id: 06512807
dt: j
an: 2016a.00586
au: Worrall, Charles
ti: An Archimedean balance: polygons on the head of a pin.
so: Math. Teach. (Reston) 107, No. 4, 313-317 (2013).
py: 2013
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: G40
ut: physics; centroids; balancing points; elementary geometry; plane geometry;
triangles; quadrilaterals; polygons; medians; balancing lines;
discovery learning; dilations
ci:
li: http://www.nctm.org/Publications/mathematics-teacher/2013/Vol107/Issue4/Delving-Deeper_-An-Archimedean-Balance_-Polygons-on-the-Head-of-a-Pin/
ab: From the text: Archimedes’s famous exposition “Mechanical Method", in
which he provides the “mechanical" derivation of the area formula for
a parabolic sector, is one of the great pieces of mathematical
literature. Since reading this exposition years ago, I have been
intrigued by the potential interplay between pure mathematical theorems
and physical properties. The traditional relationship between the two
is to have mathematics describe and justify physical truths, which
usually happens in physics class. But at times the process is reversed
‒ that is, physical truths can be used as postulates to derive
mathematical ones. I will illustrate this phenomenon with a small
example of similar reasoning, showing its accessibility to high school
students and its place in a mathematics classroom. Years before I read
Archimedes’s derivation, I became fascinated by the purely
mathematical properties of special points in triangles, but I was only
mildly interested to learn that the centroid is a triangle’s
balancing point, the point at which a uniformly thick and dense
triangle would balance on a pin (a key fact in Archimedes’s parabolic
sector derivation). But now the idea intrigued me. I wondered, How can
we logically justify its truth? And how can we deduce the location of
the balancing point for a more complicated polygon?
rv: