Dividing a sphere into twelve "identical" shapes.

[tar]

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, 2017 by tar

[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

[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.

[tar]

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]

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

 

[tar]

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

[tar]

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

 

[Trurl]

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.

[tar]

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.

 

 

[tar]

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.