Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet

M. J. Begue1 D. J. Kelleher2 *G. J. H. Khan3 *D. M. Raisingh4 University of Connecticut Mathematics REU 2011

1Department of Mathematics University of Maryland

2Department of Mathematics University of Connecticut

3Department of Mathematics Boston University

4Department of Mathematics Cornell University

YMC at Ohio State University, August 19, 2011

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 1 / 41 Simplexes and Simplicial Complices

A d-dimensional simplex is the convex hull of d + 1 points in general position.

A simplicial complex is a collection of simplices with a few conditions about how simplices intersect.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 2 / 41 Simplexes and Simplicial Complices

A d-dimensional simplex is the convex hull of d + 1 points in general position.

A simplicial complex is a collection of simplices with a few conditions about how simplices intersect.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 2 / 41 Simplicial complexes

A simplicial complex is a collection of simplexes with a few added conditions. Condition 1: If a simplicial complex K contains a simplex J, then it contains all of the simplices J 0 ⊂ J Condition 2: If two simplices intersect, they intersect on a subsimplex.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 3 / 41 Simplicial complexes

These are valid simplicial complexes:

These are not:

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 4 / 41 Simplicial complexes

These are valid simplicial complexes:

These are not:

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 4 / 41 What is a barycenter?

A barycenter is the center of gravity of a simplex. 1 P In algebraic terms, the barycenter of a simplex J is d+1 i vi where vi are the vertices of J.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 5 / 41 What is a barycenter?

A barycenter is the center of gravity of a simplex. 1 P In algebraic terms, the barycenter of a simplex J is d+1 i vi where vi are the vertices of J.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 5 / 41 What is barycentric subdivision?

Barycentric subdivision is deﬁned recursively, but its formal deﬁnition is a bit tedious. Intuitively, we subdivide all of the faces of a simplex and connect the subdivision of the faces to the barycenter of that simplex.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 6 / 41 What is barycentric subdivision?

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 6 / 41 3-D Barycentric Subdivision

This is what the barycentric subdivision of a tetrahedron (3-simplex) looks like

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 7 / 41 Repeated Barycentric Subdivision

We can repeatedly barycentrically divide a simplex K. We denote the n-th barycentric subdivision as Kn.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 8 / 41 Adjacent simplices

We say that two d-simplexes in Kn are adjacent if they intersect along a d − 1 simplex.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 9 / 41 Approximating graphs

Given Kn, we deﬁne a graph by assigning a vertex to each of simplices and an edge if those vertexes are adjacent.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 10 / 41 Approximating Graphs (Cont.)

Here is the fourth level approximation:

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 11 / 41 How to Determine Graphs

Number of vertices of this graphs grows as ((d + 1)!)n where d is the dimension of the original simplex. So we use an alphabet to create words of length n to specify the cells of Kn.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 12 / 41 How to Determine Graphs

Number of vertices of this graphs grows as ((d + 1)!)n where d is the dimension of the original simplex. So we use an alphabet to create words of length n to specify the cells of Kn.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 12 / 41 Words and Alphabets

00 01 11 10

05 15 02 12 04 14 03 13 55 54 24 53 23 25 52 22 50 42 43 33 32 20 51 21 41 44 34 31 40 45 35 30

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 13 / 41 Words and Alphabets

So we have a way of specifying vertices. How do we ﬁnd the edges? If we think of a good naming scheme, then given a word of length n (ex. 0534134), we can ﬁnd its neighbors. But more on this later...

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 14 / 41 Words and Alphabets

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 14 / 41 Contraction and Projection Mappings

We can perform contractions. This is the same as tacking on a letter to the front of the word. We also have a projection map from a cell to it’s parent cell by deleting the last letter of the word.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 15 / 41 Contraction and Projection Mappings

We can perform contractions. This is the same as tacking on a letter to the front of the word. We also have a projection map from a cell to it’s parent cell by deleting the last letter of the word.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 15 / 41 Strichartz Hexacarpet

Let the length of the words go to inﬁnity with the same connections for each of the ﬁnite length words, we obtain the Strichartz Hexacarpet. But what does this look like?

