A mathematical glange at the “History of the Square”

Consider a random quadrangle, i.e. a 4-sided closed figure with –in general– unequal angles and sides. Join the mid-points of its four sides. Strangely enough, a regular parallelogram results! The random aspect here (and elsewhere) contains its own symmetry. Harmony is revealed in the most unlikely places!

Dimitris Tragkas’s geometrical drawings can only be understood as an organic whole. The same also goes for his accompanying phrases that constitute an integral extension of these drawings. This is not a case of cubism but, rather, anti-cubism, if such a term existed. The “active” geometry in this case is but an allegory of the archetypal “square”. Schematically, the square’s convolution over time. That is exactly why Dimitris Tragkas’s work is also “holographic”. In other words, if some part of it is extracted, it remains the same… but more “hazy”. Naturally, there are –as always– different individual references by each spectator-reader, which cover a whole spectrum of reactions: the pre-eminence of the severe form, various interpretations or over-interpretations, geometry’s raison d’être, etc., etc. However, all these comprise precisely “the History (or the myth) of the Square”, as it is unfolds by the artist.

Dimitris Tragkas attenuates the severity of the four equal lines and angles (the quadrangle), and introduces us to the plasticity of its corresponding topology. This self-transformation also brings to the fore basic algebraic structures (geometry’s twin sisters), such as e.g. automorphisms of certain shapes, the symmetries of crystals, the sonata form and fugue in music, the regularity of numbers, etc. This profound isomorphism between Algebra and Geometry has been discovered as well as demonstrated by the mathematician Pierre de Fermat (1601-1665) and the philosopher-mathematician Rene Descartes (1596-1650), the Father of Analytic Geometry, in which they inextricably link the continuous (Geometry) with the discrete (Algebra). The reflections, inversions and other symmetries are part of the algebra of the “imprisoned” square and act as to enrich our perception by their sub(un)conscious correlations.

The artist here introduces high-school memories, reveals associations, highlights the saga of the contested square, sweeps us along to such an extent of deconstructivity that our high-school geometry loses its innocence. He depicts an enchanting but also illusive reflection, like a secret that everyone knows. So, here are the so-called two sides of the coin: on the one hand, the pure Platonic square and, on the other, the trace of its history as it unfolds by the drawings of Dimitris Tragkas.

Alexis Bacopoulos, 2000

(Prologue for the book of Dimitris Tragkas “History of the Square”)