Tag Archives: Heisenberg group

More detailed argument for the nature of space buillt from artificial chemistry

There are some things which were left uncleared. For example, I have never suggested to use networks of computers as a substitute for space, with computers as nodes, etc. This is one of the ideas which are too trivial. In the GLC actors article is proposed a different thing.

First to associate to an initial partition of the graph (molecule) another graph, with nodes being the partition pieces (thus each node, called actor, holds a piece of the graph) and edges being those edges of the original whole molecule which link nodes of graphs from different partitions. This is the actors diagram.
Then to interpret the existence of an edge between two actor nodes as a replacement for a spatial statement like (these two actors are close). Then to remark that the partition can be made such that the edges from the actor diagram correspond to active edges of the original graph (an active edge is one which connects two nodes of the molecule which form a left pattern), so that a graph rewrite applied to a left pattern consisting of a pair of nodes, each in a different actor part, produces not only a change of the state of each actor (i.e. a change of the piece of the graph which is hold by each actor), but also a change of the actor diagram itself. Thus, this very simple mechanism produces by graph rewrites two effects:

  • “chemical” where two molecules (i.e. the states of two actors) enter in reaction “when they are close” and produce two other molecules (the result of the graph rewrite as seen on the two pieces hold by the actors), and
  • “spatial” where the two molecules, after chemical interaction, change their spatial relation with the neighboring molecules because the actors diagram itself has changed.

This was the proposal from the GLC actors article.

Now, the first remark is that this explanation has a global side, namely that we look at a global big molecule which is partitioned, but obviously there is no global state of the system, if we think that each actor resides in a computer and each edge of an actor diagram describes the fact that each actor knows the mail address of the other which is used as a port name. But for explanatory purposes is OK, with the condition to know well what to expect from this kind of computation: nothing more than the state of a finite number of actors, say up to 10, known in advance, a priori bound, as is usual in the philosophy of local-global which is used here.

The second remark is that this mechanism is of course only a very
simplistic version of what should be the right mechanism. And here
enter the emergent algebras, i.e. the abstract nonsense formalism with trees and nodes and graph rewrites which I have found trying to
understand sub-riemannian geometry (and noticing that it does not
apply only to sub-riemannian, but seems to be something more general, of a computational nature, but which computation, etc). The closeness,  i.e. the neighbourhood relations themselves are a global, a posteriori view, a static view of the space.

In the Quick and dirty argument for space from chemlambda I propose the following. Because chemlambda is universal, it means that for any program there is a molecule such that the reductions of this molecule simulate the execution of the program. Or, think about the chemlambda gui, and suppose even that I have as much as needed computational power. The gui has two sides, one which processes mol files and outputs mol files of reduced molecules, and the other (based on d3.js) which visualizes each step. “Visualizes” means that there is a physics simulation of the molecule graphs as particles with bonds which move in space or plane of the screen. Imagine that with enough computing power and time we can visualize things in as much detail as we need, of course according to some physics principles which are implemented in the program of visualization. Take now a molecule (i.e. a mol file) and run the program with the two sides reduction/visualization. Then, because of chemlambda universality we know that there exist another molecule which admit chemlambda reductions which simulate the reductions of the first molecule AND the running of the visualization program.

So there is no need to have a spatial side different from the chemical side!

But of course, this is an argument which shows something which can be done in principle but maybe is not feasible in practice.

That is why I propose to concentrate a bit on the pure spatial part. Let’s do a simple thought experiment: take a system with a finite no of degrees of freedom and see it’s state as a point in a space (typically a symplectic manifold) and it’s evolution described by a 1st order equation. Then discretize this correctly(w.r.t the symplectic structure)  and you get a recipe which describes the evolution of the system which has roughly the following form:

  • starting from an initial position (i.e. state), interpret each step as a computation of the new position based on a given algorithm (the equation of evolution), which is always an algebraic expression which gives the new position as a  function of the older one,
  • throw out the initial position and keep only the algorithm for passing from a position to the next,
  • use the same treatment as in chemlambda or GLC, where all the variables are eliminated, therefore renounce in this way at all reference to coordinates, points from the manifold, etc
  • remark that the algebraic expressions which are used  always consists  of affine (or projective) combinations of  points (and notice that the combinations themselves can be expressed as trees or others graphs which are made by dilation nodes, as in the emergent algebras formalism)
  • indeed, that  is because of the evolution equation differential  operators, which are always limits of conjugations of dilations,  and because of the algebraic structure of the space, which is also described as a limit of  dilations combinations (notice that I speak about the vector addition operation and it’s properties, like associativity, etc, not about the points in the space), and finally because of an a priori assumption that functions like the hamiltonian are computable themselves.

