A Graphical Disquisition Upon the IgnoverseAs Expounded by Ignotus the Mage*

And Faithfully Recorded by Paul Hertz

*A genuine Neo-Pythagorean charlatan

Construction of the Ignotiles

A hole in a square Corner points Central points

Butterfly Strider Box Windmill

We can find exactly four ways of connecting the inner square to the outer square with diagonals to three corner points and two central points...

Basic Shapes

Butterfly Strider Box Windmill

...in such a way that we use only five basic shapes.

sphinx left parallelogram

rightparallelogram

triangle trapezoid

The Ignoquad or “12-point solution”

We can create a group of four tiles where:

1. The four different tiles are all used.

2. All four possible positions of the “middle diagonal” (shown in red) are used exactly once.

3. Each of the central diagonals (in green) is used exactly once.

Permutations of the Ignoquad

A C

D BABCD ADCB ACDB ABDC ADBC ACBD

BCDA DCBA CDBA BDCA DBCA CBDA

CDAB CBAD DBAC DCAB BCAD BDAC

DABC BADC BACD CABD CADB DACB

The tiles in an ignoquad can be permuted into 24 distinct sequences, emphasized here with coloring. Rotating and reflecting the individual ignotiles yields 192 different ignoquads.

The 32 Ignotiles (Rotations and Reflections)

A B C D A B C D

The Ignosquares

A

A

A

A C

C

C

C

D

D

D

D

B

B

B

B

A Latin Square is an n x n matrix of n elements where no element is repeated in any row or column.

We can easily construct Latin Squares—which we shall call ignosquares—from the ignotiles.

We add two restrictions to the general condition of non-repitition in any row or column:

1. Each corner of the Latin Square must contain an ignoquad

2. We must use 16 different ignotiles selected from the 32 tiles formed by rotations and reflections.

(12 unique solutions)

Merging Shapes in the Ignosquares

First we merge the central square with parallelograms or trapezoids to construct “bridge” shapes that cross the tile (shown in blue). Then we'll settle on some rules for merging the resulting shapes:

1. Diagonal lines never get dissolved, only horizontal and vertical lines.

2. Bridge shapes have priority for merging. If a shape shares an edge with a bridge shape and a non-bridge shape, merge it with the bridge shape.

3. Once bridge shapes have been merged, merge any other shapes that share an edge.

4. Shapes may not have self-intersections.

A

A

A

A C

C

C

C

D

D

D

D

B

B

B

B

Classifying Merged Shapes in the Ignosquares

If we tile a planar surface with an array of ignosquares, we can continue the process of dissolving boundaries between the ignosquares following the same rules. Certain kinds of shapes will emerge. For example:

n Shapes that contain no bridges or triangles.

n Shapes that contain triangles but no bridges.

n Shapes that contain bridges but no triangles.

n Shapes that contain bridges and triangles.

Many other classificatory and coloring schemes are possible.

Large Compositions with Ignosquares

By using many ignosquares in an array, we can generate compositions. In these compositions, new forms will arise. “Prairie,” in the SIGGRAPH art show, shows two kinds of emerging forms: long “creeks” and clusters of shapes enclosed by symmetrical “diamond” outlines.

zoom

Inflating Patterns in Large Scale Compositions

Many ignosquares can be created with a broken diagonal line traversing them. Since the ignotiles and the patterns that emerge in the ignoquads are formed from diagonal lines, one can use such “diagonal” ignosquares to replace these lines, thereby “inflating” a design.

Back to Small Scale (microcosm)

While we may expect more large scale forms to emerge as we work with larger and larger arrays of ignosquares, there are some interesting possibilities to be found working on a small scale, particularly with respect to what Dick Higgins called intermedia:

•The Ignogame, a performance with audience participation

• Coloring the ignotiles

•Making music with the ignotiles

•Eating the ignotiles

The Ignogame

My first “program” for the ignosquares consisted of a deck of 32 homemade punchcards. I engaged other people in a game where they chose cards face down, turning them over at the last minute to reveal a pattern, which I would then interpret as Ignotus the Mage, a dysfunctional fortuneteller.

Ignorant of the future, forgetful of the past, Ignotus tried to report on the present. In the process I obtained the raw material for my compositions: a random sample of the 483,840 possible ignosquares.

Besides, I liked having an audience. Later I wrote programs to play the game on line.

Telling the Present

Shaded Ignotilesin an ignosquare

Earth: contraction, stability

Air: extension,infinity

Fire: transitory vertical equilibrium

Water: transitoryhorizontal equilibrium

Ignotus’s fortune-telling system is the bastard child of the I-Ching and Wassily Kandinsky’s treatise on composition, Point and Line to Plane, books I greatly admire. For a combinatorial system I use the 24 shaded patterns created by the ignoquads, four of which form an ignosquare. I interpret the positions of the ignoquads within the ignosquare according to Kandinsky’s divisions of the picture plane, and his designation of the upper left to lower right diagonal as the “dramatic axis” and the upper right to lower left diagonal as the “harmonic” or “lyrical axis.” I throw in the four elements of antiquity for good measure, and as a nod to Gaston Bachelard, whose essays have been a source of inspiration.

A C

D B

Ignotus’s Fortune-telling Framework (the Ignosigns)

We hope that you can tell high-class mumbo-jumbo when you see it. A sufficiently complex and inclusive system, you might think, would not be devoid of meaning! Or would it? I don't know, but I try to make art about the human capacity for creating meaningful systems.

