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 |
Game with |
Cullinane Blocks |
Cullinane Blocks in Action |
A Block Design (in
the usual 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
"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
Related material on two meanings of "design theory":
In the mathematical sense:
Design Theory, by Beth, Jungnickel, and LenzIn the artistic sense:
Visual Language, by Karl Gerstner
For more details on the above block designs, see
Finally, some examples of the above quarter-diamond
figures
applied to the design of quilt blocks:
Quilt Block Designs |
The Dominus Designs |
"The very man despising honest quilts
Lies quilted to his
poll in his despite."
-- Wallace Stevens, "The Comedian as the Letter C"
For a large downloadable folder
containing this and many related
web pages,
see Notes on Finite
Geometry.
Page created Feb. 1, 2004