# 3D crossings in emergent algebras

This post continues the previous one  3D crossings in graphic lambda calculus . Other relevant posts are:

For graphic lambda calculus see this, for knot diagrams and emergent algebras see this, sections 3-6.

In the previous post we saw that we can “construct” crossings   by using both the $\lambda$ abstraction operation and the composition operation from lambda calculus. These operations appear as elementary gates in graphic lambda calculus, along with other two gates, namely the FAN-OUT gate denoted by $Y$ and the $\bar{\varepsilon}$ gate (with $\varepsilon$ an element in a commutative group $\Gamma$). This last gate models the family of operations of an emergent algebra.

The FAN-OUT gate $Y$ is used in two different contexts. The first one is   as a properly defined FAN-OUT, with behaviour described by the  global fan-out move, needed in the construction which attaches to any term in untyped lambda calculus a graph in the lambda calculus sector of the graphic lambda calculus as explained in “Local and global moves on locally planar trivalent graphs … ” step 3 in section 3. The second one is in relation with the decorated knot formalism of emergent algebras, “Computing with space, …” section 3.1″.

There is an astounding similarity between the $\lambda$ and composition gates from lambda calculus, on one side, and FAN-OUT and $\bar{\varepsilon}$ gates from emergent algebras, in the context of defining crossings. I shall explain this further.

In the decorated knots formalism, crossings of oriented wires are decorated with elements $\varepsilon$ of a commutative group $\Gamma$. The relation between these crossings and their representations in terms of trivalent graphs is as following:  Comparing with the notations from the previous post, we see that in both cases the $\lambda$ gate corresponds to a FAN-OUT, but, depending on the type of crossing, the composition operation gate corresponds to one of the gates decorated by $\varepsilon$ OR $\varepsilon^{-1}$.

There is a second remark to be made, namely that the crossings constructed from FAN-OUT and $\bar{\varepsilon}$ gates satisfy Reidemeister I and II moves but not Reidemeister III move. This is not a bad feature of the construction, in fact is the most profound feature, because it leads to the “chora” construction and introduction of composite gates which “in the infinitesimal limit”, satisfy also Reidemeister III, see section 6 from “Computing with space”.

In contradistinction, the crossings constructed from the $\lambda$ abstraction and composition operation do satisfy the three Reidemeister moves.

# 3D crossings in graphic lambda calculus

Related:

Let us look again at the NOTATIONS I made in the post (A) for crossings in graphic lambda calculus:  When seen in 3D, both are the same. Indeed, the 3D picture is the following one: How to imagine the graphic beta move? Start with two wire segments in 3D, marked like this: Glue the small blue arrow (which is just a mark on the wire) which goes downwards away from the blue wire with the small red arrow which goes downwards to the red wire: That’s the graphic beta move, in one direction. For the opposite direction just rewind the film.

There is a slight  resemblance with the figures from the post (B), concerning slide equivalence, consisting in the fact that here and there we see crossings decomposed (or assembled from) two types of gates, namely one with one entry, two exits, the other with two entries, one exit.

Notice also that in graphic lambda calculus we have another two gates, namely the FAN-OUT gate and the TOP gate. We shall see how they couple together, next time.

# Slide equivalence of knots and lambda calculus (I)

Related: (Graphic) beta rule as braiding.

Louis Kauffman proposes in his book Knots and Physics  (part II, section “Slide equivalence”), the notion of slide equivalence. In his paper “Knotlogic” he uses slide equivalence (in section 4) in relation to the self-replication phenomenon in lambda calculus. In the same paper he is proposing to use knot diagrams as a notation for the elements and operation of a combinatory algebra (equivalent with untyped lambda calculus).

There is though, as far as I understand, no fully rigorous relation between knot diagrams, with or without slide equivalence, and untyped lambda calculus.
Further I shall reproduce the laws of slide equivalence (for oriented diagrams), following Kauffman’ version from Knots and physics. Later, I shall discuss about the relations with my graphic lambda calculus.

Here are the laws, with the understanding that:

1. the unoriented lines may have any orientation,

2. For any version of orientation which is depicted, one may globally change, in a coherent way, the orientation in order to obtain a valid law.

Law (I’) is this one: Law (II’) is this: Law (III’): Law (IV’): Obviously, we have four gates, like in the lambda calculus sector of the graphic lambda calculus. Is this a coincidence?

UPDATE (06.06.2014): The article Zipper logic  arXiv:1405.6095  answers somehow to this, see the post Halfcross way to pattern recognition (in zipperlogic).A figure from there is: See also the posts Curious crossings (I) and Distributivity move as a transposition (Curious crossings II).

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# 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) # Mass connected processing?

In this  Unlimited Detail technology description  appears the term “mass connected processing”. Looking on the net for this one finds this post,  I cite from it:

“By the looks of what they are saying, the areas and level of real time software performance they are talking about, it is likely to be the same methods that I came up with back around 1997 (when I was also in Brisbane), or not far off of it, as the problem reduces down to single 100% efficient methods. ”

Anybody knows what’s this all about?

# Uniform spaces, coarse spaces, dilation spaces (II)

Background:
(1) Uniform spaces, coarse spaces, dilation spaces

I shall use the idea explained in (1), in the groupoid frame of (2), classical situation of the trivial groupoid over a space $X$. In this case the uniform and coarse structures are just the classical notions.

This idea says that all we need to have is a field of dilations. With such an object we may construct an uniformity, or a coarse structure, then ask that the field of dilations has some properties with respect to the said uniformity (or coarse structure). If the field of dilations has those properties then it transforms into an emergent algebra  (in the case $0$ below; in the other case  there is a new type of emergent algebra which appears in relation to coarse structures).

Remark. Measure can be constructed from a field of dilations, that’s for later.

Fields of dilations. We have a set $X$, call it “space”. We have the commutative group $(0,\infty)$ with multiplication of reals operation, (other groups work well, let’s concentrate on this one).

A field of dilations is a function which associates to any $x \in X$ and any $\varepsilon \in (0,\infty)$
an invertible transformation $\delta^{x}_{\varepsilon} : U(x) \subset X \rightarrow U(x,\varepsilon) \subset X$ which we call “the dilation based at $x$, of coefficient $\varepsilon$“.

1. $x \in U(x)$ for any point $x \in X$

2. for any fixed $x \in X$ the function $\varepsilon \in (0,\infty) \rightarrow \delta^{x}_{\varepsilon}$ is a representation of the commutative group $(0,\infty)$.

3. fields of dilations come into 2 flavors (are there more?), depending on the choice between $(0,1]$ and $[1,\infty)$, two important sub-semigroups of $(0,\infty)$.

(case $0$) – If you choose $(0,1]$ then we ask that for any $\varepsilon \in (0,1]$ we have $U(x,\varepsilon) \subset U(x)$, for any $x$,

This case is good for generating uniformities and  for the infinitesimal point of view.

(case $\infty$) – If your choice is $[1,\infty)$ then we ask that for any $\varepsilon \in [1,\infty)$ we have $U(x) \subset U(x,\varepsilon)$, for any $x$,

This case is good for generating coarse structures and for  the asymptotic point of view.

Starting from here, I’m afraid that my latex capabilities on wordpress are below what I need to continue.

Follow this working paper to see more. Thanks for comments!

PS. At some point, at least for the case of uniformities, I shall use “uniform refinements” and what I call “topological derivative” from arXiv:0911.4619, which can be applied for giving alternate proofs for rigidity results, without using Pansu’s Rademacher theorem in Carnot groups.