A message has been posted to the discussion "math-teach".
Author: kirby urner Subject: Re: example
On Fri, Aug 23, 2013 at 11:55 PM, frank zubek <email@example.com> wrote:
> Say we have a regular block, some of these blocks with they companions are > space-fillers, they are in the family of concentric hierarchy, like > "Russian dolls"= concentric > .These blocks are now diviaded in to tetrahedrons and these tetrahedrons > are vol. identical, A rh. dodeca or any such solid can be divided in to > infinite many tetrahedrons, so infinity here is not considered a attribute, > since it can be done infinite many ways, what is important WHAT IS THE > LOWEST NUMBER OF VOL. IDENTICAL TETRAHEDRONS for any such structure. Let me > assure you before you venture in to that alley, that synergetics is one of > the infinite possible construction, whereas what I have and discovered is > that the x,y,z, cube derived tetrahedroons are THE LOWEST POSSIBLE NUMBER > OF VOL. IDENTICAL BLOCKS FOR ANY MENTIONED STRUCTURE, and here where is one > of many instances where synergetics is merely just a REDUCTIO AD ABSURDUM. > > fz >
As stated your assertions are as yet unclear and this is why:
"A rh. dodeca or any such solid can be divided in to infinite many tetrahedrons" may be interpreted to mean that you can just keep subdividing to get more and more smaller and smaller tetrahedrons.
fz, well basically yes, a tet, can be subdivided and than those pieces can be subdivided so a reg. tet. can = 4, 16, 128, and on and on,
Or it might be interpreted to mean there is an infinity number of ways to divide it into some finite small number of tetrahedrons, each time using slightly differently shaped tetrahedrons.
fz, that may be possible also, so either case would imply, yes there is many ways to divide a solid.
For example the Soma cube, divided into shapes made of smaller "polycubes" can be assembled in over a million different ways by the same seven pieces. Or if you count differently, you can reduce that to 240 basic ways.
fz, well because of the orthogonality of the individual shapes yes you can do it this way and that way start at the left and than the same piece points to right or up-down, say the 1/4 reg. tet. can have 2 , 3 , 4 in my case, 6 in your case, 8, and on and on as the minimal and the lowest number is 4 , because these fore are the most minimal in surface to vol. ratio from all the infinite consecutive possible blocks and are also continuum for the whole hierarchy of our solids the 2 or 3 blocks do not build the whole hierarchy and are larger in surface to vol. ratio so 4 are the most minimal and the LCD.
"Over a million" is hardly "infinity" but it shows how a small discrete set of shapes may assemble a larger shape in many many ways.
fz, yes, fascinating in a reverse thought the rubik's cube can be rearranged by 16 moves no matter how bad it is scrambled so it is fascinating that it has bilion plus some combinations, yet only 16 moves puts it back to color order.
'Let me assure you before you venture in to that alley, that synergetics is one of the infinite possible construction"
You should also define what you mean by a unique tetrahedron.
fz, unique is special for example my blocks can build they larger self, they are always at the lowest common denominator for any structure, they are also minimal in surface to vol. ratio compared to any tetrahedron derived from that particular solid , they are all different yet vol. identical, and they can assemble a given solid by several different ways something like to soma cube.
Do left and right handed versions of a shape count as one or two?
fz well, I guess it depends what are you doing. cube = 6 double corners or characteristic tetras, or one can say 3 because the other 3 are identical. in angles, vol. surface areas.
Fuller dissected his cube, octahedron, rhombic dodecahedron, tetrahedron, cuboctahedron using two basic shapes, each with a left and a right, so one could say four basic shapes. They all do have the same volume.
fz, yes those are the barycentric division of the 1/4 reg. tet. the A mod. and the barycentric division of the octant A plus B
But of course it takes a varying number of those two basic shapes to accomplish each dissection.
fz, yes correct, because the other shapes are of different size, that is how we count with blocks the reg. tet. takes 24, the octant only 12, the cubeoc. 60 and so on.
However, the five-fold symmetric shapes also included in his concentric hierarchy, are not amenable to dissection with these same modules. There's more to that story.
fz, yes the 1/120 rh. triconta only has the vol. equivalence to 1/6 of a cube can not be assembled by our blocks
You'll notice I have not bothered to contradict your claims. The theme has been clarity and how your posts lack sufficient precision to be parsed by other readers.
fz, OK I agree, but as you notice there is this same problem with writings of Fuller on many occasions, people say I wonder what he meant by this or that, and others have pondered that, and I'm sure you had to also, remember "typo corrections?" because most people say he was opaque as GSC put ti.
I'm trying to help you out by showing how different meanings of "infinity" are clouding up your assertions.
fz, I'm not so sure, I believe infinity is real in this case, say I dissect the 1/4 reg. tet. in to 4, than each in to another 4. and on and on.
You should slow down and explain more clearly what you mean by "infinite many tetrahedrons".
fz, Of course I have never try it, just thinking it to be possible. For this very reason, even Hilbert's third problem is concluded" "It is not all-ways possible" meaning that a terahedron indeed can be dissected in to a given number of vol. ident. terahedrons and reasembeled in to a different tet. "but it is not always possible" meaning that sometimes it is, and not all the aspects of a tetrahedrons are fully known even today. As I said some time ago, Every tetrahedron is a space filler, but Marjorie Senechal's article which I have posted and stired up lots of buzz on the synergeo, where Aristotle supposetely claimed that the reg. tet. fill space, I'm sure you remember. Well the article it's self suggests that, there is only few cases where a identical tet. fills space, I know of 3 at this time, again is the L-R considered as one? well again it depends what we want to establish. all I know that every tetrahedron has a companion and it will fill space. This was the best conversation I had with you, and it always could be this way, because all I'm doing is searching and have found some very interesting facts, perhaps known facts but they are not known to me, maybe not known to you or no one because infinity is hard to prove, but there is a way to do so least in some instances. So at this point I appreciate your comments.