A Geometric Gesture
January 15th, 2008
Three illustrations from Andre Vandenbroeck’s Philosophical Geometry.
Philosophical Geometry contrasts with axiomatic geometry: the latter “discipline is founded on a group of propositions considered self-evident or necessary, from which a chain of further propositions can be deduced” (p3), whereas Philosophical Geometry is a “property of mind in general and not a specialty of the analytic mind… For example, in ‘Meno’, Plato shows geometry as a birthright of mind in general: Meno’s slave, unhampered by his lack of background, comprehends a geometric necessity with which Socrates confronts him” (p4).
Philosophical Geometry consists of theoria and practica. “Theoria is the adequate expression of geometric experience. As such, it is dependent upon the perception of the geometer; it is a subject… Practica is a necessary structure of two-dimensional events and is independent of the geometer’s perception. It is the object of the discipline… The aim of Philosophical Geometry is the individual elaboration of theoria… Theoria depends upon practica through the perception of the geometer, and it is due to the variance in perception that differences in practica occur” (p12).
Therefore, geometry is the product of geometric gesture.
“On a homogeneous two-dimensional field best termed a plane, the stylus is posited in a gesture of inscription. The contact of stylus with plane breaks the homogeneity of the undifferentiated surface into a heterogeneity of the point of contact and the remainder of the plane… Whatever the final complexity of inscription, the [practica] must pass through this initial stage: the contact of stylus with plane” (p17).
“Practica can progress beyond the point only by motion of the stylus. Motion of the stylus produces an inscription best termed a line” (p17-8).
“To qualify the line as a two-dimensional element is a contradiction in terms. A line cannot be a surface. And yet, contrary to the point, it cannot be considered an ideal indivisible marking. Although, as line, it cannot be a surface, it clings to the plane by a dimension of length and thus forces upon contemplation a second dimension of width which is the thickness of the inscripting stylus” (p18).
“To examine the line, motion of stylus has to be arrested… If the motion of stylus is not arrested, a continuous line results. This uniform, indefinitely prolonged motion of stylus finds its perfect representation only in the circle” (p18-9).
Polygonal Numbers and Pie
November 21st, 2007
Four mathematical illustrations from Wells’ Dictionary of Curious and Interesting Numbers.
“The maximum number of pieces into which a pancake [or pie] can be cut with 6 slices” is 22 (p78):
22 is also the 4th pentagonal number — whereas “the 4th centered hexagonal number, obtained by arranging hexagonal layers around a central point,” is 37 (p99):
“By a different division [of the above] the nth centered hexagonal number is equal to 6Tn-1 + 1, where Tn is the nth triangular number” (p100). Here we see a composition using the 3rd triangular number:
And the 4th triangular number? — 10, the tetractys, a figuration “so holy [to the Pythagoreans] that they even swore oaths by it” (p61, c.f. the obverse seal):
