Finite Geometry Notes

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. 227235. Pergamon Press Ltd., 1978
Gibson is discussing Euclidean 3space rather than binary 4space, 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 2element sets, of the 4element 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 diagonallydivided 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 quarterdiamond
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"
Page created Feb. 1, 2004