This recipe itself is alike a chemlambda molecule, but consisting not only of A, L, FI, FO, FOE but also of some (two perhaps)  dilation nodes, with moves, i.e. graph rewrites which allow to pass from a step to another. The symplectic structure itself is only a shadow of a Heisenberg group structure, i.e. of a contact structure of a circle bundle over the symplectic manifold, as geometric  prequantization proposes (but is a mathematical fact which is, in itself, independent of any interpretation or speculation). I know what is to be added (i.e. which graph rewrites which particularize this structure among all possible ones). Because it connects to sub-riemannian geometry precisely. You may want to browse the old series on Gromov-Hausdorff distances and the Heisenberg group part 0, part I, part II, part III, or to start from the other end The graphical moves of projective conical spaces (II).

Hence my proposal which consist into thinking about space properties as embodied into graph rewriting systems, inspred from the abstract nonsense of emergent algebras, combining  the pure computational side of A, L, etc with the space  computational side of dilation nodes into one whole.

In this sense space as an absolute or relative vessel does not exist more than the  Marius creature (what does exist is a twirl of atoms, some go in, some out, but is too complex to understand by my human brain) instead the fact that all beings and inanimate objects seem to agree collectively when it comes to move spatially is in reality a manifestation of the universality of this graph rewrite system.

Finally, I’ll go to the main point which is that I don’t believe that
is that simple. It may be, but it may be as well something which only
contains these ideas as a small part, the tip of the nose of a
monumental statue. What I believe is that it is possible to make the
argument  by example that it is possible that nature works like this.
I mean that chemlambda shows that there exist a formalism which can do this, albeit perhaps in a very primitive way.

The second belief I have is that regardless if nature functions like this or not, at least chemlambda is a proof of principle that it is possible that brains process spatial information in this chemical way.


Discrete or continuous, no other option? That’s a lack of imagination. (I)

The dilemma “discrete or continuous universe” is as old as philosophy. Now it is central to modern physics, a field whose practitioners don’t care much about philosophy.

As  a mathematician, hence belonging to the “learners” pythagorean school — cite from wikipedia on pythagoreanism:

According to tradition, Pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi Μαθηματικοι (“learners”) and the akousmatikoi Ακουσματικοι, (“listeners”).

— I shall strike back and accuse modern physicists of lack of imagination in tackling the discrete-continuous dilemma.

In the same time, and that is the more interesting part, I advance the following thesis:

Reality emerges from a more primitive, non-geometrical, substratum  by the same mechanism   the brain uses to construct  the image of reality, starting from intensive properties (like  a bunch of spiking signals sent by receptors in the retina), without any use of extensive (i.e. spatial or geometric)  properties.

Therefore understanding vision may give us new ideas for physics.


1. for the lack of imagination part, I argue that making an experiment (which in particular may probe the discreteness or continuity of a piece of reality) is like making a map of a territory. However, there are mathematical results which put a priori bounds on the accuracy of any map (aka Gromov-Hausdorff distance), thus making irrelevant the distinction between a discrete or a continuous territory. See this for an introduction, also see this for the particular case of the Heisenberg group.

2. for the thesis part, I shall explain why it is a reasonable speculation based on the same mathematical results.

This is based on the paper arXiv:1011.4485.

Heisenberg group, convex analysis, alike or not?

I posted on mathoverflow a question, with the purpose of clarifying the feelings I have concerning the formal resemblance between Heisenberg group and cyclically monotone operators from convex analysis. The question got an answer which made me realize something trivial (but sometimes we need somebody else to point to the obvious). However, I still think there is something worthy of further consideration here, that’s why I post it.

Setting: Let S be a semigroup (i.e. has an associative operation with neutral element e) and let (A,+) be a commutative group (with neutral element 0).

Let’s say that a semigroup extension of S with A is any operation on S \times A, of the form

(s,a)(s',a') = (s s' , a+a'+ \lambda(s,s'))

where \lambda: S \times S \rightarrow A is a function such that S \times A with the mentioned operation is a semigroup with neutral element (e,0). Obviously, the operation is encoded by the function \lambda, which has to satisfy:

\lambda(s s', s") + \lambda(s,s') = \lambda(s, s's") + \lambda(s',s")  (from associativity)

\lambda(e,s) = \lambda(s,e) = 0  (from the neutral element condition).

Here are two examples. Both are written in the setting of bipotentials, namely   X and Y are topological, locally convex, real vector spaces of dual variables x \in X and y \in Y, with the duality product \langle \cdot , \cdot \rangle : X \times Y \rightarrow \mathbb{R}.

The spaces X, Y have topologies compatible with the duality product, in the sense that for any continuous linear functional on X there is an y \in Y which puts the functional into the form x \mapsto \langle x,y\rangle (respectively any continuous linear functional on Y has the form y \mapsto \langle x,y\rangle, for an x \in X).

Example 1: (Heisenberg group) Let S = X \times Y with the operation of addition of pairs of vectors and let A = \mathbb{R} with addition. We may define a Heisenberg group over the pair (X,Y) as H(X,Y) = S \times A with the operation

(x,y,a)(x',y',b) = (x+x', y+y', a+a'+ \langle x, y'\rangle - \langle x', y \rangle)
Fact: There is no injective morphism from (X\times Y, +) to H(X,Y).

Example 2: (convex analysis) Let this time S = (X \times Y)^{*}, the free semigroup generated by X \times Y, i.e. the collection of all finite words with letters from X \times Y, together with the empty word e, with the operation of concatenation of words.

Let A be the set of bi-affine real functions on X \times Y, i.e. the collection of all functions a: X \times Y \rightarrow \mathbb{R} which are affine and continuous in each argument. A is a commutative group with the addition of real valued functions operation.

We define the function \lambda: S \times S \rightarrow A by:

\lambda(e, c)(x,y) = \lambda(c,e)(x,y)=0 for any c \in S and any (x,y) \in X \times Y.
– if c, h \in S are words c = (x_{1},y_{1})...(x_{n}, y_{n}) and h = (u_{1},v_{1})...(u_{m}, v_{m}), with m,n \geq 1, then

\lambda(c,h)(x,y) = \langle u_{1} - x , y_{n} - y \rangle for any (x,y) \in X \times Y.

This \lambda induces a semigroup extension operation on S \times A.

Fact: there is an injective morphism F: S \rightarrow S \times A, with the expression F(c) = (c, E(c)).

Here, for any c = (x_{1},y_{1})...(x_{n}, y_{n}) the expression E(c)(x,y) is the well known circular sum associated to the “dissipation during the discrete cycle” (x_{1},y_{1})...(x_{n}, y_{n})(x,y), namely:

E(c)(x,y) = \langle x_{1},y\rangle + \langle x,y_{n}\rangle - \langle x,y \rangle + \sum_{k=1}^{n-1}\langle x_{k+1}, y_{k}\rangle - \sum_{k=1}^{n} \langle x_{k},y_{k} \rangle

which appears in convex analysis, related to cyclically monotone operators.

The main difference between those two examples is that the example 2. is a direct product of a semigroup with a group. There are many resemblances though.

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 3

This post continues the previous one “Gromov-Hausdorff distances and the Heisenberg group, PART 2“.

We have seen that small enough balls in physicist’ Heisenberg group G are like balls in the mathematician’ Heisenberg group H(1) and big balls in G become more and more alike (asymptotically the same) as balls in the euclidean vector space \mathbb{R}^{2}.

What is causing this?

Could it be the choice of an euclidean norm on the generating set D = \mathbb{R}^{2} \times \left\{ 1 \right\}? I don’t think so, here is why. Let us take any (vector space) norm on \mathbb{R}^{2}, instead of an euclidean one. We may repeat all the construction and the final outcome would be: same for small balls, big balls become asymptotically alike to balls in \mathbb{R}^{2} with the chosen norm. The algebraic structure of the limits in the infinitesimally small or infinitely big is the same.

Remember that the group norm is introduced only to estimate quantitatively how the set D generates the group G, so the initial choice of the norm is a kind of a gauge.

Could it be then the algebraic structure (the group operation and choice of the generating set)? Yes, but there is much flexibility here.