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 16 / 41 Permutohedron

0.5

0.4

0.3

0.2

0.1

0 0.4

0.2

0 1 −0.2 0.8 0.6 0.4 −0.4 0.2

Cayley graph of Sn with adjacent elements swapped

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 17 / 41 Permutohedron

0.5

0.4

0.3

0.2

0.1

0 0.4

0.2

0 1 −0.2 0.8 0.6 0.4 −0.4 0.2

Cayley graph of Sn with adjacent elements swapped

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 17 / 41 Permutohedron

n+1 Take all the permutations the vector (1, 2, . . . , n + 1) ∈ R . Its convex hull of these vectors is the permutohedron

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 18 / 41 Permutohedron

n+1 Take all the permutations the vector (1, 2, . . . , n + 1) ∈ R . Its convex hull of these vectors is the permutohedron

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 18 / 41 2nd Level approximation

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.4

0.2

1 −0.2 0.8 0.6 0.4 −0.4 0.2 0

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 19 / 41 Deﬁning the Laplacian, Eigenvalues, and Eigenfunctions

On a ﬁnite graph approximation, the Laplacian, ∆, is deﬁned as X −∆nu(x) = (u(x) − u(y)) (1) x∼y n

Using the Laplacian, we can then determine the eigenvalues and eigenfunctions using the equation

−∆nu(x) = λu(x) (2)

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 20 / 41 An Example on Determining the Laplacian

P The Laplacian equation again −∆nu(x) = x∼y (u(x) − u(y)) n

1 1 2 3 4 1 3 −1 −1 −1  2 −1 2 −1 0  2 4   3 −1 −1 3 −1  4 −1 0 −1 2 3

This is the negative matrix of the Laplacian.

So we can see, ﬁnding the neighbors of each cell is very important.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 21 / 41 Working on the 3-Simplex

We have developed an algorithm to ﬁnd the neighbors of any cell in a arbitrary d-simplex. We will present the method on a tetrahedron so that you see the intuition behind the algorithm.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 22 / 41 Deﬁning the Naming Scheme

Deﬁnition Any cell, A, in one Barycentric subdivision of a tetrahedron has one letter word A = (a1, a2, a3, a4)

with {a1, a2, a3, a4} = {0, 1, 2, 3}, each necessarily unique. We call {0, 1, 2, 3} characters

a1 tells us the vertex A shares with its parent cell

a1a2 gives us the edge of the parent on which the second vertex is

∆a1a2a3 gives us the face of the parent on which the third vertex is

a4 is the unused character. It is the vertex opposite to the face of K0 that A intersects

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 23 / 41 Deﬁning the Naming Scheme

Deﬁnition Any cell, A, in one Barycentric subdivision of a tetrahedron has one letter word A = (a1, a2, a3, a4)

with {a1, a2, a3, a4} = {0, 1, 2, 3}, each necessarily unique. We call {0, 1, 2, 3} characters

a1 tells us the vertex A shares with its parent cell

a1a2 gives us the edge of the parent on which the second vertex is

∆a1a2a3 gives us the face of the parent on which the third vertex is

a4 is the unused character. It is the vertex opposite to the face of K0 that A intersects

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 23 / 41 Deﬁning the Naming Scheme

Deﬁnition Any cell, A, in one Barycentric subdivision of a tetrahedron has one letter word A = (a1, a2, a3, a4)

with {a1, a2, a3, a4} = {0, 1, 2, 3}, each necessarily unique. We call {0, 1, 2, 3} characters

a1 tells us the vertex A shares with its parent cell

a1a2 gives us the edge of the parent on which the second vertex is

∆a1a2a3 gives us the face of the parent on which the third vertex is

a4 is the unused character. It is the vertex opposite to the face of K0 that A intersects

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 23 / 41 Deﬁning the Naming Scheme

Deﬁnition Any cell, A, in one Barycentric subdivision of a tetrahedron has one letter word A = (a1, a2, a3, a4)

with {a1, a2, a3, a4} = {0, 1, 2, 3}, each necessarily unique. We call {0, 1, 2, 3} characters

a1 tells us the vertex A shares with its parent cell

a1a2 gives us the edge of the parent on which the second vertex is

∆a1a2a3 gives us the face of the parent on which the third vertex is

a4 is the unused character. It is the vertex opposite to the face of K0 that A intersects

