Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Smallest Shape Spaces. III. Triangles in 2- and 3-d   This is an innovative treatise on triangles, resting upon 1) 3-body problem
techniques including mass-weighted relative Jacobi coordinates. 2) Part I's
detailed layer-by-layer topological and geometrical study of Kendall-type shape
spaces - configuration spaces of all possible shapes - which, for triangles,
are (pieces of) spheres. 3) Hopf mathematics. Triangles are moreover
prototypical through being the smallest models which carry relative-angle as
well as length-ratio information. Both 1) and 3) produce insightful new
versions of Heron's formula, 3)'s simultaneously providing new foundations for
2). Medians, and regular triangles bounding between tall and flat triangles,
also play prominent roles. Right triangles form three kissing cap-circles on
the shape sphere, from which a shape-theoretic answer to the well-known
conundrum of what is the probability that a triangle is obtuse very readily
follows: 3/4. The differential-geometric aspects of this answer moreover
generalize to numerous variant problems.
  Hopf mathematics additionally gives a general bundle section interpretation
to Kendall's iconic spherical blackboard of vertex-unlablelled
mirror-image-identified triangles, and of its two variants where one of these
two conditions are dropped. We attribute a monopole to each of these spaces and
to the full shape sphere, one due to Dirac, one to Iwai and the other two are
new to this paper. We finally make insightful comparison of triangles in 2-$d$
with a) Part II's 4 points on the line. b) Triangles in 3-$d$, which are
particularly significant as the smallest model exhibiting stratification.
Stratified manifold-sheaf pairs - sheaves adding useful local and global
structure to general bundles - lie at the heart of Shape Theory's future
development.

17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int