Expansion on dramatic axis, contraction on harmonic axis: meeting with resistance

Contraction on dramatic axis, expansion on harmonic axis: moving with the current

Vortex of creation, source, birth

One of the signs of material direction

A complex sign of transition:material and spiritual ascent,clarity of purpose.

One of the signs of mental or spiritual direction

Material circumstance weighingupon mental state, directed towards action: conflict arises

Vortex of return, drain, death

The Ignosigns, continued: Zones of Influence

The signs and their positions provide cues to the improvising performer who plays the role of Ignotus the Mage. Other cues come from the audience, also know as the mark (Ignotus is said to be a former carnival mentalist). To my horror, I found people took Ignotus seriously. That is why I am at pains to represent him as a genuine fraud.

With a little more formalization, it should be possible to have a computer tell fortunes for participants, much like on-line versions of the I-Ching.

Divisions of plane to represent zones of influence for signs, indebted to Mr. Kandinsky.

visible

hidden

active potential

The Ignosigns, a Sample Reading

Let's say you played the Ignotus Game and have as a result the ignosquare at the left. The wise mage gazes at you and informs you that the most visible and active element of your life right now is a sense of stability. How did he know that? The sign of stability was in the visible and active quadrant. He can further inform you that you may be experiencing the potential for volatility, but not to worry: hidden creative forces are bringing that

into a harmonious relationship. Not only that, but your stability is complemented by a dynamic rhythm of expansion and contraction that moves with the flow around you. Doesn't that make you feel good? Now cross the mage’s palm with silver.

Coloring the Ignotiles

Butterfly Strider Box Windmill

Rules:

1. Use four colors.2. The sphinx, trapezoid and triangle receive each of the four colors. 3. The parallelograms can be distinguished as right- and left-handed and similarly receive each color once.4. The sphinx color is not shared by another shape in a tile.

Is it possible to color all 24 ignosquares in such a way that no two adjacent shapes have the same color? It turns out that it's possible for 23 of them, and the 24th one only forces us to modify rule 4.

Coloring the Ignosquares

A Greco-Latin Square is an n x n matrix of unique pairs of n different elements where no element is repeated in any row or column.

It's often used in agriculture for combining, say, corn hybrids and fertilizers in a test plot.

If we think of the four tiles as hybrids and the four colors as fertilizers, we can make a test plot of an ignosquare.

Note how the colors of the sphinx shapes form a Latin Square. We could actually add a third parameter, such as texture, in one more Latin Square to create unique triplets (the problem can be generalized to arrays of n x n unique n-1tuplets).

Musical Parameters

AB B

C

C

D

E EFF

G

A

A B

B

C

C

D

E

EF

F

G

A

Since there are 12 diagonal lines in the ignotiles, let's associate them with the 12 pitch classes of the diatonic scale of the European art music tradition. We can assign the pitch classes to different shapes, and add durations based on the area of each shape.

Topological Transformation from Map to Graph

Elementary topology states that every map has a dual graph. With a few additional rules for graph formation beyond adjacency of shapes (treating shapes with rests as a kind of “no-man’s land” through which shapes with pitch-class nodes can be connected), we can derive a graph of musical parameters from a given ignotile.

Consider the graph as a statement of the transition probabilities between notes. We can then construct musical compositions which have a structural isomorphism with paintings. Nor is it necessary to use musical parameters—the values in the nodes could be any sort of event we choose.

A B

B

C

C

D

E

EF

F

G

A

The Ignotones and Ignochords

A slightly different mapping of pitch class to lines in the ignotiles yields some interesting results. The ignotiles can be interpreted as representing triadic chords constructed of perfect, augmented and diminished fourths. Rotations and reflections of the tiles represent transpositions and intervalic inversions. This situates us squarely within contemporary practices in harmony, and incidently provides many jazz-like chords similar to bop harmonies. I am using these chords in current works (2000-2001).

AB B

C

C

D

E E FF

G

A

& &

& &

A D

B

1#3#

œ#œ

3 1

œ3 1

F

C A

CF

B

A

GE

C

œb3b 1

E D

B

Pond I, 1997

The interactive installation “Pond” is one example of an intermedia composition derived from the processes associated with the ignotiles. The sound material in Pond consists of sampled voices of 16 persons; the image material is their faces and patterns which they generated with the Ignogame. Visitors to the installation can trigger events by waving their hands over lucite rods.

General view Close up of floor sculpture

Why bother?

That’s a good question. Ignotus says:

•Production of useless things is the true sign of freedom.

•Complex new forms emerge from simple structures and it’s fun to see what’s going to happen.

•The social and performance element keeps me from getting stuck in the studio.

•The computer does things real fast that used to take me a long time by hand. I can go into mass production when the fad catches on.

•It’s fantastic, I can't stop.

Didn’t You Mention Food?The Ignocake, folks. First baked in 1981 with the aid of my friend the pastry chef, Marcelino Chacon, who studded it with four different kinds of

dried fruit and nuts over four shades of pastel �icing, the pattern on the ignocake is a map of a floor installation executed in tape.

Participants take a �piece of cake to the corresponding location �on the floor, and proceed to eat their location in space.

Some of them I never �see again.

Hoping to See You Again,