17,471 research outputs found
Diagrammatic Reasoning and Modelling in the Imagination: The Secret Weapons of the Scientific Revolution
Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Reconstructing 4-manifolds from Morse 2-functions
Given a Morse 2-function , we give minimal conditions on the
fold curves and fibers so that and can be reconstructed from a
certain combinatorial diagram attached to . Additional remarks are made in
other dimensions.Comment: 13 pages, 10 figures. Replaced because the main theorem in the
original is false. The theorem has been corrected and counterexamples to the
original statement are give
From singularities to graphs
In this paper I analyze the problems which led to the introduction of graphs
as tools for studying surface singularities. I explain how such graphs were
initially only described using words, but that several questions made it
necessary to draw them, leading to the elaboration of a special calculus with
graphs. This is a non-technical paper intended to be readable both by
mathematicians and philosophers or historians of mathematics.Comment: 23 pages, 27 figures. Expanded version of the talk given at the
conference "Quand la forme devient substance : puissance des gestes,
intuition diagrammatique et ph\'enom\'enologie de l'espace", which took place
at Lyc\'ee Henri IV in Paris from 25 to 27 January 201
- …