Global communication patterns in a hyperbolic space of negative curvature

Engaging with Hyperreality through Demonique and Angelique? (Part #8)

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Section uncompleted: work in process !

Comprehension and communication: Following the reference above to the inspiration offered by Escher drawings, the argument may be further developed through mathematical consideration of the Poincaré disk and the tessellation of the hyperbolic plane with which it is so closely associated.

The focus in this argument is on communication and comprehension as they may be more appropriately and richly envisaged using hyperbolic tessellation and the Poincaré disk model. Contrasting images can therefore be usefully represented to suggest the degree of connectivity between participants (or perspectives) at any table around which they are gathered (as discussed below).

The question to be explored is the degree of complexity which can be individually or collectively comprehended and communicated. This relates to the concern in the main paper with the number of factors to be understood to avoid the failure of a complex viable system. How to engage with a greater number of factors when human cognitive and communication capacity is constrained to seven (plus or minus two)?

Expressed otherwise, the issue is the number of distinctions required to ensure sustainability -- and how they might be organized for optimal memorability. The limited number conventionally presented in relation to global governance at this time can be fruitfully called into question by the larger sets typical of traditional articulations in various cultures. Given the present dynamics between perceived problems and advocated solutions, traditional articulations of sets of 72 demons and sets of 72 angels suggest that a number of that order may enable intuitive insight into comprehensible patterns of greater complexity. Hence the framing here of a Demonique and an Angelique as potentially offering more widespread comprehensibility and engagement.

Mathematical intuition: For Szivos Eszter (On the boundaries of mathematics and psychology. Poincaré's notion of intuition, Magyar Pszichológiai Szemle, 19 November 2013):

"Intuition" is a central term in Henri Poincaré's writings on the philosophy of mathematics. Poincaré -- as opposed to the logicists (above all Russell) -- states that mathematics cannot be reduced to logic: without its intuitive component, mathematics would become tautological and sterile. But what exactly does the Poincaréan term of intuition mean and what sort of role does it play in mathematics and in mathematical history, study, and invention?In the present analysis -- due to the complexity of the concept "intuition". I shall review the different meanings of the expression and the linkages between these meanings. Furthermore, I shall point out that Poincaré's view on mathematical invention and intuition very much resembles that of several psychologists of his age and that Myers' theory of the unconscious had a strong influence on Poincaré's reasoning. Lastly, I shall examine the question of how closely Poincaré's theory is linked to psychology. Can his theses function within the philosophy of mathematics, or are they committed -- from a philosophical point of view -- to psychologism? I shall argue that in Poincaré's writings mathematical and psychological issues are not to be confused: some important and relevant arguments in the philosophy (rather than the psychology) of mathematics are grounded in the Poincaréan concept of intuition.

Tessellation: There are many commentaries on hyperbolic tessellation. That of Dmitry Brant (Hyperbolic Tessellations 2007), with helpful images, includes the comment:

A tessellation refers to a uniform tiling of a plane with polygons, such that an equal number of identical polygons meet at each vertex. For example, the tiles in a bathroom, the squares of linoleum on an office floor, or the honeycomb pattern in a bees' nest are all tessellations of the Euclidean plane.

However, tessellations are also possible on non-Euclidean spaces, such as the elliptic plane (like the stitching pattern on a soccer ball), and the hyperbolic plane (like... nothing you'd find around the house). In fact, the Euclidean plane has only three regular tessellations (with squares, hexagons, and triangles), while the hyperbolic plane can be tessellated in infinitely many ways.

Since we do not exist in hyperbolic space, we cannot truly "see" hyperbolic tessellations. We can only "represent" them in Euclidean form. A common way of doing this is on the Poincaré disk, which is a finite circle that represents the boundary of the (infinite) hyperbolic plane that is contained inside. The image on the right is a hyperbolic tessellation drawn on the Poincaré disk.

Since tessellations of the hyperbolic plane are especially interesting and mesmerizing to look at, I wrote a small program that generates them, with a great deal of configurable options.

For A. A. Goodenough and A. Schlamm (Interactive visualization of hyperspectral images on a hyperbolic disk. Connecting Minds for Global Solutions, 2011):

Visualization of the high-dimensional data set that makes up hyperspectral images necessitates a dimensionality reduction approach to make that data useful to a human analyst. The expression of spectral data as color images, individual pixel spectra plots, principal component images, and 2D/3D scatter plots of a subset of the data are a few examples of common techniques. However, these approaches leave the user with little ability to intuit knowledge of the full N-dimensional spectral data space or to directly or easily interact with that data. In this work, we look at developing an interactive, intuitive visualization and analysis tool based on using a Poincaré disk as a window into that high dimensional space. The Poincaré disk represents an infinite, two-dimensional hyperbolic space such that distances and areas increase exponentially as you move farther from the center of the disk. By projecting N-dimensional data into this space using a non-linear, yet relative distance metric preserving projection (such as the Sammon projection), we can simultaneously view the entire data set while maintaining natural clustering and spacing. The disk also provides a means to interact with the data; the user is presented with a "fish-eye" view of the space which can be navigated and manipulated with a mouse to "zoom" into clusters of data and to select spectral data points. By coupling this interaction with a synchronous view of the data as a spatial RGB image and the ability to examine individual pixel spectra, the user has full control over the data set for classification, analysis, and instructive use.

