I'm assembling this posting from my contributions to one of the Telegram faculty lounge channels.
I think a good way to picture the CCP is to start with a 3D checkerboard of cubes, like you’d get in XYZ, but only put a ball at the center of every other cube, like all the black or all the white cubes, to continue the chess board analogy. Let the balls swell / bulge to such a size as they touch their cube cage confinement cells at the 12 mid-edges. That’s where they touch their neighboring balls in the 12-around-1 arrangement.
From the point of view of one of the “empty cubes” (no ball at its center), the six neighboring cubes have bulging balls intruding inward into my “empty space”. That’s the six-bulges-around-a-void type of void. The other type of void is at the cube cage corners, the duo-tet cube tips. Those are the four-bulges-around-a-void type of void.
Four-bulge and six-bulge voids are depicted here:
From the point of view of one of the “empty cubes” (no ball at its center), the six neighboring cubes have bulging balls intruding inward into my “empty space”. That’s the six-bulges-around-a-void type of void. The other type of void is at the cube cage corners, the duo-tet cube tips. Those are the four-bulges-around-a-void type of void.
Four-bulge and six-bulge voids are depicted here:
http://rwgrayprojects.com/synergetics/s10/figs/f3212.html
This and other GIF animations are available here, most of them towards the end of the scroll.
https://github.com/4dsolutions/School_of_Tomorrow/blob/master/QuadCraft_Project.ipynb
"I think a good way to picture the CCP is to start with a 3D checkerboard of cubes, like you’d get in XYZ" -- but then comes the caveat. In XYZ we take for granted that all cube edges are length 1, what else? But in the CCP checkerboard, it's the cube face diagonals that are length 1, where 1 is likewise the diameter of a CCP ball.
This and other GIF animations are available here, most of them towards the end of the scroll.
https://github.com/4dsolutions/School_of_Tomorrow/blob/master/QuadCraft_Project.ipynb
"I think a good way to picture the CCP is to start with a 3D checkerboard of cubes, like you’d get in XYZ" -- but then comes the caveat. In XYZ we take for granted that all cube edges are length 1, what else? But in the CCP checkerboard, it's the cube face diagonals that are length 1, where 1 is likewise the diameter of a CCP ball.
In going mid-edge across to the opposite mid-edge, each ball is spanning a cube face diagonal (in terms of same length) with its diameter D. D = 2R. If you wanna measure in Radiuses, the distances double because the units are halved.
To derive the Synergetics Constant S3 1.06066... we actually do go to the smaller XYZ cube of edges 1, meaning R, of volume 1 (in XYZ), and compare it with the IVM tetrahedron of edges D, also of volume 1 in tetravolumes. The D-edged tet is actually a little bit less capacious than the R-edged cube such that
To derive the Synergetics Constant S3 1.06066... we actually do go to the smaller XYZ cube of edges 1, meaning R, of volume 1 (in XYZ), and compare it with the IVM tetrahedron of edges D, also of volume 1 in tetravolumes. The D-edged tet is actually a little bit less capacious than the R-edged cube such that
D^3 * 1.06066... = R^3
Translation: what we call our unit volume on the left (D to the 3rd power), made a bit bigger by this conversion constant, has the volume of what the XYZers use, their unit cube, R to their way of 3rd powering. S3 may be computed as an algebraic number: sqrt(9/8).
a screen shot from the old days
