[Back to nmaths MAIN page] [^^The Pond Normal School]## GEOMetry

See also: -[zix teaching methods]- (teaching folder) -[GEOM: b1]- -[GEOM: b2]- -[GEOM: b3]- -[GEOM: b4]- -[GEOM: b5]- -[GEOM: b6]- (under construction) On this Page: {Intro} {Topology} {Rational Fractals} Rubik's Cube as a fractal, etc} {}## GEOMetry

## Stuff

Going back to Montessori's idea of manipuables. What if we had (like triominos, pentaminos, and of course dominoes)... Plasitc trianges - various sizes in multiples of a "unit length"... Squares, etc. This puzzle, "Pythagoras"... yes, but mulitples and multiples and then of course an infinite staging area -- you get this in tessalations like with the 3-6 hex grid. But, why not a 7-3-? grid or a .... Take the platonic solids and net them out, then fill the spaces tileing 2-d Yes: But, what about 3d? You'll have to get someone else to help. But, i'm ready to work in 4d, 5d, and of course 6d - which i can almost sometimes "see" - (but that might just be hae'g done too much LDS back in the '60's... ;) - but of course 6d is predicted by the Borges Equations. tesseracts - and then KonNex... Tinker toys, and that Kenner "sky scraper" construction kit..... buckminister fuller and the geo-desics and not-quite-perfect fits - gravel in the glass, and then bee-bees, and then sand, and then water, and then ethanol/methanol??? and then..... hold infinity within a grain of sand, and infinity in an hour - Wm Blake via Bronowski, but a side trip to BF Skinner and the self-programming mode... Warf: And commander, Targ: Yes?!? Warf: Welcome to the twenty-fourth century. ... Warf: Comfortable chair. --42--

## Topology

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## Rational Fractals

google: "rational fractals" Also: -[Rubik's cube as a fractal-filling object!]- BEGIN BLOCK QUOTE ====================== Tree Formalism for Fractal Description S. Duval and M. Tajine We study a new notion of fractals based on the embedding of trees in a metric space (E, d). Actually, if R is a tree over an alphabet with arity (F, rho) then we associate to each symbol f of F a contracting mapping if rho(f)>0 and a compact of E otherwise. The embedding of R is obtained by the union of embeddings of its branches. To embed an infinite branch, we compose the mappings along it. We obtain, in this case, a point which belongs to E. To embed a finite branch, we successively apply the mappings along the branch to the leaf associated compact. The modeling of trees is done by using grammars of any order. The grammars of order 0, 1 and 2 generate respectively rational, algebraic, and functional fractals. More generally, a n-fractal is generated by a grammar of order n. This allows to classify the fractals according to the grammar order used. The rational fractals contain the IFS generated fractals. We can describe the high level operations most often found in object modeling. That way, given two fractals G and H, we model G u H, G x H, G + H and h(G) where x represents the cartesian product, + represents the sum of Minkowski and h is a homogeneous function in E. We are also able to describe the geometric simplicial complexes. The fractal geometry introduced here, is closely linked to discrete geometry in the sense that the model contains its own discretization. In fact, if a fractal G is the embedding of a tree R and R' is a tree obtained from R by truncation, then the embedding of R' is a discretization of G. It should be noticed that for the modeling of a segment, an adequate truncation permits to find the Bresenham's discretization. In the particular case of rational fractals, it is possible to effectively calculate the Hausdorff dimension. If G is a fractal obtained by the embedding of a rational tree R then the Hausdorff dimension of G is a function of the convergence radius of the generative series corresponding to R. END BLOCK QUOTE ======================== -[]- -[]- -[]- -[]- -[]- -[]-