifies the origin of an X Y type grid in each diamond where the degrees go up to the right in the one colors wheel and up toward the top in the other color's wheel.

Each color thus spans six diamonds around the sphere and intersects with each of the other color wheels twice.

This system is potentially useful because the square degrees are all named and of the same area.

spheres have 41,253 square degrees

Compare that with this system that shows 43,300 diamond degrees

also notice the 3 point under each equatorial diamond

Red, yellow, green and blue axis points respectively

Mit have gotten my lowers and uppers and lefts and rights fouled up.

 

Like this, is the way the current number system goes.

      10       7         4       1

     11       8       5        2      

   12     9         6        3

Regards, TAR2

 

Your work on dividing the sphere is interesting. As a graphic artist I appreciate all the lines drawn on the volleyball. What are your applications? The one I think of is antenna signal propagation. The shape of impedance and conductance on the antenna would be the division of a sphere that changes size and shape. If you use the division of the sphere as a reference, you have the 3D way to explain electricity, magnetism, and waves; like a sine curve is a reference to the 2D.

Here is a balloon I made for my Grandson.

Not just any balloon but the whole of what I have been working on since the start of this thread.

The green marker shows a cube.

The blue marker shows a cuboctblahedron.

The red marker with the brown squiggles on it shows  a tetrahedron.

The red marker with the black squiggles on it shows an opposite tetrahedron.

Together the tetrahedrons make the twelve sections of the sphere.

The red line connects the center of the numbered diamond sections.

 

 

Here is a toothpick and clay version of the twelve sections of the sphere, made up of two tetrahedra.

The yellow tetrahedra is one

the red,green and blue is the opposite 

 

Note that the combination yields a cube and the center of each of the twelve diamonds is on the center of one of the edges of the cube.

I have been still working with this figure, clay versions, toothpick versions, drawing on balls and such over the years since I posted here. I sort of got chased off the board for political reasons but still have my thoughts in private.

Here I want to sort of update my progress by showing how garnet crystals tesselate space and give my numbering system for the twelve sections of the sphere.

Here each garnet crystal, 12 sided with diamond shapes, represent the twelve sections of the sphere.  The crystals can be achieved by flattening the edges of a cube until they meet at a 4 point on the faces of the cube and a three point at the corners.  The arrangement is exactly the same as if you surrounded a ball with 6 others stacked three on the top and three underneath.   A stack of cannonballs would achieve this arrangement starting with a square base plane or a hexagonal base plane.  The garnet shaped clay pieces fit together exactly and each element solidly touches at its faces with 12 similar garnet shaped elements of the same size.

One can imagine this arrangement of space by sitting in a cubicle room and imagining a direction from the center going through the midpoint of each edge of the room.   There are twelve equally spaced directions defined.   four where the walls meet the floor..  Four where the walls meet each other and 4 where the walls meet the ceiling.

 

For understanding of the difference between the directions, since they are all relatively similar I have established a convention where they are numbered 1 through 12, and arbitrarily  picked the directions as if you are sitting in the center of the room facing North with the walls of the room going North to South and East to West.

number one is up and behind you

number two is left and up

number three is right and up

number four in up and front

five is front and left 

6 is forward and right

7 is forward and down

8 is down and left

9 is right and down

10 is down and behind

11 ia left and behind 

12 is behind and right.

Of course these directions are not universal as the Earth is spherical and your room could be anywhere on Earth at any elevation and the Earth rotates on its axis and revolves around the Sun and the equator of the Solar system is tilted forward 60 degrees hurting around the Center of the Milky Way and we are not particularly certain of how the Milkyway is moving and oriented to the great attractor, so the directions can be used from any observer facing north in a cubical room in general but to be used universally a reference room needs to be arbitrarily assigned.

Being as I am now at the announcement of the scheme near Radford VA at an elevation of 2000 ft

I propose the reference position be an observer at 37.1485degees N

                                                                                      80.5784degrees W

On NOON EST January 20th to give a universally recognized time.

Thusly the position of the Earth and the position of the reference room on the Earth will be established.

Of course for any particular purpose the numbering scheme can be used and you just need to establish which direction is up, and forward.  Being a person left and right is universally agreed upon.

 

What I just recently noticed is that you can with this system define every point in space from a starting point and direction without using degrees or pi.   That is, as shown in the thread the dense packed spheres build out layer by layer and form a cube with the corners cut off.

