1. Lecture 5: Triangulations & simplicial complexes (and cell complexes).
in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, business, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc.
Lecture 5: Triangulations and simplicial complexes (with a brief intro to cell complexes).
Isabel K. Darcy
Mathematics Department/Applied Mathematical & Computational Sciences
University of Iowa

2. Building blocks for oriented simplicial complex
0-simplex = vertex = v
1-simplex = oriented edge = (v1, v2) e v1 v2
2-simplex = oriented face = (v1, v2, v3) v2 e2 e1 e3 v1 v3
A simplicial complex starts with 0 dimensional vertices. Since they are 0 dimensional we call them 0 simplices. So a vertex is called a 0-simplex.
we have oriented edges which are 1-dimensional, so we call them 1-simplices.
When using Z mod 2 coefficients we can ignore orientation, but with any other type of coefficients such as the integers, our simplices will be oriented.
We will always use pictures or a symbol such as e to denote an edge. However, we could also use an ordered pair (v1, v2) to denote an edge that points from v1 to v2.
A 2-dimensional face is a 2-simplex. Again we will use pictures or a symbol such as f to denote a 2-simplex, but one can also use an ordered triple such as (v1, v2, v3), but multiple triples can denote the same oriented face.

3. Building blocks for oriented simplicial complex
2-simplex = oriented face =
f = (v1, v2, v3) = (v2, v3, v1) = (v3, v1, v2)
– f = (v2, v1, v3) = (v3, v2, v1) = (v1, v3, v2) v2 e2 e1 e3 v1 v3
For example (v1, v2, v3) , (v2, v3, v1), (v3, v1, v2) all denote the same oriented face while (v3, v2, v1) denotes a face with the opposite orientation.
2 triples correspond to the same orientation if one is an even permutation of the other,
If I perform an odd number of permutations, I get a face with the opposite orientation
For example if I permute v1 and v2 I go from a clockwise orientation to a counterclockwise orientation.
the orientation v1, v2, v3 is clockwise while the orientation v1, v3, v2 is counter clockwise
If I perform a second permutation, for example exchanging v1 and v3, I now return my orientation to clockwise
Every time we perform a permutation of two elements that are next to each other, we reverse the orientation.
Thus three different ordered triples represent the same face

4. Building blocks for oriented simplicial complex
3-simplex =
σ = (v1, v2, v3, v4) = (v2, v3, v1, v4) = (v3, v1, v2, v4)
= (v2, v1, v4, v3) = (v3, v2, v4, v1) = (v1, v3, v4, v2)
= (v4, v2, v1, v3) = (v4, v3, v2, v1) = (v4, v1, v3, v2)
= (v1, v4, v2, v3) = (v2, v4, v3, v1) = (v3, v4, v1, v2)
– σ = (v2, v1, v3, v4) = (v3, v2, v1, v4) = (v1, v3, v2, v4)
= (v2, v4, v1, v3) = (v3, v4, v2, v1) = (v1, v4,v3, v2)
= (v1, v2, v4, v3) = (v2, v3, v4, v1) = (v3, v1, v4, v2)
= (v4, v1, v2, v3) = (v4, v2, v3, v1) = (v4, v3, v1, v2)
While 12 different ordered quadruples represent a three simplex v4 v3 v1 v2

5. Building blocks for oriented simplicial complex
0-simplex = vertex = v
1-simplex = oriented edge = (v1, v2)
Note that the boundary of this edge is v2 – v1 e v1 v2
2-simplex = oriented face = (v1, v2, v3) v2 e2 e1 e3 v1 v3
Note that the boundary
of this face is the cycle
e1 + e2 + e3
= (v1, v2) + (v2, v3) – (v1, v3)
= (v1, v2) – (v1, v3) + (v2, v3)
Note that the boundary of the 2-dimensional face is the 1-dimensional cycle e1 + e2 + e3. While the boundary of the 1-dimensional edge is the difference between it 0-dimensional boundary vertices v2 – v1 . A zero dimensional vertex does not have boundary. Boundary v = 0.
Note that when we use an ordered triple to represent a face, we can calculate its boundary via an alternating sum of its two dimensional boundary edges.
If I remove vertex v3, I get the edge v1v2. If I remove the edge v2, I get the edge minus v1 v3. removing the vertex v1, I get the edge V2 V3
Note that when I remove a vertex with an odd subscript, the resulting edge is added, while if I remove a vertex with an even subscript we subtract the resulting edge.
1 and 3 are both odd, so Removing v3 results in plus v1 v2, and removing v1 results in plus v2 v3
while removing v2 results in minus v1 v3 since 2 is even
Similarly when calculating the boundary of the edge v1v2, removing v1 results in plus v2 since 1 is odd while removing v2 results in minus v1 since 2 is even.
The boundary of a vertex is zero since if we remove the vertex we have nothing, zero
Note this also means that the dimension of the boundary of an object is 1 less than that of the original object.
The boundary of the 2dim face is a 1 dimensional cycle, while the boundary of 1 dimensional edge is 0-dimensional. The boundary of a 0 dimensional vertex is empty.

