1. Euler Characteristics and Genus
Florida 1999
CS 39, 2017
Euler Characteristics and Genus
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
A chose a very mathematical title …

2. Surface Classification Theorem:
Granada 2003
Topological Analysis
Surface Classification Theorem:
All 2-manifolds embedded in Euclidean 3-space can be characterized by 3 parameters:
Number of borders, b: # of 1D rim-lines;
Orientability, σ: single- / double- sided;
Genus, g, or: Euler Characteristic, χ, specifying “connectivity” . . .
The “Surface Classification Theorem” says: All 2-manifolds embedded in Euclidean 3-space can be characterized by 3 parameters: The number of its borders, b: i.e., # of 1D rim-lines; Orientability, σ: whether the surface is single- or double-sided;
and its connectivity, specified either by the Genus, g, or by the Euler Characteristic, χ, -- Let’s look at these three characteristics more closely: . . .

3. Determining the Number of Borders
Granada 2003
Determining the Number of Borders
Run along a rim-line until you come back to the starting point;
count the number of separate loops.
Here, there are 4 borders:
This is a sculpture by Charles Parry. Determining the number of borders is the easiest of the 3 tasks:
Run along a rim-line until you come back to the starting point and count this as ONE border. Count the number of separate loops.
In this sculpture, there are 4 borders, all circular rings, marked by 4 different colors.

4. Determining the Surface Orientability
Granada 2003
Determining the Surface Orientability
Flood-fill_paint the surface without stepping across rim.
If whole surface is painted, it is a single-sided surface (“non-orientable”).
If only half is painted, it is a two-sided surface (“orientable”). The other side can then be painted a different color.
The orientability of a surface can be determined by starting to flood-fill-paint the surface without ever stepping across the rim.
If, in the end, all surface areas have been painted, then the surface is single-sided.
If only half of it got painted, and the rest of the surface can be painted with a different color, then the surface is double sided.
If something else happens, then the surface was not a proper 2-manifold in the first place.
A double-sided surface

5. Determining Surface Orientability (2)
Granada 2003
Determining Surface Orientability (2)
A shortcut: If you can find a path to get from “one side” to “the other” without stepping across a rim, it is a single-sided surface.
Sometimes there is a quick way to determine surface orientability: If you can find a path to get from “one side” to “the same point on other side” without stepping across a rim, it is a single-sided surface.
You only need to find ONE such path and you are done!

6. Determining the Genus of a 2-Manifold
Granada 2003
Determining the Genus of a 2-Manifold
The number of independent closed-loop cuts that can be made on a surface, while leaving all its pieces connected to one another.
Closed surfaces
(e.g., handle-bodies)
Surfaces with borders
(e.g., disks with punctures)
The genus of a surface specifies its connectivity. It is defined as: The number of independent closed-loop cuts that can be made on the surface, while still leaving all its pieces connected to one another.
On the left, I show 3 examples of handle-bodies: A sphere has genus 0: any closed loop will cut out a disconnected region. On the two-hole torus, we can make two cuts shown by the red lines, and the surface will still be connected into a single piece. On the genus-4 handle-body, we can make the 4 red loop-cuts, and still move from anywhere to anywhere on this surface On the right, I show a few disks with a varying number of punctures or holes. All of them have genus 0: any closed loop will cut out a disconnected region. The number of punctures does not change anything!
genus 0
genus 2
All: genus 0
genus 4

7. Euler Characteristic of Handle-Bodies
Granada 2003
Euler Characteristic of Handle-Bodies
For polyhedral surfaces:
Euler Characteristic = χ = V – E + F
Platonic Solids:
V: E: F: χ:
Sometimes it is simpler to determine first the Euler Characteristic, and then calculate the genus from that.
One way to find the E.C. is to draw a mesh onto the surface and then apply the formula that E.C. is equal to # of V-E+F.
 Genus = (2 – χ – b) / 2 for double-sided surfaces.

8. “ EC-Mathematics”  Genus = (2 – χ – b) / 2 for double-sided surfaces.
Granada 2003
“ EC-Mathematics”
Fuse the two together:
EC=2; g=0
EC=1; b=1; g=0
Sometimes it is simpler to determine first the Euler Characteristic, and then calculate the genus from that.
One way to find the E.C. is to draw a mesh onto the surface and then apply the formula that E.C. is equal to # of V-E+F.
EC=2; b=0; g=0
Punch multiple holes: the missing faces are replaced by additional borders.
 Genus = (2 – χ – b) / 2 for double-sided surfaces.

9. Granada 2003
EC-Mathematics (2)
Two or more cylinders stuck together end-to-end:
EC=2; g=0
At each joint, n vertices and n edges are removed;
 EC does not change!
Without end-faces:
Closed into torus:
EC=0; b=0; g=1
Sometimes it is simpler to determine first the Euler Characteristic, and then calculate the genus from that.
One way to find the E.C. is to draw a mesh onto the surface and then apply the formula that E.C. is equal to # of V-E+F.
EC=0; b=2; g=0
EC=0; b=2; g=0
 Genus = (2 – χ – b) / 2 for double-sided surfaces.