Thusly you can exactly define any direction by naming the position of the garnet crystal that you wish to draw a vector through the center of.

 

Which layer and which garnet crystal on the layer can be defined as a position on a certain face of the cuboctahedron structure that thusly builds out.   I have not yet come up with a simple way to define this because my 71 year old brain does not nimbly jump between the dual figures involved.   One has 12 faces and 14 vertices and the other has 14 faces and twelve vertices but it is the same figure directionally speaking as each Vertice of the one corresponds exactly with the center of a face of the other

I post a thought in progress in the hopes that other nimbler minds can help me work out the best scheme and the corresponding math involved.

My hope is that these positions can be defined EXACTLY and PI to a certain number of decimal places plays no role whatsoever.

As I continue to work on this rhombic dodecahedral I find it is VERY possibly the basis of a new way to describe space. Not in 3 dimensions, but in 7. The seven axis are

up/down,

forward/back,

left/right

up-right-forward/down-left-backward

up-left-forward/down-right-backward

up-left-backward/down-right-forward

up-right-backward/down-left-forward

Basically the corners and the center of each side of a cube, describing 14 directions. If you place a ball on the center of each edge of the cube you will have the 12 balls around a center ball arrangement I have been talking about in this thread. The twelve sections of the sphere. If you continue to build balls out, each ball with the same angular relationship to the other balls, a truncated octahedron begins to take shape.

A truncated octahedron is a permutohedron and fills space. Turns out you can take a central "ball" in the shape of a truncated octahedron, and put 12 truncated octahedrons around it. The center of each the same distance away from the center of all neighboring rhombic dodecahedra.

Notice from right to left starting with the tetrahedron on the right, that exactly the same seven axis are present in the tetrahedron, the truncated tetrahedron or octagon, the truncated octagon, the cube, the cuboctahedron, and the rhombic dodecahedron. Next to the truncated octagon is the 12 sections of the sphere built out several layers and it builds out into a truncated octagon shape. It consists of 3 square planes and 4 intersecting hexagonal planes all tilted to each other in the exact same arrangement. This arrangement appears when you put twelve balls around a center ball, building a hexagon equator, a three ball north pole and a 3 ball south pole with the triangle of the three north balls opposite the three south balls.

Imagining a truncated octahedron as your origin, and a truncated octahedron of the same volume as your unit of volume you can build out in all directions, with no gaps.

This give us several advantages in describing space. You don't need PI or Sines and Cosines to describe lengths and distances, to find every location in Cartesian Coordinates.

You can find every location in space based on a direction (one of 14) from the home truncated octahedron, and the number of volumes out to a second "home" truncated octahedron from which a direction and number of volumes will get you to ANY truncated octagon in space. Perhaps more coordinates are required to describe a volume, but you don't have any approximations. You can deal in whole numbers.

I also see this matrix of truncated octagons useful in 3D computer imagery as you can build ANY shape out of these identical volumes. If you can give the coordinates of any truncated octagon in space than you can build ANY shape in space. If you need more precision, you start your home truncated Octagon at a smaller volume. or distance between parallel sides.

The truncated octagon can be used as a 7D pixel and you could render any "Three Dimensional" shape.

I mixed up two packing permutahedrons in that one of my sentences there because both the truncated octagon and the rhombic dodecahedron seem to be based on the same 7 axis, however I think when we talk about centers and positions, which is required if we are going to use unit volumes, I think the truncated octagon works out better than the rhombic Dodecahedron. I did not mean you use rhombic Dodecahedron in that sentence, I meant to use Truncated Octahedron as that is the Shape you build as you build out spheres from a central sphere.

the7 axis in the rhombic dodecahedron goes through the corners between the flat diamond faces and you cant line the corners up and fill space. Has to be faces. So if you want to build the required unit volume shape, it needs 14 faces. each face the same distance from the center of the shape. This IS the truncated octagon..

