_{1}

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.

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. Progress in the field is to a large extent stimulated by its relevance for computer graphics and mathematical physics [

On the other hand, [

Bringing in a differential structure on the mesh, we could consider the global structure of closed trajectories of triangles. As an example, we consider the paradox of an Escher-style trick art shown in

Ascending and Descending is a lithograph made by dutch artist M. C. Escher, who gets the idea from an article by L. S. Penrose and R. Penrose [

Let’s consider a flow of triangles on the mesh shown in

In

occur.

A regular triangle is a triangle with one heavy edge. A terminal triangle is a triangle with two heavy edges. An isolated triangle is a triangle with three heavy edges. A branch triangle is a triangle with no heavy edge. A singular triangle is a triangle which is not regular.

Assigning heavy edges to all the triangles in the mesh, we obtain a flow of triangles. A flow without singular triangles is called a regular flow.

Discrete differential structure is defined on the triangular mesh to obtain a discrete version of Riemannian manifold as follows. Firstly we consider a three-dimensional cube over the mesh as shown in

By piling up these cubes diagonally, we obtain a mountain and valley-like structure, which is called an affine cube covering of the mesh. 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. In particular the gradient of the slant triangle over a flat triangle indicates the moving direction of a trajectory at the flat triangle (note that we could not associate a singular flat triangle with a slant triangle because the latter has only one heavy edge).

Locally the corresponding slant triangle over a regular flat triangle is obtainable as shown in

As for the problem of extension of a “local” affine cube covering over the whole flow on the mesh, we have the following theorem.

Theorem 1 (Continuation Theorem) An affine cube covering is extendable over a closed trajectory of a regular flow.

See Appendix for the sketch of the proof.

Now let’s look at the Escher-style trick art given in

Using the discrete differential geometry of triangles (i.e., 2-simplices) proposed in [

With this approach at our disposal, the interaction between global and local structures of n-dimensional discrete objects (

the paradox of the Escher-style trick art is attributable to a singular triangle at the center.

Naoto Morikawa, (2016) Discrete Differential Geometry of Triangles and Escher-Style Trick Art. Open Journal of Discrete Mathematics,06,161-166. doi: 10.4236/ojdm.2016.63013

Proof. We could construct a cube covering over a closed trajectory in the following four steps.

[Step 1] Choose any triangle T on the closed trajectory and construct a local cube covering

[Step 2] Extend

[Step 3] Solve local contradictions in

[Step 4] Solve local contradictions in