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Design Theory and...

Block Designs

in Art and Mathematics

by Steven H. Cullinane on February 1, 2004

Hartshorne's principle: "Whenever one approaches a subject from two different directions, there is bound to be an interesting theorem expressing their relation." - Robin Hartshorne, AMS Notices, April 2000, p. 464.

For some aesthetic background, see

Wechsler Blocks for
Psychological Testing

Wechsler Block Design Test
Click on picture for details.

Game with
Wechsler Blocks

Cullinane Blocks
(Click on picture for details.)

Cullinane Blocks in Action
(Click on picture for details.)

A Block Design

(in the usual
mathematical sense)

with ( v, k, lambda, r, b) =

(7, 3, 1, 3, 7)

(Click on picture for details.)

Cullinane Block Design

with ( v, k, lambda, r, b) =

(16, 4, 7, 35, 140)

(Click on picture for details.)

Note that the 4x4 arrays in the picture at bottom right may serve as the basis for patterns like those in the picture at top left. The 35 structures in the picture at bottom right may be regarded as exemplifying the aesthetics of James J. Gibson in his 1978 essay "The Ecological Approach to the Visual Perception of Pictures" --

"What modern painters are trying to do, if they only knew it, is paint invariants."

-- James J. Gibson, Leonardo,
    Vol. 11, pp. 227-235. Pergamon Press Ltd., 1978

Gibson is discussing Euclidean 3-space rather than binary 4-space, but his remarks on invariants are still relevant.

An example of invariant structure:

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn.  Taken as a set, these three line diagrams describe the structure of the bottom colored figure.  After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.

For another sort of invariance of the colored figure, try applying a symmetry of the square to each of the set of four diagonally-divided squares from which the figure's entries are drawn, and observe the induced effect on the figure itself.

A more remarkable invariance -- that of symmetry itself -- is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2x2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry. The cause of this symmetry-invariance in the colored patterns is the symmetry-invariance of the line diagrams under a group of 322,560 binary affine transformations.

Related material on two meanings of "design theory":

In the mathematical sense:
Design Theory, by Beth, Jungnickel, and Lenz

In the artistic sense:
Visual Language, by Karl Gerstner

For more details on the above block designs, see

Theme and Variations,

Geometry of the 4x4 Square,

Galois Geometry,

Latin-Square Geometry,

Walsh Functions

The Diamond 16 Puzzle, and

Diamond Theory.

Finally, some examples of the above quarter-diamond
figures applied to the design of quilt blocks:

Quilt Block Designs
by Mark Jason Dominus


The Dominus Designs
in Action

"The very man despising honest quilts
Lies quilted to his poll in his despite."
-- Wallace Stevens, "The Comedian as the Letter C"

Page created Feb. 1, 2004