If anybody uses this idea to build a unit volume for computer rendering, I would appreciate a mention. Perhaps call a unit volume "the 7 D TAR"

interestingly if you want to build a truncated octagon out of spheres, to get all the right angles and distances, you can do it in the following manner

put three spheres together in an equilateral triangle

put another three together and put them on top other three turning the three 60 degrees so they fit into the other three to make a octagonal shape

build two stacked hexagonal planes by putting spheres right around the three you started with on each plane. You should then have two hexagonal planes with 12 spheres in each, 9 around the center three. Put the planes together with the three ball sides mating with the two ball sides of each 12 ball hexagon plane section. Then build a 7 ball hexagon plane section and put it on top of the two and another 7 ball hexagonal plane section for the top. The top and bottom sections will fit onto the neighboring plane just right to make the truncated octagonal shape. Make each of 8 square sides flat as well as each of 6 hexagonal sides and you have a 7 D TAR unit volume. 14 sides all pointing in one of the required directions with a parallel side facing in the opposite direction. Each pentagonal side with an opposite and parallel pentagonal side and each square side with an opposite square side.

I made the shown truncated octagon out of 38 spheres 7+12+12+7 totaling 85.5g. This is the same volume of clay that made a sphere that exactly fit though a ring with a 2 inch outside diameter. All the figures on the paper were made with 85.5g of clay to visualize the relationship between the figures. So whatever volume you want as your unit volume, divide that volume into 38 equal portions and build the unit volume truncated octagon out of the 4 octagonal plane sections noted.

Copyright TAR2 (Thomas A. Roth) August 2025.

There are several figures shown that were built earlier before I used the 85.5g as the volume of the sphere that fits through the 2 inch outside diameter ring.

The truncated octagon shape made out of balls was a model I made out of 3g balls. When I made the actual truncated octagon I counted the balls in the model, found there to be 38 and divided an 85.5 gram lum[p of clay up into 38 pieces by rolling an 85.5g lump of clay into a 190mm long log, marking off 10mm sections, cutting the sections and then dividing each of the sections in half, giving me 38 lumps of clay I rolled into little balls it built into the truncated octagon shape mentioned, out of the hexagonal plane sections, pressed it all together side by parallel side between two boards.

I got my axis and directions mixed up.

The 12 directions I was talking about are in the direction of the faces of the rhombic dodecahedron, there are 14 directions using the truncated octagon. The names of these 14 directions would be

up/down

Then 60 forward from up

60 degrees to the left of that still 60 degrees lower than up

60 more counter clockwise

60 more which is 60 degrees lower than up and 60 degrees backward

60 degrees more

60 degrees more with each of the last 6 having the opposite direction.

Each direction having 6 surrounding directions 60 degrees away.

Thread,

I have been having trouble over the past years of this work I am doing on the 12 sections of the sphere, understanding the relationship between the 6 axis presented by the spheres arranged equally around a central sphere and the 7 axis created by the 4 intersecting hexagonal planes and the 3 intersecting square planes that become apparent when you build the matrix outward. As you build out layer by layer the shape takes on more and more the shape of a truncated octagon. The other day I built a truncated octagon by starting with 4 balls in a tetrahedral orientation ant built out a layer around it, staying true to stacked hexagon parallel planes. When you do this you wind up with two 12 element hexagonal plane sections with a 7 element section on top and another on the bottom. (7 element section being a central sphere surrounded exactly on the same plane by 6 identical spheres. 7+12+12+7 or 38 elements. After squishing the parallel sides flat and smoothing it all out and sharpening up the edges I got a perfect truncated octagon with 6 square sides and 8 hexagonal sides. All the sides have parallel sides of the same shape on the other side of the element which suggests one can stack elements out along each of the 14 axis. This shape, the truncated octahedron fills space. So it seems promising to be able to use a truncated octahedron as a unit volume.

The rhombic dodecahedron also fills space but the sides stack along 12 axis. I had a had time visualizing what was going on and the relationship between the shapes since they both seemed to be based on the same 4 hex, 3 square planes. Using toothpicks to show the various axis I now understand the relationship. Here I post a picture of the Space filling Polyhedra the Cube, the Triangular Prism, the Hexagonal Prism, the Truncated Octahedron and the Rhombic Dodecahedron. (green and white figures), with the seven axis shown with toothpicks to show the same 4hex,3square matrix exists in all three permutohedron. The Cube, the Truncated Octahedron, and the Rhombic Dodecahedron.

Interestingly the Dodecahedron with the pentagonal sides LOOKS like it would dense pack but it does not, when you try it, you are left with spaces, close, but no cigar.

Also the Pentagonal sided Dodecahedron is NOT consistent with the matrix that you get when you build out the 12 sections of the sphere I am discussing in this thread.

