On Fri, Aug 23, 2013 at 12:20 PM, frank zubek <email@example.com> wrote:
> GCS you know it would be of great help to me if you could pinpoint what > exactly you do not understands; > I wrote a fraction 24/24 = 1 > this fraction is a number of blocks in the unit tet. vol. > please tel me if this fraction can or can NOT be reduced any further. > > If you still do not understand, please tell me what part of the fraction > is so difficult, is it the nominator the denominator is it the decimal > form, is it the blocks them self, what is it you do not understand. > > 3/24 is 1/8 of this very fraction, we all know that is a solid ANY solid > has 1/8 portions we know that such solid is 8 times the 1/8 portion in > size/magnitude.
Make sure you remember the difference between slicing a cake into wedges, and the number of intervals around the edge. When you do it with circles and triangles, these numbers may be the same, but if your "cake" is an icosahedron and you slice it up into irregular tetrahedra, phi-scaled, as Koski does, then all bets are off as to how "number of wedges" relates to "number of surface facets" or "edge intervals".
I think in this thread you are collapsing two ideas of fraction: segments of a line, and portions of a volume. In between is sub-area within area, i.e. a portion of a "plane" ("great plain" in Lakota). The 1,2,3 power rule is that when you scale linear dimensions all by n, area increases as a factor of n * n, and volume as a factor of n * n * n. This is our entre into calculus. Check the Common Core.
In the namespace we've been yakking about, to double the frequency may just mean doubling the number of intervals without actually changing the overall size, and it may be discrete counting that's going on, as when we use spheres to "tile" (not really) the surface of a cuboctahedron: 12, 42, 92, 162... that's an important number sequence (again, Common Core should give you a clue).