Instead of G = \mathbb{R}^{2} \times S^{1} with the given group operation, we may take any contact manifold structure over the set G (technically we may take any symplectic structure over \mathbb{R}^{2} and then contactify it (with the fiber S^{1}). Sounds familiar? Yes, indeed, this is a step in the recipe of geometric quantization. (If you really want to understand what is happening, then you should go and read Souriau).

Briefly said, put a norm on the kernel of the contact form and declare all directions in this kernel as horizontal, then repeat the construction of the sub-riemannian distance and metric profiles. What you get is this: small balls become asymptotically like balls in the mathematician’ Heisenberg group, big balls are alike balls in a normed vector space.

Therefore, it is not the algebraic structure per se which creates the phenomenon, but the “infinitesimal structure”. This will be treated in a later posting, but before this let me mention an amazing phenomenon.

We are again in the group G and we want to make a map of the small (i.e. of a small enough ball in G) into the big (that is into a ball in the vector space \mathbb{R}^{2}, which is the asymptotically big model of balls from G). Our macroscopic lab is in the asymptotically big, while the phenomenon happens in the small.

A good map is a bi-lipschitz one (it respects the “gauges”, the group norm) from a ball in the vector space \mathbb{R}^{2} to a ball in the Heisenberg group H(1). Surprise: there is no such map! The reason is subtle, basically the same reason as the one which leads to the algebraic structure of the infinitesimally small or infinitely large balls.

However, there are plenty of bi-lipschitz maps from a curve in the ball from the lab (one dimensional submanifold of the symplectic \mathbb{R}^{2}, this are the lagrangian submanifolds in this case) to the small ball where the phenomenon happens. This is like: you can measure the position, or the momentum, but not both…

If there are not good bi-lipschitz maps, then there are surely quasi-isometric maps . Their accuracy is bounded by the Gromov-Hausdorff distance between big balls and small balls, as explained in this pedagogical Maps of metric spaces.

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 2

As the title shows, this post continues the previous one

Gromov-Hausdorff distances and the Heisenberg group, PART 1

The Heisenberg group G is seen from the point of view of the generating set D. Quantitatively, the group norm “measures how” D generates G. The group norm has the following properties:

  • \| X \| = 0 if and only if X = E = (0,1), the neutral element of G. In general \| X\| \geq 0 for any X \in G.
  • \| X \cdot Y \| \leq \|X\| + \|Y\|, for any X,Y \in G (that is a consequence of the fact that if we want to go from E to X \cdot Y by using horizontal increments, then we may go first from E to X, then from X to X \cdot Y, by using horizontal strings).
  • \| X^{-1} \| = \| X \| for any X \in G (consequence of X \in D implies X^{-1} \in D).

From (group) norms we obtain distances: by definition, the distance between X and Y is

d(X,Y) = \| X^{-1} \cdot Y \|

This is the sub-riemannian distance mentioned at the end of the previous post.

The definition of this distance does not say much about the properties of it. We may use a reasoning similar with the one in (finite dimensional) normed vector spaces in order to prove that any two group norms are equivalent. In our case, the result is the following:

there are strictly positive constants a, c, C such that for any
X \in G (which has the form X = (x, e^{2\pi i z})) with \| X \| \leq a we have

c ( x_{1}^{2} + x_{2}^{2} + \mid z \mid) \leq \|X\|^{2} \leq C ( x_{1}^{2} + x_{2}^{2} + \mid z \mid).

We may take a = 1/3, for example.

For “big” norms, we have another estimate, coming from the fact that the S^{1} part of the semidirect product is compact, thus bounded:

there is a strictly positive constant A such that for any X \in G (which has the form X = (x, e^{2\pi i z})) we have

\| x\| \leq \|X \| \leq \|x\| + A

Let us look now at the ball B(R) = \left\{ X \in G \mbox{ : } \|X\| \leq R \right\} endowed with the rescaled distance

d_{R} (X,Y) = \frac{1}{R} d(X,Y)

Denote by Profile(R) = [B(R), d_{R}] the isometry class (the class of metric spaces isometric with … ) of (B(R), d_{R}). This is called a “metric profile”, see Introduction to metric spaces with dilations, section 2.3, for example.

The function which associates to R > 0 the Profile(R) can be seen as a curve in the Gromov space of (isometry classes of) compact metric spaces, endowed with the Gromov-Hausdorff distance.

