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.
Highlights
Discrete differential geometry studies discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces
For n = 3, we obtain the discrete differential geometry of tetrahedrons which is used for protein
Regular slant triangles on an affine cube covering are associated with flat triangles in the mesh in such a way that the heavy edges of the former are projected onto the heavy edges of the latter
Summary
Discrete differential geometry studies discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. On the other hand, [4] proposes the discrete differential geometry of n-simplices and considers flows (of n-simplices) on a specific n-dimensional mesh. In other words, it considers connection between space-filling n-simplices. (2016) Discrete Differential Geometry of Triangles and Escher-Style Trick Art. Open Journal of Discrete Mathematics, 6, 161-166. Bringing in a differential structure on the mesh, we could consider the global structure of closed trajectories of triangles. We consider the paradox of an Escher-style trick art shown, a simpler version of “Ascending and Descending”. We give a detailed global analysis of the paradox of the lithograph, using a discrete mathematical approach
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