6. Building blocks for oriented simplicial complex
3-simplex = (v1, v2, v3, v4) = solid tetrahedron
boundary of (v1, v2, v3, v4) =
– (v1, v2, v3) + (v1, v2, v4) – (v1, v3, v4) + (v2, v3, v4)
n-simplex = (v1, v2, …, vn+1) v4 v3 v1 v2
The boundary of the 3-dimensional tetrahedron is the alternating sum of four 2-dim faces
Remove v4, even subscript, and we get – (v1, v2, v3)
Remove v3, odd subscript, and we get + (v1, v2, v4)
Remove v2, even subscript, and we get – (v1, v3, v4)
Remove v1, odd subscript, and we get + (v2, v3, v4)
-(v1, v2, v3) + this back face (v1, v2, v4) - bottom face (v1, v3, v4) + the remaining face (v2, v3, v4)
The boundary of an n-simplex will be the sum of n-1-dimsional simplices, via the alternating sum of removing one vertex at a time.
v2, …, vn+1 minus v1v3, …, vn+1 + etc.

7. Note that the boundary of this edge is v2 + v1 e v1 v2
Building blocks for an unoriented simplicial complex using Z2 coefficients
0-simplex = vertex = v
1-simplex = edge = {v1, v2}
Note that the boundary of this edge is v2 + v1 e v1 v2
2-simplex = face = {v1, v2, v3} v2 e2 e1 e3 v1 v3
If I use Z2 coefficients, the Building blocks for a simplicial complex using are unoriented
Since my complexes are unoriented, we now use set notation to indicate our complexes. So an edge can be denoted by the unordered set v1v2, while a face is denoted by the unordered set v1v2v3. Since orientation doesn’t matter with Z2 coefficients, the order of the vertices in my set also does not matter.
Since 1 = -1 mod 2, I don’t need minus signs and thus we can calculate boundary using only addition and no subtraction. Thus to calculate the boundary of a face, I can remove v3 to get the boundary edge v1v2, remove v1 to get
The sum of these 3 edges is the boundary of this face. I don’t have alternating signs since 1 = -1.
Similarly the boundary of an edge, remove v1, I get v2, remove v2, I get v1 so the boundary of the edge v1v2 is the sum v2 + v1
As before the boundary of a vertex is 0 since if I remove the vertex, I have the empty set.
Note that the boundary
of this face is the cycle
e1 + e2 + e3
= {v1, v2} + {v2, v3} + {v1, v3}

8. Creating a simplicial complex
To create a simplicial complex, we start by adding 0-simplices (ie 0-dimensional vertices). So our step zero will be to add 0-simplices.
0.) Start by adding 0-dimensional vertices
(0-simplices)

9. Creating a simplicial complex
While the next step, step one will be to add 1-simplices. Ie, 1-dimensional edges
Note: These edges must connect two vertices. Thus I cannot have any loops. Nor can I have multiple edges between a pair of vertices. This is because I want to use these vertices to identify the edge. If I have an edge connecting vertices v1 and v2, I want to be able to use the unordered set {v,1 v2} to uniquely denote this edge. If I had multiple edges or loops then my edge would not be uniquely identified by an unordered pair of vertices.
1.) Next add 1-dimensional edges (1-simplices).
Note: These edges must connect two vertices.
I.e., the boundary of an edge is two vertices

10. Creating a simplicial complex
1.) Next add 1-dimensional edges (1-simplices).
Note: These edges must connect two vertices.
I.e., the boundary of an edge is two vertices

11. Creating a simplicial complex
1.) Next add 1-dimensional edges (1-simplices).
Note: These edges must connect two vertices.
I.e., the boundary of an edge is two vertices

12. Creating a simplicial complex
Step 2.) Add 2-dimensional triangles (2-simplices). But we can only add a 2-dimensional triangle where 3 edges form the boundary of a triangle.
The boundary of a simplex must exist before we can add that simplex.
Note we can also use a set of three vertices to identify a 2-simplex. But to add a 2-simplex, we must have its three boundary edges in the complex.
2.) Add 2-dimensional triangles (2-simplices).
Boundary of a triangle = a cycle consisting of 3 edges.