10. Determining the Euler Characteristic of Tubular Structures
Granada 2003
Determining the Euler Characteristic of Tubular Structures
Cutting a tube (“circular” cut) does not change χ :
χ = V – E + F = Euler Characteristic
How many cuts to obtain a tree-like connected graph?
A tree-like tubular graph structure has genus = 0  It is a “sphere” with (n “cuts”) = n punctures.
Each cut done adds +1 to the genus of the structure,  e.g., a tubular Tetrahedral frame has genus = 3
If the structure was single-sided (Klein bottle), multiply by 2 the calculated genus
Reverse operation: Cutting open a toroidal ring (or any other tubular branch) does NOT change the EC.
Those cuts will change the genus of the handle-body; -- and also the number of borders!
The two changes cancel out to so that the EC stays the same.

11. Determining the Euler Characteristic of Disks with Punctures & Borders
Granada 2003
Determining the Euler Characteristic of Disks with Punctures & Borders
χ = V – E + F
E=V; F= V+F=E V=E=F=1  Disk: χ = 1
Closing the gap
eliminates 1 edge and 2 vertices;
EC := EC 1
b := b +1
Genus unchanged
“Disk”
One way to find the E.C. is to draw a mesh onto the surface and then apply the formula that E.C. is equal to # of V-E+F. This also works if we do not have a handle-body, but just some 2-manifold piece of surface, as the above “Disk”.
For such surfaces, there is a short-cut; particularly for surfaces that are composed from many interconnected ribbon elements: We simply count the number of ribbons that we have to cut, until we are left with a jagged piece of surface with a single rim, that is topologically equivalent to a disk.
The Euler characteristic of our original surface is then 1 minus the # of cuts made.
Genus = (2 – χ – b) / 2 for double-sided surfaces.

12. EC-Mathematics (3) Genus = 2 – χ – b for non-orientable surfaces;
Granada 2003
EC-Mathematics (3)
A ribbon is a “disk”; EC =1
Closing the ribbon into a loop yields: EC = 1  1 = 0
Odd number of half-twists  1-sided Möbius band with 1 border; genus = 1
Even number of half-twists  2-sided annulus/cylinder with 2 borders; genus = 0
As seen on the last slide, when we close a loop, we reduce the EC by 1.
High interconnectivity corresponds to strongly negative EC !
Genus = 2 – χ – b for non-orientable surfaces;
Genus = (2 – χ – b) / 2 for double-sided surfaces.

13. Determining the Euler Characteristic
Granada 2003
Determining the Euler Characteristic
Sometimes a simpler approach: Find χ
χ = V – E + F = Euler Characteristic
How many cuts (rim-to-rim) are needed to obtain a single connected disk?
Disk: χ = 1; do the inverse process: every cut re-join lowers χ by 1;  thus “Tetra” ribbon frame: χ = –2
The Euler characteristic of our original surface is 1 minus the # of cuts needed to turn the surface into a disk.
When we apply this to this tetrahedral ribbon-frame, we have to make 3 cuts; thus its E.C. is -2.
From the E.C. we can then calculate the genus of the surface according to these two formulas.
If some of the ribbon branches, the total number of borders may change, and the number of sides may change; thus the genus may change
From this:
Genus = 2 – χ – b for non-orientable surfaces;
Genus = (2 – χ – b)/2 for double-sided surfaces.

14. “Endless Ribbon” Max Bill, 1953, stone
Granada 2003
“Endless Ribbon” Max Bill, 1953, stone
Single-sided (non-orientable)
Number of borders b = 1
E.C. χ = 2 – = 0
Genus g = 2 – χ – b = 1
Independent cutting lines: 1
Let’s analyse some sculptures in this manner. Here is “Endless Ribbon” by Max Bill. This is a non-orientable Moebius-band; it has a single border. Its E.C. can be found most easily by laying a simple mesh on this surface, consisting of 2 vertices, 3 edges and a single rectangular, twisted face. It’s E.C. thus is 0, and from this we calculate a genus of 1.
And indeed there is one cut we can make on a Moebius band that keeps the surface still connected. You probably have seen this trick how a Moebius band can be cut down the middle, resulting in a double-sized, 2-sided loop with a full twist in it. (If we cut this again along the center line, we obtain two interlinked loops.

15. Costa_in_Cube Carlo Séquin, 2004, bronze
Granada 2003
Costa_in_Cube Carlo Séquin, 2004, bronze
Double-sided (orientable)
Number of borders b = 3
Euler characteristic χ = –5 (make 6 cuts to turn into disk)
Genus g = (2 – χ – b)/2 = 2
Independent cutting lines: 2
Playing this same game on this more complicated surface, we find 3 borders: a 3-segment curve on the upper left, and equal one on the lower right; and a 6-segment border around the equator. We have to make six cuts to open up all the tunnels, and from this we calculate a genus of 2, which means we can make two loop cuts that leave the surface connected.
>>> Now let’s look at some non-orientable surfaces. . .

16. “Möbius Shell” Brent Collins, 1993, wood
Granada 2003
“Möbius Shell” Brent Collins, 1993, wood
Single-sided (non-orientable)
Number of borders b = 2 (Y,R)
Euler characteristic χ = –1 (need 2 cuts to turn into disk)
Genus g = 2 – χ – b = 1
Independent cutting lines: 1 blue or green – but not both, they would intersect!
Brent Collins’ “Moebius Shell” is also single-sided (as shown earlier) but it has two borders (outlined in yellow and red). Its E.C. is -1, (2 cuts are required to break all the ribbon loops). And from this we calculate again a genus of 1.
>>> We can make ONE cut – either along the blue OR the green curve and keep the surface still connected – but not both! Since the two curves intersect each other, the second one would no longer be a closed loop.
NOTE: the blue and green curves are indeed CLOSED loop if we see this sculpture as a thin mathematical 2-manifold, where the curves are “in” the surface.