On Wed, Sep 5, 2012 at 12:16 PM, Joe Niederberger <email@example.com> wrote: > Kirby says: >>What I have been showing is "repeated addition" will take us a long way, if we extend it to include what others call "scaling" -- we're repeatedly adding length. > > I see it a bit differently. We have pre-mathematical (perhaps proto-mathematical) intuitions about the world. > Replication, and growing (or shrinking) are quite different from that viewpoint. We can connect both of these as abstracted concepts to what we learn about the formal symbolic manipulations we call multiplication (pick your numbers N - C) -- which have course evolved over a long period. Its perhaps a bit surprising that two seemingly dissimilar concepts should both map to mult. > Think of babies: replicating babies, growing babies, rather different. Once we have found a connection, we can amuse ourselves by finding deeper connections, such as the fact that growing babies involves replication of cells, which is something that people long ago didn't know for sure. Perhaps all growing involves replication in this physical world. > > Joe N
Yes, it is easy to connect these two (scale and replication), especially in volume, where when we talk about "a growing supply of grain" we mean more and more grain particles.
Cell growth: the body gets bigger as there are more and more of them. Exponential growth important. Relation between linear, areal, and volumetric growth as powers 1,2,3 also important.
Studying this on a growing / shrinking sphere, variously divided (surface-wise, volume-wise) is a focus (as Polyhedra are), in a STEM-worthy K-12 math/geometry/geodyssey. Calculus gets a foothold here, in terms of the differences, the rates of change, in these growth numbers.
The sequence/pattern 1, 12, 42, 92... is one I keep harping on (a 2nd power function): a nuclear ball surrounded by 12, in the form of six in a plane, 3 on top, 3 below, such that it could be viewed as four intersecting hexagons (parallel to the 4 planes of the regular tetrahedron). A next layer of balls has 42, then 92 and so on.
The thing about this ball pattern is it's both uniform and optimum density and is a basic "holodeck" for chemistry, where hexagons are more common than squares, 60 and 120 degree angles more common than 90 degree (although this sphere packing matrix has 90 degrees also).
In STEM (which includes architecture-engineering) we spiral back into the above packing matrix in the form of the octet truss, a commonly used space frame originally pioneered by Alexander Graham Bell (who didn't patent it) and Bucky Fuller (who did). This was before crystallographers had such a clear idea about the FCC, though they were getting there, through Barlow etc.