13. Creating a simplicial complex
Step 2.) Add 2-dimensional triangles (2-simplices). But we can only add a triangle where 3 edges form a triangle since the Boundary of a triangle is a cycle consisting of 3 edges.
The boundary of a simplex must exist before we can add that simplex.
Note we can also use n set of three vertices to identify a 2-simplex. But to add a 2-simplex, we must have its three boundary edges in the complex.
2.) Add 2-dimensional triangles (2-simplices).
Boundary of a triangle = a cycle consisting of 3 edges.

14. Creating a simplicial complex
Continuing to increase in dimension, in step 3, we Add 3-dimensional tetrahedrons (3-simplices). But we can only add the tetrahedron if it’s faces are already part of the simplicial complex that we have created so far. That is we can only add a 3-simplex if there are 4 2-dimensional triangles that form the boundary of a tetrahedron, then we can add a 3-simplex by filling it in.
I can identify a 3-simples with four vertices.
Note all the 2-dimensionals faces must be triangles. Thus I cannot fill in this pyramid because it has a square base.
3.) Add 3-dimensional tetrahedrons (3-simplices).
Boundary of a 3-simplex
= a cycle consisting of its four 2-dimensional faces.

15. Creating a simplicial complex
Continuing to increase in dimension, in step 3, we Add 3-dimensional tetrahedrons (3-simplices). But we can only add the tetrahedron if it’s faces are already part of the simplicial complex that we have created so far. That is we can only add a 3-simplex if there are 4 2-dimensionals that form the boundary of a tetrahedron, then we can add a 3-simplex by filling it in.
I can identify a 3-simples with four vertices.
Note all the 2-dimensionals must be triangles. Thus I cannot fill in this pyramid because it has a square base.
3.) Add 3-dimensional tetrahedrons (3-simplices).
Boundary of a 3-simplex
= a cycle consisting of its four 2-dimensional faces.

16. 4.) Add 4-dimensional 4-simplices, {v1, v2, …, v5}.
Boundary of a 4-simplex
= a cycle consisting of 3-simplices.
= {v2, v3, v4, v5} + {v1, v3, v4, v5} + {v1, v2, v4, v5}
+ {v1, v2, v3, v5} + {v1, v2, v3, v4}
We can only add a 4-simplex if it’s boundary which is a sum of three dimensional 3-simplices exists. We can calculate its boundary by removing one vertex at a time. Thus the boundary of the 4-simplex {v1, v2, …, v5} is the sum of the three dimensional tetrahedron, {v2, v3, v4, v5} + {v1, v3, v4, v5} + {v1, v2, v4, v5}
+ {v1, v2, v3, v5} + {v1, v2, v3, v4}
Thus we would need these five tetrahedron in order to add a four symplex. We only have one tetrahedron so we cannot add a four Symplex.

17. Creating a simplicial complex
In step n, we can add n-simplices, but only if its boundary, a sum of (n-1) dimensional simplices was created in the previous step, the n-1 step where we added n-1 - simplices.
In our example, the highest dimensional simplex is a 3-simplex, this solid tetrahedron, so we have a 3-dimensional simplicial complex.
This partitioning of an object into 0-simplices, 1-simplices, 2-simplices, etc. is called a triangulation of the object. An object is called a simplicial complex if it has a triangulation.
Let us now triangulate a couple of familiar examples.
n.) Add n-dimensional n-simplices, {v1, v2, …, vn+1}.
Boundary of a n-simplex
= a cycle consisting of (n-1)-simplices.

18. Example: Triangulating the circle.
circle = { x in R2 : ||x || = 1 }
Let’s first triangulate the circle.

19. Example: Triangulating the circle.
circle = { x in R2 : ||x || = 1 }
I can try adding just 2 vertices,

20. circle = { x in R2 : ||x || = 1 }
Example: Triangulating the circle.
circle = { x in R2 : ||x || = 1 }
Two vertices define a single edge
Not a triagulation
But then I would have mutliple edges, ie 2 edges joining the same pair of vertices. Remember for a simplicial complex, we are not allowed multiple edges.

21. Example: Triangulating the circle.
circle = { x in R2 : ||x || = 1 }
Thus I need a third vertex.

22. Example: Triangulating the circle.
circle = { x in R2 : ||x || = 1 }
Adding edges joining pairs of vertices, we now have a triangulation of the circle consisting of three vertices and three edges. Note my edges do not need to be straight lines segments. A curve is topologically equivalent to a straight line segment. So these three green arcs are all edges.

