TITLE:
Discrete Differential Geometry of Triangles and Escher-Style Trick Art
AUTHORS:
Naoto Morikawa
KEYWORDS:
Discrete Differential Geometry, Triangle Mesh, Global Analysis, Singular Point, Penrose Stairs
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.6 No.3,
July
5,
2016
ABSTRACT: This paper shows the usefulness of discrete differential geometry in
global analysis. Using the discrete differential geometry of triangles, we could
consider the global structure of closed trajectories (of triangles) on a
triangular mesh consisting of congruent isosceles triangles. As an example, we
perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After
defining the local structure on the trick art, we analyze its global structure
and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless
“Penrose stairs” is described as a closed trajectory around the isolated
singular point. The approach fits well with graphical projection and gives a
simple and intuitive example of the interaction between global and local
structures. We could deal with higher dimensional objects as well by
considering n-simplices (n > 2) instead of triangles.