**Conditions**

Duration is order of time.

A video is a duration of time.

Each video is a collection of clips.

A clip is a three second duration.

Each duration contains exactly two endpoints.

**Terms:**

A model of time discipline is a mapping of real time onto representational time. A model is expressed as an ordered pair G = (V, E) comprising a set V of vertices and a set E of edges which are two ordered element subsets of V.

A vertex is an endpoint of a duration.

An edge is a pair of vertices with a one second duration.

**Definition:**

- A
*sequence*is an ordered set of edges.

**Example:**

**Notation: **A = {0'' - 3''} = {03}

Vertices of the clip: {0, 3}

**Edge of Clip A: **

**Notation: **{1'' - 2''} = {12}

Vertices of the edge {1, 2}

Three edges comprise Clip A: A = {0-1, 1-2, 2-3} = {01, 12, 23}

{01}

{12}

{23}

**Graph of Clip A**

**Inversion**

**Definition: **

- An
*inversion*of an edge maps the first vertex onto the second vertex, and the second onto the first.

**Notation: **(01)'=10 ** **

**Property:**

- Inversion is distributive to a set of edges.

**Notation: **{01, 23}' = {(01)', (23)'} = {10, 32}

**Definitions: **

- A
*clip-sequence*is an ordered set of edges with the domain {01, 12, 23, 10, 21, 32}. In other words, it is a sequencing of edges from Clip A and from the inverted edges of Clip A.

We call {0''-1'', 1''-2'', 2''-3''} = {01, 12, 23 } the

*original*for each clip-sequence.

A’ is called the inverted clip-sequence of Clip A.

Vertices of A’ : {0, 1, 2, 3}

Note: A’ is not a clip because it has more than two vertices and thus is not a duration.

A’ = {01, 12, 23}' = {(01)', (12)', (23)' = {10, 21, 32}

**Edges of A’: {10, 21, 32} **

{10}

{21}

{32}

**Graph of A’**

**Definition: **

- The vertices of a sequence that produce a duration are said to
*coincide*. The vertices of a clip coincide by definition. The vertices of A’ do not coincide, as displayed in the graph above.

**Discussion: **

*Cutting up three second clips into one second durations and re-sequencing the durations is our method for distorting the appearance of an action in real time. We use these one second durations (“edges”) to establish new relations in represented time among different points (“vertices”) in the action. In effect, a familiar action such as swinging a pickax becomes disjointed and, in a sense, undirected — swinging a pickax may not end with the earth being tilled. However, the entire action is displayed in the course of the clip-sequence, though not in its familiar, real-time sequence. (In Example 2 the earth is tilled, but this outcome occurs before the whole swing is displayed.) An effect of the re-sequencing and reorientation of a clip is the loss in priority of the practical consequence of an action. A viewer can now attend to the structure of actions — their sequencing, continuity, and orientation — under a new discipline of time that is metered in one second durations.*

**Reflection**

**Definition: **

- A
*reflection*is a re-sequencing of edges in a clip so that the first edge is last and the last edge is first.

**Notation: -**A= -{01, 12, 23} = {23, 12, 01}

- -A is called the reflected clip-sequence of Clip A.

**Example:** -(A' ) = -{10, 21, 32} = {32, 21, 10}

**Example:** (-A)' = {23, 12, 01}' = {32, 21, 10} = -(A')

**-**(A' )is a clip with vertices {0, 3}

**Theorem: **

- Inversion and reflection yield exactly four distinct sequences of Clip A:

A={01, 12, 23}

A'={10,21,32}

-A={23, 12, 01}

-A' ={32, 21, 10}

** **

**A**

**A'**

**-A**

**-A'**

**Definition: **

- We define two permutations of Clip A that each yield four distinct clip-sequences:

B = {12, 01, 23}

B' = {21, 10, 32}

-B = {23, 01, 12}

-B' = {32, 10, 21}

C = {01, 23, 12}

C' = {10, 32, 21}

-C = {12, 23, 01}

-C' = {21, 32, 10}

**Examples:**

**Clip B**

**B'**

**-B**

**-B'**

Twelve clip-sequences in total are constructed from an original clip A: {A, A', -A, -A', B, B', -B, -B', C, C', -C, -C'}

**Discussion:**

*For the video installation, these twelve clip-sequences are constructed out of a variety of clips from different videos. The sequences of different actions in the videos — pick-swinging, shoveling, pouring, etc. — are displayed simultaneously on multiple screens. A challenge in visual effect is, how to draw aesthetic relationships among multiple screens with disparate actions?*

*An action can be visually identified through an oriented, continuous motion with an outcome or endpoint. However, when an action is transformed — i.e., cut into durational units and re-sequenced — so that its continuity is broken and its endpoint relocated, the action becomes less identifiable. This opens up possibilities for relating a re-presented action to other actions that are disparate in original motion and outcome, but similar in the timing and extent of discontinuity. In other words, this effect relaxes the viewer focus on the purpose of specific action while drawing attention to the relations-in-time among a group of actions. In the following sections, we develop several tools that will draw out these relations-in-time and control the visual impact of installation***.**