23. Example: Triangulating the disk.
disk = { x in R2 : ||x || ≤ 1 }
In the next example, we will triangulate the two dimensional disk

24. Example: Triangulating the disk.
disk = { x in R2 : ||x || ≤ 1 }
We start by adding three vertices

25. Example: Triangulating the disk.
disk = { x in R2 : ||x || ≤ 1 }
We then add three edges

26. Example: Triangulating the disk.
disk = { x in R2 : ||x || ≤ 1 }
We then fill in the circle with a two dimensional face. Note that this 2-dimensional face really is a topological triangle. It’s boundary consists of three edges.

27. Example: Triangulating the disk.
disk = { x in R2 : ||x || ≤ 1 } =
Recall that a triangle is topologically equivalent to a circular disk. So as long as I have 3 edges forming a triangle, I can add a 2-dimensional face.

28. Example: Triangulating the sphere.
sphere = { x in R3 : ||x || = 1 }
For the next example, we will triangulate the sphere.

29. Example: Triangulating the sphere.
sphere = { x in R3 : ||x || = 1 }
Again we start by adding vertices

30. Example: Triangulating the sphere.
sphere = { x in R3 : ||x || = 1 }
We then add edges connecting pairs of vertices.

31. Example: Triangulating the sphere.
sphere = { x in R3 : ||x || = 1 }
And finally, we fill in the topological triangles with two dimensional faces, We fill in this front face with a triangle. We fill in this side face with the triangle, as well as this back face. The bottom hemisphere is also a topological triangle. It’s boundary consists of three edges so we can also fill it in with a face. In case you don’t see that,

32. Example: Triangulating the circle.
disk = { x in R2 : ||x || ≤ 1 } =
Recall that a circular disk is topologically equivalent to a triangle .

33. Example: Triangulating the circle.
disk = { x in R2 : ||x || ≤ 1 }
So we have a topological triangle. If you ever find topology frustrating, well there is an outlet.

34. Example: Triangulating the circle.
disk = { x in R2 : ||x || ≤ 1 }
I now get to punch my topological triangle in order to form the bottom hemisphere.
Fist image from

35. Example: Triangulating the sphere.
sphere = { x in R3 : ||x || = 1 }
So this bottom hemisphere is a circular disk which is topologically equivalent to a triangle, i.e. a face. So we have a triangulation of the sphere,

36. Example: Triangulating the sphere.
sphere = { x in R3 : ||x || = 1 } =
Which, by the way is topologically equivalent to the boundary of a tetrahedron.

37. Creating a cell complex
Building block: n-cells = { x in Rn : || x || ≤ 1 }
Examples: 0-cell = { x in R0 : ||x || < 1 }
1-cell =open interval ={ x in R : ||x || < 1 } ( )
2-cell = open disk = { x in R2 : ||x || < 1 }
We will now describe cell complexes. You don’t need to know anything about cell complexes (so you can sstop this lecture here if you like), but it is good to know that there are different ways of building things. In some ways, cell complexes are easier, but most computational software use simplicial complexes.
Note that the building blocks for cell complexes are similar to that for simplicial complexes. We still have 0 dimensional vertices, one dimensional edges, two dimensional faces (but instead of a triagle, we use a disk, but they are topologically equilvalent,
We also have three-dimensional balls which are topologically equivalent to 3-dimensional tetrahedron.
3-cell = open ball = { x in R3 : ||x || < 1 }

38. Building blocks for a simplicial complex
0-simplex = vertex = v
1-simplex = edge = {v1, v2}
Note that the boundary of this edge is v2 + v1 e v1 v2
2-simplex = triangle = {v1, v2, v3} v2 e2 e1 e3 v1 v3
For simplicial complexes, we also had 0 dimensional vertices, one dimensional edges, two dimensional triangles,
Note that the boundary
of this triangle is the cycle
e1 + e2 + e3
= {v1, v2} + {v2, v3} + {v1, v3}

39. Creating a cell complex
Building block: n-cells = { x in Rn : || x || ≤ 1 }
Examples: 0-cell = { x in R0 : ||x || < 1 }
1-cell =open interval ={ x in R : ||x || < 1 } ( )
2-cell = open disk = { x in R2 : ||x || < 1 }
Thus the building blocks are basically the same for both simplicial and cell complexes. But the rules for building cell complexes are much more lax than those for building simplicial complexes. We will illustrate with a couple of examples
3-cell = open ball = { x in R3 : ||x || < 1 }

