Finite
Geometry Notes

1. Background
2. Other related web pages
3. Related philosophical remarks
Background:
For nonmathematicians...
See some selected quotations on Hesse's Glass Bead Game, and the way it combines music and mathematics. Then see the combination of music and mathematics in Timothy A. Smith's Shockwave movie (or his pdf essay) combining patterns like those above with an analysis of a Bach fugue.
For mathematicians...
Kaleidoscopes are, of course, intimately related to reflection groups. For the connection to the geometry of the above patterns, see The Diamond Theorem. Switching two rows, columns, or quadrants in the "kaleidoscope" above is equivalent to performing an affine reflection (an automorphism of an affine space fixing a hyperplane pointwise) in a 16point finite geometry. For some context, see Reflection Groups in Finite Geometry.
Other related web pages
Clifford Geertz on LeviStrauss, from The Cerebral Savage: "Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns.... The patterns consist in the disposition of the chips visavis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately). And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation.... ... LeviStrauss generalizes this permutational view of thinking to savage thought in general. It is all a matter of shuffling discrete (and concrete) images... ... And the point is general. The relationship between a symbolic structure and its referent, the basis of its meaning, is fundamentally 'logical,' a coincidence of form not affective, not historical, not functional. Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.' Or like linguistics." Edward Sapir on Linguistics, Mathematics, and Music: "...
linguistics has also that profoundly serene and satisfying quality
which inheres in mathematics and in music and which may be described as
the creation out of simple elements of a selfcontained universe of
forms. Linguistics has neither the sweep nor the instrumental
power of mathematics, nor has it the universal aesthetic appeal of
music. But under its crabbed, technical, appearance there lies
hidden the same classical spirit, the same freedom in restraint, which
animates mathematics and music at their purest." Edward
Sapir, "The Grammarian and his Language," Robert de Marrais on LeviStrauss and Derrida, from "Catastrophes, Kaleidoscopes, String Quartets: Deploying the Glass Bead Game," Part II: "...underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the nonprofessionals call . . . kaleidoscopes! * (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name...) * H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an ndimensional analog of a polygon or polyhedron. Chapter V of this book is entitled 'The Kaleidoscope'....)" De Marrais attacks Derrida for ignoring the "kaleidoscope" metaphor of LeviStrauss. Here is a quotation from Derrida himself: "The time for reflection is also the chance for turning back on the very conditions of reflection, in all the senses of that word, as if with the help of an optical device one could finally see sight, could not only view the natural landscape, the city, the bridge and the abyss, but could view viewing. (1983:19)  Derrida, J. (1983) 'The Principle of Reason: The University in the Eyes of its Pupils,' Diacritics 13.3: 320." The above quotation comes from Simon Wortham, who thinks the "optical device" of Derrida is a mirror. The same quotation appears in Desiring Dualisms at thispublicaddress.com, where the "optical device" is interpreted as a kaleidoscope. Derrida's "optical device" may be compared with Joyce's "collideorscape." For a different connection with Derrida, see The 'Collideorscape' as Differance. From Noel Gray, The Kaleidoscope: Shake, Rattle, and Roll: "... what we will be considering is how the ongoing production of meaning can generate a tremor in the stability of the initial theoretical frame of this instrument; a frame informed by geometry's long tradition of privileging the conceptual ground over and above its visual manifestation. And to consider also how the possibility of a seemingly unproblematic correspondence between the ground and its extrapolation, between geometric theory and its applied images, is intimately dependent upon the control of the truth status ascribed to the image by the generative theory. This status in traditional geometry has been consistently understood as that of the graphic ancilla a maieutic force, in the Socratic sense of that term an ancilla to lawful principles; principles that have, traditionally speaking, their primary expression in the purity of geometric idealities.* It follows that the possibility of installing a tremor in this tradition by understanding the kaleidoscope's images as announcing more than the mere subordination to geometry's theory yet an announcement that is still in a sense able to leave in place this selfsame tradition such a possibility must duly excite our attention and interest. * I refer here to Plato's utilisation in the Meno of graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave." See also Noel Gray, Ph.D. thesis, U. of Sydney, Dept. of Art History and Theory, 1994: "The Image of Geometry: Persistence qua Austerity Cacography and The Truth to Space." 
Page created August 9, 2005. Last modified August 23, 2010.