Regards, TAR2

This recent breakthrough of mine, that 38 elements of the spherical close packing matrix can be used to build a truncated octagon, together with the fact that the truncated octagon can fill space leads me to a yet to be proven hypothesis that you can use the truncated octagon as a unit of volume.

Work in progress.

simultaneously I am working on whether you could use the rhombic dodecahedron as a unit of volume.

Looking for the most elegant solution.

I am keeping my profile picture because it has close loved ones two of which are no longer alive, but still a solid part of me.

Here is me 5 years ago, I look about the same, just 5 years older.

66 or 67 in the picture, 71 now, but I have been working on this thread for a number of years and plan on continuing my work toward a math that will handle "3d" space without the use of PI, and/or arrive at a better understanding of what PI is. I know its the ratio of the diameter of a circle to its diameter, but its just a ratio. And any figuring you do with PI or Trig is only an approximation. I am thinking it would be helpful to develop a math where you could use whole numbers and repeating fractions and not need to use approximations to a given number of decimal places. In the past, in this thread when I was working with spherical angles they would never add back up exactly to 180 or 60 or 90 or 360 because the component angles were approximations based on the use of PI taken out to a certain decimal place.

If you calculated the volume of one of the 12 sections of the sphere you would not get exactly Volume/12, if you base your calculations on the Radius of the sphere and PI and various trigonometric and spherical geometric calculations.

I am looking for a way to find the exact volume, based on an exact unit volume.

for instance, if I take my 85.5g sphere of clay and divide it into 12 equal size identical shapes, each section will weigh exactly 7.125 grams. Pi would not be required to arrive at this answer.

Thread, sorry for the misdirection but I was trying to find a shape that would both tessellate space along the 7 axis and be an element of itself. This I have not yet achieved. Seems the best candidate for tessellating space for my purposes is the rhombic dodecahedra which has 6 axis if you put the toothpicks in the center of each of the 12 faces. And 7 axis if you put the toothpicks at the vertices.

I do not have a clear understanding of how and why this works out. Maybe it does not. Maybe I cannot use the Rhombic Dodecahedra to build out into the same matrix as spheres build out into.

Seems this idea is still a work in progress. But as of now, I don't know if I have a good candidate for a unit volume that can tessellate space AND be arranged in 3 intersecting square planes and 4 intersecting hexagonal planes.

Seems there SHOULD be a shape that one could make, starting with a sphere in the matrix and "adding" clay in the voids between the spheres, filling the void just halfway to the next sphere.

I just have not come up with that shape yet. I try mashing spheres together around a central sphere and then taking the 12 outside balls off and see what shape I have left but its always been hard to discern the exact shape of the elemental shape that fills the voids.

I will check back in if I make any progress.

Regards, TAR

The truncated octahedron is looking good again.

the Truncated Octagon and the Rhombic Dodecahedra are duals of a sort and you can find both the six axis arrangement and the 7 axis arrangement is both figures The exact same direction coming perpendicularly off the one of the faces in the one figure, coming off a apes or an edge in the other. So using either figure you can define 26 directions, either end of 13 axis. You can find these same Azis as well in a cube. The center of each edge of cube defines the center of one of the diamonds in the twelve sections of the sphere, or in the rhombic dodecahedron. The other 7 axis can be found going through the center of each face of the cube, that is three axis and through the corners, which is another 4. So the cube has 13 simple axis, easy to find, so the some 26 directions belong to each of these three space filling figures. The Cube, the Truncated Octagon, and the Rhombic Dodecahedra. So space can be thought of as consisting of 4 intersecting hexagon planes and 3 intersecting square planes.

apex not apes

The red toothpicks show the 12 sections of the sphere centers, 6 axis, understandable as starting at the center of a cube and going out through the center of an edge of the cube.

The truncated octagon is the olive green figure, the white one is the rhombic dodecahedron.

Red tipped toothpicks denote the six axis directions, unmarked toothpicks denote the 7 axis arrangement. The cube behind has the seven axis denoted, three though the center of the 6 sides and 4 through the eight corners.

If the red tipped toothpicks were added to the cube they would go through the center of each of the 12 edges.

Regards, TAR

Thread, This morning I was looking at the tuncated octagon with the toothpicks spinning it in my hand and noticed that were 8 toothpicks spaced around the equator, eight midway between the equator and the poles, one at each pole and another 8 spaced around the area between the south pole and the equator. I found this interesting because both the 7 plain toothpick axis and the 6 red tipped axis are derived from the spacing of equal sized spheres around a center sphere. Much the same type of thing you might find when an electron cloud is looking for a "volume" to inhabit. I then looked up a few shell arrangement of some of the elements and the numbers 1 and 2 and 8 and 16 come up a lot. Also 10 which is 8 and two.

