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Extending the Coordination of Cognitive and Social Perspectives
Cognitive analyses are typically used to study individuals, whereas social analyses are typically used to study groups. In this article, I make a distinction between what one is looking with?one’s theoretical lens?and what one is looking at?e.g., an individual or a group?. By emphasizing the former, I discuss social analyses of individuals and cognitive analyses of groups, additional analyses that can enhance mathematics education research. I give examples of each and raise questions about the appropriateness of such analyses
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Complexity of triangulations of the projective space
It is known that any two triangulations of a compact 3-manifold are related
by finite sequences of certain local transformations. We prove here an upper
bound for the length of a shortest transformation sequence relating any two
triangulations of the 3-dimensional projective space, in terms of the number of
tetrahedra.Comment: 10 pages, 3 figures. Revised version, to appear in Top. App
How to make a triangulation of S^3 polytopal
We introduce a numerical isomorphism invariant p(T) for any triangulation T
of S^3. Although its definition is purely topological (inspired by the bridge
number of knots), p(T) reflects the geometric properties of T.
Specifically, if T is polytopal or shellable then p(T) is `small' in the
sense that we obtain a linear upper bound for p(T) in the number n=n(T) of
tetrahedra of T.
Conversely, if p(T) is `small' then T is `almost' polytopal, since we show
how to transform T into a polytopal triangulation by O((p(T))^2) local
subdivisions. The minimal number of local subdivisions needed to transform T
into a polytopal triangulation is at least .
Using our previous results [math.GT/0007032], we obtain a general upper bound
for p(T) exponential in n^2. We prove here by explicit constructions that there
is no general subexponential upper bound for p(T) in n. Thus, we obtain
triangulations that are `very far' from being polytopal.
Our results yield a recognition algorithm for S^3 that is conceptually
simpler, though somewhat slower, as the famous Rubinstein-Thompson algorithm.Comment: 24 pages, 17 figures. Final versio
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