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 23 / 41 Deﬁning the Naming Scheme

Deﬁnition Any cell, A, in one Barycentric subdivision of a tetrahedron has one letter word A = (a1, a2, a3, a4)

with {a1, a2, a3, a4} = {0, 1, 2, 3}, each necessarily unique. We call {0, 1, 2, 3} characters

a1 tells us the vertex A shares with its parent cell

a1a2 gives us the edge of the parent on which the second vertex is

∆a1a2a3 gives us the face of the parent on which the third vertex is

a4 is the unused character. It is the vertex opposite to the face of K0 that A intersects

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 23 / 41 An Example of Naming a Level 1 Subdivision: (0, 1, 2, 3)

3 1

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 24 / 41 Naming Higher Level Subdivisions

Deﬁnition The vertices of a cell A that results from a Barycentric subdivision are 0 labeled by looking at each vertex of K that has is named character ai then ﬁnding the vertex of A that is closest to it using the Euclidean metrix. This vertex of A is then named ai.

2 3

3 1

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 25 / 41 Naming Higher Level Subdivisions

As we wish to reference cells in further Barycentric subdivisions, Kn, the words become n × 4 matrices where each ith row, (ai,1, ai,2, ai,3, ai,4), gives the level-i cell that contains A. 0 1 2 3 An example of this: 0 1 2 3

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 26 / 41 0 1 2 3 An Example of Naming Higher Levels: 0 1 2 3

0 0

3 1

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 27 / 41 The Number of Neighbors

A cell can have at most 4 neighbors because it has 4 faces. All cells have 4 neighbors, but some have 3 neighbors if they are boundary cells which are deﬁned as: Deﬁnition A cell is an boundary cell if one of its faces intersects the face of K0

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 28 / 41 Finding the First Three Neighbors

Corollary Every cell in B1 is a boundary cell.

Proposition Any cell A ∈ Kn, n ≥ 1, has three neighbors in it’s (n − 1) parent cell.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 29 / 41 Finding the First Three Neighbors

Lemma Any cell A ∈ Kn has 3 inner neighbors in the same (n − 1)-cell obtained by applying the transposition σi = (ani, an(i+1)) to the nth row of A for i = 1, 2, 3. 0 1 2 3 Example: Let A = 0 1 2 3 Its 3 neighbors are: 0 1 2 3 0 1 2 3 0 1 2 3 , , 1 0 2 3 0 2 1 3 0 1 3 2 This transposition gives us the permutahedron.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 30 / 41 Finding the First Three Neighbors - Some Intuition

Let us consider (0, 1, 2, 3) which has neighbors (1, 0, 2, 3), (0, 2, 1, 3), (0, 1, 3, 2)

3 1

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 31 / 41 Finding the Fourth Neighbor

Proposition

If A 6∈ ∂Kn then A has a fourth neighbor in a j cell for some j ≤ n − 1.

Suppose that A, B ∈ Kn and A ∼ B (B is the fourth neighbor), then A’s parent is touching B’s parent.

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 32 / 41 Finding the Fourth Neighbor

Using these ideas, we can develop an algorithm to ﬁnd the fourth neighbor (details omitted) (1) Find the parent cell that neighbors A. (2) Find a homomorphism that converts the characters from A’s coordinates to the neighbor’s coordinates (3) Apply this homomorphism and obtain the neighbor

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 33 / 41 The Heat Equation

The heat equation is ∂u − c∆u(x, t) = 0 (3) ∂t where t > 0 where ∆ is a Laplacian in the space variable x and c is a positive constant.

For a level 3 subdivision of a tetrahedron we get the resistance is ρ = 13.4274

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 34 / 41 Eigenfunction Video - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 35 / 41 Eigenfunction Pictures - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 36 / 41 Eigenfunction Pictures - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 37 / 41 Eigenfunction Pictures - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 38 / 41 Eigenfunction Pictures - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 39 / 41 Eigenfunction Pictures - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 40 / 41 Eigenfunction Pictures - Level 4

G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 41 / 41