Thought it interesting enough to mention here. Just another neat discovery looking at the 12 sections of the sphere.

Lots of neat symmetries you get with the four intersecting hexagonal planes and the three intersecting square planes.

Regards, TAR

Thread, Uranium is spaced out in shells, 2,8,18,32,21,9,2 fpr a total of 92 electron clouds. Oganesson is spaced out in shells of 2,8,18,32,32,18,8 for a total of 118 electrons.

Interestingly if you inspect the above figures derived by the close packing of the 12 spheres around a central sphere, that builds out to a matrix of 4 intersecting hexagon planes and 3 intersecting square planes and you combine the 6 axis of the 12 segments of the sphere with the 4 axis of the intersecting hex planes and the three axis of the intersecting square planes you get the toothpick arrangement shown above with 14 plain toothpicks and 12 red tipped toothpicks nicely spaced around the center voluume.. You have eight around the equator, one at the top and 8 spaced around the sphere midish beween the North pole and the equator, Same arragement opposite in the south pole since each toothpick represents one end of the same axis. as belongs to its opposite toothpick.

Interestingly when you look at this arrangement you see that you have spaces between the toothpicks that could hold a sphere, either between three neighboring toothpicks in q triangle formation or 4 neighboring toothpicks in a diamond arrangement. Counting these spaces we have 8 triangular spaces around the North pole, 8 diamond shaped spaces between the mid range toothpicks and the equator and the exact same opposite structure on the South end of the Truncated OCTOGON, Totaling 32.

Regards, TAR

Hear we have 32 pearls added in the cradles between the axis in the manner I suggested before 8 in the triangular cradles provided at the poles and 8 in the diamond shaped cradles provided above the equator and 8 below. Total 32.

2 hours ago, tar said:

Thread, Uranium is spaced out in shells, 2,8,18,32,21,9,2 fpr a total of 92 electron clouds. Oganesson is spaced out in shells of 2,8,18,32,32,18,8 for a total of 118 electrons.

Interestingly if you inspect the above figures derived by the close packing of the 12 spheres around a central sphere, that builds out to a matrix of 4 intersecting hexagon planes and 3 intersecting square planes and you combine the 6 axis of the 12 segments of the sphere with the 4 axis of the intersecting hex planes and the three axis of the intersecting square planes you get the toothpick arrangement shown above with 14 plain toothpicks and 12 red tipped toothpicks nicely spaced around the center voluume.. You have eight around the equator, one at the top and 8 spaced around the sphere midish beween the North pole and the equator, Same arragement opposite in the south pole since each toothpick represents one end of the same axis. as belongs to its opposite toothpick.

Interestingly when you look at this arrangement you see that you have spaces between the toothpicks that could hold a sphere, either between three neighboring toothpicks in q triangle formation or 4 neighboring toothpicks in a diamond arrangement. Counting these spaces we have 8 triangular spaces around the North pole, 8 diamond shaped spaces between the mid range toothpicks and the equator and the exact same opposite structure on the South end of the Truncated OCTOGON, Totaling 32.

Regards, TAR

Leaves room for a nice symmetrical 120th element 2,8.18,32,32,18,8,2. We can name it dodecaium when we make it.

Thead.

Major breakthroughs' this morning over the past couple hours. My mind is going a mile a minute in 6 different directions but I will try and put into words what has "clicked" for me this morning.

First the rhombic dodecahedron DOES build out into the seven axis spherical close packing arrangement of 4 intersecting hex planes and 3 intersecting square planes.

Second realization, the spheres build out in this pattern to the shape of a cuboctahedron and so does the rhombic dodecahedron.

Third aha moment for me this morning. If you inscribe a unit sphere within a rhombic dodecahedron the distance from the center of the sphere to its surface is 1 r and the distance from the center of any of the 12 faces of the dodecahedron is 1 r.

The implications of this are many and my mind is off on a number of tracks that I will try and flesh out on this thread as I build models and descriptions and flesh out the math and relationships with the Cartesian coordinate system. The center of the twelve faces, of a dodecahedron centered on the origin, in no particular oder would be

1,0,0

0,1,0

-1,0,0

0,-1,0 making center points of the four equatorial faces and the four north polar faces would be at 1,1,1 1,-1,1 -1,-1, 1 and -1,1,1. Southern four would be the center of the negative Z octants.

