Intriguing Relationship between Topology and

Geometry

Ergun Akleman & Jianer Chen

A Story of Our Discovery that involves

three continents and countless of people

Ilhan Koman Exhibition

• I read an article in an airplane magazine while flying to Istanbul

• About a retrospective exhibition of Koman’s Sculptures.

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.

Koman’s Saddle Shaped Developable Sculptures

I did not know he used

mathematics in his

sculptures.

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

Exhibition in Beyoglu

We were both excited about the work. There were

lots of fresh ideas.

Exhibition was in Beyoglu, near to this place

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.

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.

Back to Beyoglu

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

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.

Back to USA

Jianer and I decided to

investigate more about regular

meshes.

Regular Meshes

• Regular Meshes came out of our latest collaboration.

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)

Regular Meshes

• We had shown existence and construction of infinitely many regular meshes.

• We had not completed the list, yet.

(4,6,2)

Regular Meshes for g=2

1. (3,7,2)

2. (3,8,2)

3. (3,9,2)

4. (3,12,2)

5. (4,5,2)

6. (4,6,2)

7. (5,5,2)

8. (6,6,2)(5,5,2)

Complete List for Regular Meshes for g=2

1. (3,7,g)

2. (3,8,g)

3. (3,9,g)

4. (3,10,g)

5. (3,12,g)

6. (4,5,g)

7. (4,6,g)

8. (5,5,g)

9. (5,10,g)

10. (6,6,g)

11. (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…

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…

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.

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.

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.

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.

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 4as I expected.

For instance: Cube, each vertex deviates from 2for /2. Total deviation turns out to be 4= 8 (/2) = 2 (2-2g)

If we assume regular meshes can have regular faces

• It was easy to show that the result will also turned out to be 2 (2-2g)

• However, we cannot have regular planar faces for regular meshes.

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.

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.

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.

Practical Impacts

• Most important impacts are practical.

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.

# 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.

This happens regardless of the operation we use

• Any Subdivision

• Extrusion

• Wrinkle

• Any other homogenous operation that do not change topology

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.

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.

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.

Future Work:Discrete Approach

Here, clearly angle deviation

gives a better intuition about

the local behavios than

“Gaussian curvature.”

Future Work:Generalization from 1-Manifold Meshes

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

Questions?

Ilhan Koman, working.