Dividing a sphere into twelve "identical" shapes.

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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 by tar

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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

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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.

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