40. Creating a cell complex
Building block: n-cells = { x in Rn : || x || ≤ 1 }
Examples: 0-cell = { x in R0 : ||x || < 1 }
1-cell =open interval ={ x in R : ||x || < 1 } ( )
2-cell = open disk = { x in R2 : ||x || < 1 }
To distinguish cell complexes from simplicial complexes, I’ll now color my 2-dimensional faces, light brown instead of blue since one can use any color one wants.
3-cell = open ball = { x in R3 : ||x || < 1 }

41. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
Cell
complex =
U U
( )
The simplicial complex for a disk contains 3 vertices, 3 edges, and 1 2-dimensional face. For the cell complex for the same disk, we only need one vertex, one edge, and one 2-dimensional face.
For both simplicial and cell complexes, we still follow the rule of adding objects one dimension at a time starting with lower dimensional objects.

42. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
Cell
complex =
U U
( )
Thus we first add the 0-dimensional vertex.

43. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
Cell
complex =
U U
( )
( ) U
Going up one dimension, we then add the one dimensional edge. For simplicity I will first take the closure of the edge, i.e. throw in its boundary.

44. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
Cell
complex =
U U
( )
[ ] U
The main gluing rule is that I need to glue the entire boundary of the cell to the previously constructed lower dimensional cell complex.
Since I started off with a vertex, to add the edge, I must glue the entire boundary of the edge, ie its two endpoints to the vertex. Note the rule when attaching a cell is that we must glue the entire boundary to the previously constructed lower dimensional complex.

45. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
Cell
complex =
U U
( )
[ ] U
In this case, we get a a circle. Note the circle consists of the vertex plus the open edge. =

46. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
Cell
complex =
U U
( )
[ ] U
Going up another dimension, I can now add the 2-dimensional disk (the 2-cell) to the 1-dimensional circle that I just created. I follow the same rule of gluing the entire boundary of the 2-cell to the circle, the lower dimensional object created in the previous step. = U

47. Example: disk = { x in R2 : ||x || ≤ 1 }
Simplicial complex
3 vertices, 3 edges, 1 triangle =
Cell complex
1 vertex, 1 edge, 1 disk. =
U U
( )
[ ] U
For example,
For the simplicial complex of a disk we have three vertices -3 edges plus s one 2-dimensional equals to one, so the euler characteristic of a disk is one. We also get an Euler characteristic of one for the cell complex of a disk one vertex -1 edge plus one 2-dimensional equals to 1. = = U

48. Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex
4 vertices, 6 edges, 4 triangles =
Cell
Complex
1 vertex, 1 disk = U
Let’s now look at the sphere. The simplicial complex of a sphere contains four vertices, six edges, and four 2-dimensionals, so it’s Euler characteristic is = 2.
For the cell complex for the same sphere, we only need one vertex and one 2-dimensional. 1 vertex - 0 edges plus one 2-dimensional also gives an Euler characteristic of 2.
To create the sphere as a cell complex, as before we first start by adding the zero dimensional vertex.

49. Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex =
Cell
complex = U
, as before we first start by adding the zero dimensional vertex.

50. Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex =
Cell
complex = U
We then add the next higher dimensional cell, in this case, a 2-cell U

51. Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex =
Cell
complex = U
We then take the entire boundary of this 2-cell, this infinite number of points creating the boundary circle and glue them to this single vertex. = U

52. Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex =
Cell
complex = U
That is we hit this disk so hard that it forms almost the entire two sphere except for this one vertex. Note that the sphere minus a vertex is topologically equivalent to an open disk. I can topologically deform the sphere minus the vertex by increasing the size of the hole created by removing the vertex.
This cell complex is not a simplicial complex but the simplicial complex above is also a cell complex.
We created it by = U
Fist image from

53. Example: sphere = { x in R3 : ||x || = 1 }
Cell
complex = U
Adding four vertices. = U

54. Example: sphere = { x in R3 : ||x || = 1 }
Cell
complex = U
Then adding six edges gluing the boundaries of the six edges to the vertices. We now have a one dimensional complex. = U

55. Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex
Cell complex =
Cell
complex = U
To which we can add 4 two dimensional disks, gluing the entire boundaries of the two dimensional disks to the one-dimensional complex created on the previous slide.
So this is both a cell complex as well as a simplicial complex.
So why do we use simplicial complexes instead of cell complexes? Well the answer will become obvious in the next lecture but in the meantime think about how you would encode simplicial complexes in a computer program versus how you would encode a cell complex. = U