Tag Archives: Pythagoras

Right angles everywhere (II), about the gnomon

In this post I shall write about the gnomon. According to wikipedia,

The gnomon is the part of a sundial that casts the shadow. Gnomon (γνώμων) is an ancient Greek word meaning “indicator”, “one who discerns,” or “that which reveals.”

In the next figure are collected the minimal ingredients needed for understanding the gnomon: the sun, a vertical shape and its horizontal shadow.

That is the minimal model of the ancient greek visual universe: sun, a man and its shadow on the beach. It is a speculation, but to me, a gnomon seems to be a visual atom.

Pythagoreans extracted from this minimal visual universe the pattern and used it for giving an explanation for the human vision, described by the next figure.

Here the sun is replaced by the eye (of a god, initially, but the pattern might apply to a mortal also), the light rays emanated by the sun are assimilated with the lines  of vision (from here the misconception that the ancient greeks really believed that the eyes shoot rays which illuminate the field of vision) and the indivisible pair man-shadow becomes the L-shape of a gnomon.  An atom of vision.

Here comes a second level of understanding the gnomon, also of pythagoreic flavor. I cite again from the wiki page:

Hero defined a gnomon as that which, added to an entity (number or shape), makes a new entity similar to the starting entity.

This justifies the Euclid’ picture of the gnomon, as a generator of self-similarity:

(image taken from the wiki page on gnomon)
So maybe the word “atom” is less appropriate than “generator”. In conclusion, according to ancient greeks, a gnomon (be it a triple sun-man-shadow or a pair eye – elementary L-shape) is the generator of the visual perception, via the mechanism of self-similarity.

In their architecture, they tried to make this obvious, readable.  Because it’s scalable (due to the relation with self-similarity), the architectural solution of constructing with gnomons  invaded the world.

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Entering “chora”, the infinitesimal place

There is a whole discussion around the key phrases “The map is not the territory” and “The map is the territory”. From the wiki entry on the map-territory relation, we learn that Korzybski‘s dictum “the map is not the territory” means that:

A) A map may have a structure similar or dissimilar to the structure of the territory,

B) A map is not the territory.

Bateson, in “Form, Substance and Difference” has a different take on this: he starts by explaining the pattern-substance dichotomy

Let us go back to the original statement for which Korzybski is most famous—the statement that the map is not the territory. This statement came out of a very wide range of philosophic thinking, going back to Greece, and wriggling through the history of European thought over the last 2000 years. In this history, there has been a sort of rough dichotomy and often deep controversy. There has been violent enmity and bloodshed. It all starts, I suppose, with the Pythagoreans versus their predecessors, and the argument took the shape of “Do you ask what it’s made of—earth, fire, water, etc.?” Or do you ask, “What is its pattern?” Pythagoras stood for inquiry into pattern rather than inquiry into substance.1 That controversy has gone through the ages, and the Pythagorean half of it has, until recently, been on the whole the submerged half.

Then he states his point of view:

We say the map is different from the territory. But what is the territory? […] What is on the paper map is a representation of what was in the retinal representation of the man who made the map–and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all.

Always the process of representation will filter it out so that the mental world is only maps of maps of maps, ad infinitum.

At this point Bateson puts a very interesting footnote:

Or we may spell the matter out and say that at every step, as a difference is transformed and propagated along its pathways, the embodiment of the difference before the step is a “territory” of which the embodiment after the step is a “map.” The map-territory relation obtains at every step.

Inspired by Bateson, I want to explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using tangle diagrams.

Let us imagine that the exploration of the territory provides us with an atlas, a collection of maps, mathematically understood as a family of two operations (an “emergent algebra”). We want to organize this spatial information in a graphical form which complies with Bateson’s footnote: map and territory have only local meaning in the graphical representation, being only the left-hand-side (and r-h-s respectively) of the “making map” relation.

Look at the following figure:

In the figure from the left, the “v” which decorates an arc, represents a point in the “territory”, that is the l-h-s of the relation, the “u” represents a “pixel in the map”, that is the r-h-s of a relation. The relation itself is represented by a crossing decorated by an epsilon, the “scale” of the map.

The opposite crossing, see figure from the right, is the inverse relation.

Imagine now a complex diagram, with lots of crossings, decorated by various
scale parameters, and segments decorated with points from a space X which
is seen both as territory (to explore) and map (of it).

In such a diagram the convention map-territory can be only local, around each crossing.

There is though a diagram which could unambiguously serve as a symbol for
“the place (near) the point x, at scale epsilon” :

In this diagram, all crossings which are not decorated have “epsilon” as a decoration, but this decoration can be unambiguously placed near the decoration “x” of the closed arc. Such a diagram will bear the name “infinitesimal place (or chora) x at scale epsilon”.