# Gnomonic cubes: a simultaneous view of the extended graphic beta move

Recall that the extended  beta move is equivalent with the pair  of  moves :

where the first move is the graphic beta move and the second move is the dual of the beta move, where duality is (still loosely) defined by the following diagram:

In this post I want to show you that it is possible to view simultaneously these two moves. For  that I need to introduce the gnomonic cube. (Gnomons appeared several times in this blog, in expected or unexpected places, consult the dedicated tag “gnomon“).

From the wiki page about the gnomon,     we see that

A three dimensional gnomon is commonly used in CAD and computer graphics as an aid to positioning objects in the virtual world. By convention, the X axis direction is colored red, the Y axis green and the Z axis blue.

(image taken from the mentioned wiki page, then converted to jpg)

A gnomonic cube is then just a cube with colored faces. I shall color the faces of the gnomonic cube with symbols of the gates from graphic lambda calculus! Here is the construction:

So, to each gate is associated a color, for drawing conveniences. In the upper part of the picture is described how the faces of the cube are decorated. (Notice the double appearance of the $\Upsilon$ gate, the one used as a FAN-OUT.)  In the lower part of the picture are given 4 different views of the gnomonic cube. Each face of the cube is associated with a color. Each color is associated with a gate.

Here comes the simultaneous view of the pair of moves which form, together, the extended beta move.

In this picture is described a kind of a 3D move, namely the pair of gnomonic cubes connected with the blue lines can be replaced by the pair of red lines, and conversely.

If you project to the UP face of the dotted big cube then you get the graphic beta move. The UP view is the viewpoint from lambda calculus (metaphorically speaking).

If you project to the RIGHT face then you get the dual of the graphic beta move. The RIGHT view is  the viewpoint from emergent algebras (…).

Instead of 4 gates (or 5 if we count $\varepsilon^{-1}$ as different than $\varepsilon$), there is only one: the gnomonic cube. Nice!

# 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.

# Right angles everywhere (I)

Look at almost any building in the contemporary city, it’s constructed from right angles, assembled into rectangles, assembled into boxes. We expect, in fact,  a room to have a rectangular floor, with vertical walls meeting in right angles. Exceptions are either due to architectural fancies or to historical constraints or mistakes.

When a kid draws a house, it looks like a rectangle, with the  triangle of the roof on top.

Is this normal? Where does this obsession of the right angle comes from?

The answer is that behind any right angle is hidden a gnomon. We build like this because we  are Pythagoras children, living by the rules and categories of our cultural ancestors, the ancient greeks.

Let’s see:
(I) In ancient times,  or in  places far from the greeks  (and babylonians), other architectural forms are preferred, like the  roundhouse. Here’s a Scottish broch (image taken from this wiki page)

and here’s a Buddhist stupa (image taken from the wiki page)

Another ancient building form is the step pyramid , like the Great Ziggurat of Ur (image taken from the last wiki page)

or the egyptian pyramids, or any other famous  pyramid in the world (there are plenty of them, in very different cultural frames).

Here is a Sardinian Nuraghe

Conclusion: round, conical, pyramidal is the rule, there are no right angles there!

Until the greeks: here’s the Parthenon

It is made of gnomons, here’s one (from the wiki page)

# The gnomon in the greek theater of vision, I

In the post Theatron as an eye I proposed the Greek Theater, or Theatron (as opposed to the “theater in a box”, or Cartesian Theater, see further) as a good model for   vision.

Any model of vision should avoid the homunculus fallacy. What looks less understood is that any good model of vision should avoid the scenic space fallacy. The Cartesian Theater argument against the existence of the homunculus is not, by construction, an argument against the scenic space. Or, in the Cartesian Theater, homunculus and scenic space come to existence in a pair. As a conclusion, it seems that there could not be a model of vision which avoids the homunculus but is not avoiding the scenic space. This observation is confirmed by facts: there is no good, rigorous  model of vision up to date, because all proposed models rely on the a priori existence of a scenic space. There is, on the contrary, a great quantity of experimental data and theoretical partial models which show just how complex the problem of vision is. But, essentially, from a mathematician viewpoint, it is not known how to even formulate the problem of vision.