This curve parameterized with R roams in this huge abstract space.
I want to see what happens when R goes to zero or infinity. The interpretation is the following: when R is small (or large, respectively), how the small (or large) balls look like?

Based on the previous estimates, we can answer this question.

When R goes to infinity, the profile Profile(R) becomes the one of the unit ball in \mathbb{R}^{2} with the euclidean norm. Indeed, this is easy, because of the second estimate, which implies that for any X = (R x, e^{2 \pi i z}) and Y = (R y, e^{2 \pi i u}) which belong to B(R), (thus \|x\|, \|y\| \leq 1) we have:

d_{euclidean}(x, y) \leq d_{R}(X,Y) \leq d_{euclidean}(x, y) + \frac{A}{R}.

Therefore, as R goes to infinity, we get the isometry result.

On the other side, if R is small enough (for example smaller or equal to 1/3, then Profile(R) becomes stationary!

Indeed, let me introduce a second Heisenberg group, baptized H(1) = \mathbb{R}^{2} \times R, with the group operation

(x, z) \cdot (y, u) = (x+ y, z + u + \frac{1}{2}\omega(x,y))

Remark that the function (x, e^{2 \pi i z}) \mapsto (x,z) is a group morphism (in fact a local group isomorphism), for z small enough! That means locally the groups G and H(1) are isomorphic. If you don’t know what a local group is then see the post Notes on local groups by Terence Tao.

By exactly the same procedure, we may put a group norm on H(1).

OK, so small balls in G are isometric with small balls in H(1). What about the rescaling with \frac{1}{R}? Well, it turns out that the group H(1) is selfsimilar, moreover, is a conical group (see for example section 6 from the paper Braided spaces with dilations… and check also the references, for the notion of conical group). Conical means that the group has a one parameter family of self-similarities: for any R > 0 the function

\delta_{R} (x,z) = (R x, R^{2} z)

is an auto-morphism of H(1) and moreover:

\| \delta_{R} (x,z) \| = R \| (x,z)\| for any (x,z) \in H(1).

As a consequence, all balls in H(1) look alike (i.e. the metric profile of the group H(1) is stationary, a hallmark of null curvature…). More precisely, for any R > 0 and any X,Y \in H(1), if we denote by d the distance in H(1) induced by the group norm, we have:

d_{R}( \delta_{R} X, \delta_{R} Y) = d(X,Y).

Conclusion for this part: Small balls in G look like balls in the Heisenberg group H(1). Asymptotically, as R goes to infinity, balls of radius R in the group G look more and more alike balls in the euclidean space \mathbb{R}^{2} (notice that this space is self-similar as well, all balls are isometric, with distances properly rescaled).

Example: Gromov-Hausdorff distances and the Heisenberg group, PART 1

This post continues the previous one “Quantum physics and the Gromov-Hausdorff distance“.

Let me take an example. We are in the following Heisenberg group (this is really a physicist Heisenberg group): the semidirect product G = \mathbb{R}^{2} \times S^{1}. Elements of the group have the form

X = (x, e^{2\pi iz}) with x \in \mathbb{R}^{2} and z \in \mathbb{R} / \mathbb{Z}

(think that z \in [0,1] with the identification o = 1).
The group operation is given by:

X \cdot Y = (x,e^{2\pi i z}) \cdot (y, e^{2 \pi i u}) = (x+y, e^{2 \pi i (u+z+ \frac{1}{2} \omega(x,y))})

where \omega: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R} is the “symplectic form” (or area form)

\omega(x,y) = x_{1} y_{2} - x_{2} y_{1}

Remark 1. The pair (\mathbb{R}^{2}, \omega) is a symplectic (linear) space. As we well know, the Hamiltonian mechanics lives in such spaces, therefore we may think about (\mathbb{R}^{2}, \omega) as being a “classical”, or “large scale” (see further the justification, in PART 2) phase space of a mechanical system with one degree of freedom.

The group G is generated by the subset D = \mathbb{R}^{2} \times \left\{ 1 \right\}, more precisely any element of G can be expressed as a product of four elements of D (in a non-unique way).

What is the geometry of G, seen as generated by D? In order to understand this, we may put an euclidean norm on D (identified with \mathbb{R}^{2}):

\| (x, 1) \| = \| x\|, where \|x\|^{2} = x_{1}^{2} + x_{2}^{2} for example.

