*In this section we use the concept of isomorphism to correspond structure among clip-sequences. The visual impact of isomorphic relationships is synchronicity in distinct actions. In other words, the visual content appears different but the relations-in-time among actions are equivalent in a definite sense. We develop several types of isomorphisms that gauge and meter synchronicity, so there are stronger and weaker variations of structural equivalence. From the viewer’s perspective, the effect is a sort of symphonic movement in and out of visual harmony that overrides the specificity of each individual action.*

*The sample videos in this section display isomorphic relations between several screens. However, it helps to bear in mind that we plan to display videos simultaneously on 64 screens. Given this scale, the aesthetic relationships — an interplay of patterns, continuity, and discord in the movement of bodies — will take on greater variation and complexity. Orchestration will emerge in the passage of time as well as within subsets of screens that comprise the spacial arrangement of the installation.*

**Definition:**

- Two tables are
if they have equivalent values in corresponding cells.

**Example 1**:

The following pairs of clip-sequences have isomorphic distance tables:

- -A = {23,12,10} and A’ = {10, 21, 32}
- A = {10, 12, 23} and -A’ = {32, 21, 10

Diagonalized Sequences:

- -A x A’ = {23, 10, 12, 10, 23, 21, 23, 32, 12, 21, 01, 10, 01, 21, 12, 32, 01, 32}
- A X -A’ = {01, 32, 12, 32, 01, 21, 01, 10, 12, 21, 23, 32, 23, 21, 12, 10, 23, 10}

**Note**:

These sequences can be constructed from the same *or* different original clips. In other words, two sequences that display distinct actions can be isomorphic.

**Example 2:**

The tables in Example 1 also have *diagonalized* isomorphic distance tables: the standard script and subscript values in corresponding cells are equivalent.

**Definition:**

- Two tables are
*congruent*if they are isomorphic and have identical table entries (“factors”). Again, note that the table entries may be identical, though they may represent distinct actions.

**Example 3:**

Congruent sequences (A X A’) of distinct actions.

**Example 4:**

The following pairs of sequences are isomorphic in connection but not in distance.

- A’ = {10, 21, 32} and A = {01, 12, 23}
- B’ = {21, 10, 32} and B = {12, 01, 23}

Non-Isomorphic Distance Tables:

Isomorphic Connection Tables:

Diagonalized products:

- A’ x A = {10, 01, 01, 21, 10, 12, 10, 23, 21, 12, 32, 01, 32, 12, 21, 23, 32, 23}
- B’ x B = {21, 12, 10, 12, 21, 01, 21, 23, 10, 01, 32, 12, 32, 01, 10, 23, 32, 23}

**Discussion: **

- The tables in Example 1 are isomorphic in terms of distance and diagonalized distance. The connection tables in
**Example 4**are isomorphic in terms of connection and diagonalized connection. Diagonalized isomorphisms are very strong structural equivalence; specifically, it implies that each sequential pair of edges or clips in the diagonalized product will have equivalent distance and connection. It is clear that isomorphic diagonalized tables imply isomorphic tables; however, the validity of converse is not as clear.

**Open Question:**

- Are two 3 x 3 distance (or connection) tables isomorphic if and only if they are diagonalized isomorphic?

A counterexample to this biconditional statement using 2 x 2 tables with edge table entries is given below. However, it has been surprising difficult to construct a counterexample with 3 x 3 tables; indeed, we wonder if it is even possible. It would be a surprising and useful result if the statement proved true, for then there would be no need to construct and analyze diagonalized distance or connection tables, which would reduce data and increase the efficiency of our program.