In the influent paper “The brain a geometry engine”  J. Koenderink proposes that (at least a part of) the visual mechanism is doing a kind of massively parallel computation, by using an embodiment of the geometry of jet spaces (the euclidean infinitesimal geometry of a smooth manifold)  of the scenic space. Jean Petitot continues along this idea, by proposing a neurogeometry of vision based essentially on the sub-riemannian geometry of those jet spaces. This an active mathematical area of research, see for example “Antropomorphic image reconstruction via hypoelliptic diffusion“, by Ugo Boscain et al.

Sub-riemannian geometry is one of my favorite mathematical subjects, because it  is just a  particular model of a metric space with dilations.  Such spaces are somehow fundamental for the problem of vision, I think. Why? because there is behind them a purely relational formalism, called “emergent algebra“, which allow to understand “understanding space” in a purely relational way. Thus I hope emergent algebras could be used in order to formulate the problem of vision as the problem of computing with space, which in turn could be used for getting a good model of vision.

To my surprise, some time ago I have found that this  very complex subject has a respectable age, starting with Pythagora  and Plato!  This is how I arrived to write this blog, as an effort to disseminate what I progressively understand.

This brings me back to the theater and, finally, to gnomon. I cite from previous wiki link:

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

In the greek theater, a gnomon sits in the center of the orchestra (which is the circular place where things happen in the greek thater, later replaced by the scene in the theater in a box). Why?

# Theatron as an eye

I want to understand what “computing with space” might be. By making  a parallel with the usual computation, there are three ingredients which need to be identified: what are the computing with space equivalents of

1. the universal computing gate (in usual computing this is the transistor)

2. the universal machine (in usual computing this is the Turing machine)

3. what is the universal machine doing by using its arrangement of universal computing gates (in usual computing this is the algorithm).

I think that (3) is (an abstraction of) the activity of map making, or space exploration. The result of this activity is coded by a dilation structure, but I have no idea HOW such a result is achieved. Once obtained though, a mathematical model of the space is the consequence of  a priori assumptions (that we can repeat in principle indefinitely the map making operations) which lead to the emergent algebraic and differential structure of the space.

The universal gate (1), I think, is the dilation gate, or the map-territory relation.

Today I want to pave the way to the discovery of the universal machine (2). This is related to my previous posts The Cartesian Theater: philosophy of mind versus aerography and Towards aerography, or how space is shaped to comply with the perceptions of the homunculus.

My take is that the Greek Theater, or Theatron (as opposed to the “theater in a box”, or Cartesian Theater) is a good model for an universal machine.

For today, I just want to point to the similarities between the theatron and the eye.

The following picture represents the main parts of the theatron (the ancient greek meaning of “theatron” is “place of seeing). In black are written the names of the theatron parts and in red you see the names of the corresponding parts of the eye, according to the proposed similarity.

Let me proceed with the meaning of these words:

– Analemmata means the pedestal of a sundial (related with analemma and analemmatic sundial; basically a theatron is an analemmatic sundial, with the chorus as the gnomon). I suggest to parallel this with the choroid of the eye.

– Diazomata (diazoma means “belt”), proposed to be similar with the retina.

Prohedria (front seating) is a privilege to sit in the first few rows at the bottom of the viewing area. Similar with the fovea (small pit), responsible for sharp central vision.

Skene (tent), the stage building, meant to HIDE the workings  of the actors which are not part of the show, as well as the masks and other materials. When a character dies, it happens behind the skene. Eventually, the skene killed the chorus and  became the stage. The eye equivalent  of this is the iris.

Parodos (para – besides, counter, and ode – song) entrance of the chorus. Eye equivalent is the crystalline lens.

– Orchestra, the ancient greek stage, is the place where the chorus acts, the center of the greek theater. Here we pass to abstraction: the eye correspondent is the visual field.