I will work on some figures showing the build out of the layers of these "unit dodecahedral volumes"..

One adjustment I have to make is the 85.5g unit volume I am working with is NOT the volume of the unit sphere. The volume of the unit sphere is less than the volume of the unit dodecahedra because the unit sphere dense packs but leaves spaces between the spheres that the corners of the dodecahedrons fill. So I have not yet found the relationship between the sphere that, inscribes a 85.5g dodecahedra, the 85.5g sphere that fits inside the 2 inch diameter outside dimension ring, and the sphere that circumscribes the 85.5g dodecahedron.

I suppose it is going to be a matter of determining the distance from the center of a dodecahedron to any of its faces. That is equal to the r of the inscribed sphere.

Thread,

Here the third figure over from the left in the front is the 12 balls arranged around the center ball in the 7 axis configuration. Works out to put a pearl in every location that was NOT where the 8 pearl arrangement on the cuboctahedron to the left of it had a pearl. There are 18 pearls that found a nice nest in between spheres.

From left to right you have a sensible arrangement where electrons are trying to stay away from each other at the same time they are attracted to the nucleus.

from left to right in the front row you have 2,6,18,32 the same way the elemental shells work out.

Nice.

I have a somewhat smaller sphere behind and to the right of the 85.4g sphere with the ring around it. It is the sphere that inscribes the 85.4g rhombic dodecahedron, the 85.5g cuboctahedron and the 85.5g cube. It is, for my purposes the unit sphere in the 7 axis coordinate system I am developing. This however is not the unit volume for the system. The unit volume is the dodecahedron with distance of 1 between the center of dodecahedron and any of its 12 faces. Also it appears the unit sphere inscribes the 85.5 gram cuboctahedron and cube as well so it appears mathematically things will work out nicely..

I am not yet ready to set the units though because the clay I am using seems to have a density of about 1.75 so I can not use a cubic centerment = a gram conversion. I would need a polyetheline dodecaheron or some other ideal material with the density of 1 to true everything up.

However, the scheme is looking good. Still have work to do but it makes sense that the unit circle would not equal the unit volume as the sphere does not completely fill space but leaves space, even in the close packed arrangement. My measurements might be a bit off because it does not make any sense that the unit cuboctahedron would have the same volume as the unit cube because a cuboctocheron is cure with its corners cut off....anyway, I have work to do, but it looks promising. It looks like you might be able to put a unit sphere in the unit cube, and put 8 such spheres around the origin, one in each octant.

I am hoping this works because then you could just shift the unit sphere to the center of any rhombic dodecahedron in the 6 d matrix of dodecahedra for figuring purposes. OR make measurements in the direction of woe of the six axis going through the 12 faces of the rhombic dodecahedron. You could measure out x number of radii in any of the six directions, which will get you to an exact location in space that is the center of another rhombic dodecahedron from which you could again measure out exactly in any of 11 directions to the center of another rhombic dodecahedron. You could use simple integer and fractional distance and simple and exact angles. Although you might still be able to use trig and radians and such, it would not be essential to get you to the area in space where you want to be. I am thinking there are some advantages to using a volume because you are not really starting at a point, you are starting at a volume and you are ending at a volume so approximations are never required and Pi is never absolutely necessary to use. And the whole system is completely scalable. You could make your unit rhombic dodecandrian you are using as the origin any size you want. tiny or huge. So whatever precision you need could determine what unit radius/distance from center to face of origin rhombic dodecahedra, you start with.

I have not designed the procedure yet for getting to an area in space but there are nice possibilities since every rhombic dodecahedra in space is unique, yet identical and can be reached by numerous routes.

Regards TAR

Sorry forgot the picture

Thread

Actually the center of your dodecahedra are, not in the next octant they are in the next dodecant There are 4 nothern dedahdra, four suuthern a four equatorial. Each axis x.y.z rims throgh the cornern of a diamond on the long diaganal of each diamond the ends of diamonds along the short diagnal make the three poing in the middle of the octant. So the center of the face is not centered in the cartesian octant. It is centered midway between two axis, talking about the xyz

remembering that if you are in the center of a cube the axis that are normal to your twelve faces are running through the center of the twelve edges of the cube