Then we define “horizontal strings” and their “length”: a string w = X_{1} ... X_{n} of elements of G is horizontal if for any two successive elements of the string, say X_{i}, X_{i+1} we have

X_{i}^{-1} \cdot X_{i+1} \in D, where X^{-1} denotes the inverse of X with respect to the group operation. Also, we have to ask that X_{1} \in D.

The length of the horizontal string w = X_{1} ... X_{n} is defined as:

l(w) = \|X_{1}\| + \| X_{1}^{-1} \cdot X_{2}\| + .... + \|X_{n-1}^{-1} \cdot X_{n}\|. The source of the string w is the neutral element s(w) = E = (0,1) and the target of the string is t(w) = X_{1}\cdot ... \cdot X_{n}.

OK, then let us define the “group norm” of an element of G, which is an extension of the norm defined on D. A formula for this would be:

\| X\| = \, inf \left\{ l(w) \mbox{ : } t(w) = X \right\}.

Small technicality: it is not clear to me if this definition is really good as it is, but we may improve it by the following procedure coming from the definition of the Hausdorff measure. Let us introduce the “finesse” of a horizontal string, given by

fin(w) = \max \left\{ \|X_{1}\| , \| X_{1}^{-1} \cdot X_{2}\| , ... , \|X_{n-1}^{-1} \cdot X_{n}\| \right\}

and then define, for any \varepsilon > 0, the quantity:

\| X\|_{\varepsilon} = \, inf \left\{ l(w) \mbox{ : } t(w) = X \mbox{ and } fin(w) < \varepsilon \right\}.

The correct definition of the group norm is then

\| X\| = \, sup \left\{\| X\|_{\varepsilon} \mbox{ : } \varepsilon > 0 \right\}.

With words, that means: for a given “scale” $\varepsilon > 0$, take discrete paths from E to X, made by “small” (norm smaller than \varepsilon) horizontal increments, and then take the infimum of the length of such curves. You get \| X\|_{\varepsilon}. Go with \varepsilon to o and get the norm \| X\|_{\varepsilon}.

Up to some normalization, the bigger is the norm of an element of G, the bigger is the infimal length of a horizontal curve which expresses it, therefore the group norm gives a quantitative estimate concerning how the group element is generated.

In disguise, this norm is nothing but a sub-riemannian distance!

Quantum physics and the Gromov-Hausdorff distance: more than discrete or continuous

In face of the question “is reality digital or analog?”, my first reaction was “how can one be so unimaginative?”. Actually, I know that there is at least another possibility, thanks to some mathematical results, like Pansu’ Rademacher theorem for Carnot groups and to the Gromov-Hausdorff distance.
I saw this question announced as the subject of a FQXI essay contest and I decided to give a try to explain a bit what I have in mind. It was interesting to do this. I think it was also a bit disconcerting, because it feels a lot like social interacting during the participation in a MMORPG.

Anyway, my contribution was this (see arXiv link also)

More than discrete or continous: a bird’s view

The idea is that any experiment is like making a map of a part of the reality (say at the quantum scale, for example) into another (the results, as we read them in the lab). Estimating distances are proposed as a good analogy of a measurement. Or, it may be possible that there are a priori limits of the precision (accuracy, etc) of the map (any map obtained by a exploiting a physical process) due to the fact that at different scales the space is different, like metric spaces are.

Up to some normalization, the Planck length may be just (proportional to) the Gromov-Hausdorff distance between the (unit ball in ) the Heisenberg group (as a Carnot group) and the euclidean phase space of the same system.

Reality (whatever that means, say the “real” phase space of a system) may be discrete, continuous, both or none, but at different scales, up to a (small) error, the Heisenberg group or the euclidean space, respectively, may be close to it (to the small part of reality explored). Then, as a consequence of Pansu’ Rademacher theorem, it follows that one cannot make a perfect (read “bi-lipschitz”) map of a part of reality into the other, but instead, there is a lower limit of the accuracy of any such map, quantitatively proportional with a Gromov-Hausdorff distance.

In order to explain this in more detail, I took the view that anyway the differential calculus (which is the mathematical base of physics) is an abstraction of the activity of map-making, or chorography.

The “bird’s view” is a play with a quote from Plato, I let you discover it, at the end of the paper.