For Vladimir Bulatov (Tilings of the hyperbolic space and their visualization Corvallis, Oregon, USA Joint MAA/AMS meeting, New Orleans, January 7, 2011):

Visual representation of tiling of 3D hyperbolic space attracted very little attention compare to tilings of hyperbolic plane, which were popularized by M.C.Escher circle limit woodcuts. Although there is a lot of activity on theoretical side of the problem starting from work of H.Poincaré on Kleinian groups and continuing with breakthrough of W.Thurston in the development of low dimensional topology and G.Perelman's proof of Poincaré conjecture. The book "Indra's Pearl" have popularized visualization of 2D limit set of Kleinian groups, which is located at the infinity of hyperbolic space. In this talk we present our attempts to build and visualize actual 3D tilings. We study tilings with symmetry group generated by reflections in the faces of Coxeter polyhedron, which also is the fundamental polyhedron of the group.

Poincaré disk: The hyperbolic plane cannot be metrically represented in the flat Euclidean plane. Poincaré described ways that it can however be conformally represented in the Euclidean plane -- known as the Poincaré disk (see also Poincaré Hyperbolic Disk, Wolfram MathWorld) . It is one of the most common models used to visualize hyperbolic geometry (and to print it in 2D).

The disk is a 2-dimensional model of n-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk and the straight lines consist of all segments of circles contained within that disk that are perpendicular to the boundary of the disk, plus all diameters of the disk. A straight line in the hyperbolic plane is thus represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles.

The Poincaré disc maps the point at infinity of a hyperbolic space to a circle where hyperbolic lines are represented as arcs of circles intersecting the éé disc at 90 degrees. As we move away from the origin of a hyperbolic space, the space itself expands due to negative curvature, so as we reach the perimeter of the Poincaré disc, the scale of the space changes dramatically, subdividing into an infinite number of pieces.

A Poincaré disk is a circle that represents an infinite region of space. As the circular edge is approached, the images diminish at such a rate that they appear to be infinitely small and be infinitely close to the circle's edge without ever touching it. By using this Poincaré disk model, it is possible to give the impression of an infinite array of tile images within a limited space and, unlike other disk models, the shape of the tiles stays recognizable as they approach the circular boundary.

As succinctly summarized by Wikipedia with regard to the Poincaré disk model: In plain English, this means you can squash an infinite 2D plane into a circular disc! (also see images on Wikipedia at Media in category "Poincaré disk models").

Other resources relating to Poincaré disk:

• Jeff Weeks. Kaleido Tile
• Tiling the Hyperbolic Plane with Regular Polygons Wolfram Demonstrations Project, 2013
• Eric W. Weisstein Poincaré Hyperbolic Disk Wolfram Demonstrations Project, 2013
• Borut Levart Triangles in the Poincaré Disk
• Teichmuller Navigator 2.0
• The Poincaré Disk Model of Hyperbolic Geometry
• John D. Norton Philosophical Significance of the General Theory of Relativity or What does it all mean, again? Geometric Morals and a few more 2013
• Eugenio Urdapilleta, et al: Can rodents conceive hyperbolic spaces? 2015
• Maryellen Mott Allen The Hype Over Hyperbolic Browsers InfoToday,
• Negative Curvature Mirror Applet -- hyperbolic triangles and transformations using the Poincaré and Klein models.
• The Poincaré Disk - A Western Mandala / quantum mandala
• Wolfram --- The illustration above shows a hyperbolic tessellation similar to M.C.Escher's Circle Limit IV (Heaven and Hell) (Trott 1999, pp.10 and 83).

A valuable overview is offered by David E. Joyce (Hyperbolic Tessellations: some printable tilings, 2002) through a page providing access to a range of images -- and enabling users to display their own -- vertices etc

Animation of set of regular tilings (above)	Animation of set of quasiregular tilings (above)

W. Goldman, Ultraideal Triangles, 2004

This graphics package draws objects in the hyperbolic plane using the Poincaré model. The pictures are drawn in the unit disc, but the calculations occur in the upper half plane, where the isometry group is PSL(2,R).

Ultraideal Triangles, 2004	Escher

Poincaré Disk adapted from Wolfram Demonstration Project)
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