Intriguing Relationship between Topology and Geometry

1. Intriguing Relationship between Topology and Geometry Ergun Akleman & Jianer Chen A Story of Our Discovery that involves three continents and countless of people

2. Ilhan Koman Exhibition • I read an article in an airplane magazine while flying to Istanbul • About a retrospective exhibition of Koman’s Sculptures.

3. Mediterranean I only knew a few of his sculptures before. Realized (1980) in ca. 120 pieces cut from iron sheets, this sculpture stands in Zincirlikuyu in Istanbul.

4. Koman’s Saddle Shaped Developable Sculptures I did not know he used mathematics in his sculptures.

5. Koman Exhibition When I am in istanbul, I visited his exhibition with a friend of mine, Tevfik Akgun, who is the head of the design communication department in Yildiz Technical University. A Photograph from the exhibition

6. Exhibition in Beyoglu We were both excited about the work. There were lots of fresh ideas. Exhibition was in Beyoglu, near to this place

7. Fresh Ideas • We decided to explore his work further. • Tevfik found out that his son Ahmet Koman was a Biochemistry professor in Bosphorous University; he was also head of the Koman Foundation. • Tevfik contacted Ahmet.

8. Stata Center was also developable Around that time, I returned back to USA to attend the Shape and Solid Modeling Conferences which were held next to the Stata center.

9. Back to Beyoglu • After the conference, we met Ahmet at Koman Foundation in Beyoglu, Istanbul.

10. Simit is a non-developable genus-1 surface We talked about Koman’s sculptures while drinking tea and eating simit. Koman had a Leonardo article in 1979.

11. Back to USA Jianer and I decided to investigate more about regular meshes.

12. Regular Meshes • Regular Meshes came out of our latest collaboration.

13. Regular Meshes • A regular mesh is denoted by (n,m,g) where n is the number of the sides of faces, m is the valence of vertices and g is the genus of the mesh. (5,2,0) (5,3,0) (4,5,2)

14. Regular Meshes • We had shown existence and construction of infinitely many regular meshes. • We had not completed the list, yet. (4,6,2)

15. Regular Meshes for g=2 • (3,7,2) • (3,8,2) • (3,9,2) • (3,12,2) • (4,5,2) • (4,6,2) • (5,5,2) • (6,6,2) (5,5,2)

16. Complete List for Regular Meshes for g=2 • (3,7,g) • (3,8,g) • (3,9,g) • (3,10,g) • (3,12,g) • (4,5,g) • (4,6,g) • (5,5,g) • (5,10,g) • (6,6,g) • (8,8,g) • These regular meshes exist for any genus higher than 2. • (4,5,g) and (4,6,g) can particularly be useful for texture mapping and morphing. • Now, we know how to construct all of these…

17. While investigating Regular Meshes • I was thinking about trees and others. • Regular meshes did not provide an answer for their structures: We can make a genus-0 tree…

18. First, Morse theory gave a intuition! • A branch adds to surface one saddle (negative curvature) and one minima/maxima (convex/concave) type critical point (positive curvature). • A handle adds two saddle type critical points.

19. That was where (2-2g) came from If we put –1 for saddle, +1 for minima/maxima the total adds up to 2-2g which is the right side of Euler Equation.

20. Of course, in meshes this is not that straightforward! • Meshes are discrete by nature. • The positions minima, maxima or saddle points that depend on the orientation of the shape is not really useful for meshes.

21. In meshes, local geometry around vertices is important. When I was trying to make my daughter sleep, I realized that Koman’s sculptures provided the answer: Angle deviation from the plane gave the information.

22. Regular Platonic solids supported my assumption. While I was trying to make my daughter sleep, I quickly checked the platonic solids. Their total angle deviation turned out to be the same 4p as I expected. For instance: Cube, each vertex deviates from 2p for p/2. Total deviation turns out to be 4p= 8 (p/2) = 2 p(2-2g)

23. If we assume regular meshes can have regular faces • It was easy to show that the result will also turned out to be 2 p(2-2g) • However, we cannot have regular planar faces for regular meshes.

24. But, we still did not have proof • I had a sketch of proof that suggests the results. But, it required some sort of geometric regularity. I believed that the result is general. • I discussed with Jianer several times. • He also agreed that the result is correct and it must be general.

25. Solution First Saturday of November, I got up at night and the answer came. We have to look a averages. • Average vertex angle • Average Valence • Average Sides Then, it was easy to write the proof.

26. Solution • Next weekend, Jianer took over. When he sent the document back to me at November 15, I could not believe my eyes. • We had an extremely simple argument which is to the point and clear. • It was a great joy.

27. Practical Impacts • Most important impacts are practical.

28. What happens when we increase the # of vertices, faces and edges • If average number of sides goes to x • Average valence goes to 2x/(x-2) For instance, if we repeatedly add quadrilaterals, average valence goes to 4.

29. # of vertices, faces and edges increases • If we repeatedly add triangles, average valence goes to 6. • If we repeatedly add pentagons, average valence goes to 10/3. • If we repeatedly add hexagons, average valence goes to 6.

30. This happens regardless of the operation we use • Any Subdivision • Extrusion • Wrinkle • Any other homogenous operation that do not change topology

31. That means if we gain angle somewhere, we lose it in another place. • Unexpectedly, introducing extraordinary vertices, “carefully”, is a good modeling practice. • It takes the tension away from the mesh. • Faces can be more regular looking.

32. It also tells how to approximate a smooth surface with planar meshes • Using only triangles will not guarantee to have regular looking triangles. It is better to use other polygons. • It also means better reconstruction. • It can really be done. We can automatically create beautiful meshes that today can only be created by professional modelers.

33. Future Work:Discrete Approach • Meshes are not nice analytical shapes in which we can apply differential geometry. • Things like “Discrete Gaussian curvature” that is obtained starting from analytical approach always disturbed me . • Discrete may have its own rules.

34. Future Work:Discrete Approach Here, clearly angle deviation gives a better intuition about the local behavios than “Gaussian curvature.”

35. Future Work:Generalization from 1-Manifold Meshes Total angle deviation tells us how many holes exists: p(2-2g). In this case, concave points play a role similar to saddles of piecewise linear 2-Manifold meshes. For k-manifolds, is it kp(2-2g)?

36. Questions? Ilhan Koman, working.