id: 06474542
dt: j
an: 2015e.00773
au: Gmeiner, Peter; Seri, Marcello
ti: Surprising simple geometries.
so: Math. Sch. (Leicester) 44, No. 3, 25-30 (2015).
py: 2015
pu: Mathematical Association (MA), Leicester
la: EN
cc: I90 G90
ut: iteration; piecewise linear curves; length; area; infinity; limit curves;
Koch curve; Koch snowflake; Sierpinski triangle; Sierpinski sieve; new
concept of dimension; fractal dimensions; Hausdorff dimension; fractal
geometries; calculus; fractals; geometry
ci:
li:
ab: From the text: It may seem obvious that a continuous curve that can be
inscribed by some simple figure of finite area must be of finite length
and, if it is closed, it may seem obvious that its interior must have a
positive finite area. What if we build a figure iterating some infinite
process but still being constrained in some finite space? Somebody
could think that we would just approximate some kind of limit curve
that must preserve a finite length and a positive area. Moreover one
could think that it would be difficult to extrapolate the area and the
length of such curves analytically, at least only using elementary
analysis concepts. We would like to show that all this intuition is in
principle false and that there are some really simple and astounding
counterexamples. In the few pages that follow, we want to present to
you an uncommon way to describe and use some elementary analytical
tools. Sometimes it may be hard to motivate students to study
(geometric) series and limits and it can he hard for them to accept and
understand the bizarre behaviour of mathematical infinity, thus it
could be interesting to show how some particular geometries can be
described easily using these tools. Despite their simplicity these
topics are related to many beautiful mathematical questions, most of
which are particularly old, simple and elegant but nevertheless still
open. If this is not enough, many actual technological improvements are
based on them and the simple examples that we are going to describe.
This is something that could give the students reasons to study
mathematics and physics.
rv: