Summer Happenings: An NYAS Talk and a New Publication on Quaternions and Chirality

All my quaternion research diggings and analysis, including music, cognition, and 4D, have led to some nice conclusions, articles, and presentations. This is how I described it on my website:

“Herb has a new publication, written with two collaborators: Quaternions, Chirality, Social Exchange: A Tool for Neuroscience? It appeared in October 2015 as the Feature Article in the Newsletter of the Society for Chaos Theory in Psychology and Life Sciences, Vol. 23, No. 1, p. 8-14. See Quaternions, Chirality, and Social Exchange (V20 – official newsletter copy).”

“He also gave a successful 2-hour talk about Quaternions for the New York Academy of Sciences/Lyceum Society in June 2015. See menu for THE PRESENT to access the Powerpoint slides (about 65) for the talk. The title is: The Culture of Quaternions–the Phoenix Bird of Math.” See Quaternions – Phoenix Bird presentation, v23.”

“Herb met one of his publication collaborators, Martin Hay in England, when the New York Academy of Sciences talk was posted on the ResearchGate repository site and Martin noticed it and contacted Herb.”

Martin is very bright and creative. He visited with me for two days during a week’s visit to the U.S. This gave us ample time to discuss a number of topics, including music at an evening get-together with music cognition neuroscientist Marina Korsakova-Kreyn.

The third collaborator for the article is also very bright and creative — Terry Marks-Tarlow in Santa Monica, CA. She is a psychoanalyst and researcher who is friendly to the Jung tradition. And quaternions have captured her attention and imagination. She has related quaternions to general cognition and to fractals and border processes involved in development. We have become a powerful team, in terms of idea development.

My next focus is sociable robotics and the arc back to Ada Lovelace and aesthetics….and the possible role of quaternions.


Quaternions and the World of Carl Jung and His Followers

Three months ago I began to hear about, and then read about, and then correspond with people about the world views of Carl Jung and its resonance with quaternions. I even joined the Kristine Mann Library of very  extensive Jungian materials at the Carl Jung Center on E. 38th St. in New York City.

I first probed the index books for the word quaternion — and got a false hit, where one  particular German translation should have used quaternio (of a set of 4 things). A related word was quaternity, meaning a square of four entities facing and relating to each other, or a division into four principles or elements, similar to the word duality for two related   elements. A quaternity was at a somewhat higher level of generality than the mathematical system of quaternions, and simply meant relationships of four-fold principles or aspects of reality. Also, one element of the four principles is usually seen as dominating the other three.

I also went through the Jung-Pauli correspondence (1932-1958 in book form) and did find a discussion by Jung of the space-time continuum quaternity, looking at it in two interesting ways — one, the traditional three dimensions of space and one of time, the other a nontraditional one dimension of fixed space and three of time — past, present, and future in parallel.

I concluded that Jung had probably seen examples of applied quaternions from Pauli, Piaget, or other researchers with whom he was in contact, but did not try to develop conceptual applied systems in cognition and personality with them. But some of his later generations of followers did, particularly the analyst/researcher, Terry Marks-Tarlow. We correspond, and learn from each other.

Terry is in Santa Monica, CA and, in her therapy, specializes in helping artists, writers, and other creative and professional people to become unblocked in their work.

Terry uses quaternions to represent 4D symbolic and integrative thinking. She also use mathematical fractal theory (interactional processes occurring across borders and between dimensions) to model play and development.

In the Jungian framework, inexact borders and paradoxes are rich opportunities for growth experiences in the interpersonal domain. Also, in line with fractal math, the Jungian world focuses on repeated patterns and looks to understand the mental archetype driving it within the person’s individuality.

In fractals, patterns are found repeated at different levels of depth and detail.

One of the striking things about quaternions is that they have been use to model processes at every level of nature, from quantum mechanics to DNA strands, from molecular docking to (animal and human) bio-logging, and from cognitive psychology to celestial mechanics.

My conjecture is that quaternion systems are reflective of an operational toolbox of nature, recurring at every level in some form, with a linkage or at least resonance between the levels.

This is also consistent with Mike Mair’s construct and theory of The Universal Template in the field of social anthropology, first developed in the 1970s under a grant arranged by Mary Douglas, who wrote Purity and Danger. Mike is hoping to resume work soon on his construct and its expression.


Marks-Tarlow, T. (2004). Semiotic Seams: Fractal Dynamics of Reentry. Cybernetics & Human Knowing, 11, 1

Fractal Self at Play, AJP

Marks-Tarlow, T. (2010). Fractal Self at Play. American Journal of Play, Summer 2010

Jung-Pauli Correspondence — letters 14J, 21P, on dimensionality interpretation of space-time.

Atom and Archetype:
The Pauli/Jung Letters, 1932-1958
Edited by C. A. Meier, 2001, Princeton University Press

Mair, M. W. (original paper 1981, plus revisiting later). The melody of the text – revisited (after 2002). Semioticon website

“Even though the speech trajectories capture virtual world models rather than actual objects on four-dimensional trajectories (like a prey animal moving in the environment), I suggest that the trajectory of speech with movement is non-verbal, the product of the core brain forming the core to the speech act. The ‘point’ is the point. A growth point is defined as the ‘initial form of thinking out of which speech-gesture organization emerges’. (McNeill) It might also be called the ‘projection point’.

The core brain mechanisms underlying human natural story telling can now be glimpsed. Damasio’s core brain text generator in action describes the nonverbal internal structure of gesturing behaviour in speech with movement. It may have functioned projectively on 4D-space time for probably billions of years. Additional control of outcomes is achieved by adding more dimensions or variables to the modeling process, up to our present limit of 7+/-2.

I think that the ‘material’ of speech with movement, via the mechanism of mirror neurons and the shape of the trajectory itself, makes a ‘synctium’ of the communicating minds. They directly collaborate to control the outcomes of the topics in which they are immersed. The core brains are entrained, and the story is shared. However the logic is ‘projective’ on 4D space-time, and I have termed our everyday ‘logic in use’ as ‘visual projective logic’.”

The Neuroscience of Sensing Space

In our last blog, Rivers of Music Flow into the Mind, one of the contributing rivers we listed was a sense of space, and our movement within it.

We have no direct sense of space. There are no space sensors in our body — only indirect information from touch, sight, hearing.

The brain conducts a theater inside us (in the parietal lobe of the brain) which Arnold Trehub calls the retinoid space (The Theater of Consciousness), with ourselves in the center, in which 3D objects (and theoretically also time-based 4D objects) are constructed as models around us and then are “painted” with various multisensory attributes – sound, touch, taste – and displayed (become part of our consciousness).

  • Part of this display preparation results from mathematical transformations of perceptual objects to compensate for viewing point (for example, turning ellipses caused by perspective into circles) or to perform imaginary mental changes such as mental rotation.
  • The other part results from integration of the flow of multisensory and body information entering the theater through the thalamus structure, which is like a network of intelligent cables connecting all parts of the brain.

Other researchers with similar or overlapping views are Ravi Jerath & Molly Crawford, Marina Korsakova-Kreyn, Daniel Wolpert, and Daniela Dentico.

  • Jerath & Crawford emphasize the role of the thalamus in creating consciousness and the images for display in the “3D theater.”
  • Korsakova-Kreyn emphasizes the importance of supramodal (beyond individual senses) general processes for shaping our perceptions and mentally created objects, as in rotation transformations, which seem to treat musical and visual objects in a similar way.
  • Wolpert says that brains are made for walking, so to speak, that is, motion control, not for purely perceiving or feeling. Animals have brains because they are mobile and need them. They need to know how to negotiate an environment, not just live in it. Plants do not need to do this. So he studies sensorimotor integration and finds this function is active in the inferior parietal lobe. Control of motion — and control of frame of reference — is vital in negotiating an environment, and that is what quaternions excel in computing. So I suspect that life forms learned how to tap the principles of quaternion relationships and operations in space through some handy analog process or mechanism involving organic structures, including molecules, that relate action to perceived and imagined movement — the logic of animal life.
  • Dentico and associates demonstrate that electric flow of patterns from the occipital (initial vision-gathering) lobe during perception of real objects reverses direction during creation of imagined images, going from the prefrontal (planning) lobe to the parietal lobe (presumably for construction and awareness) and then to the occipital lobe (the reason is not clear).

One pattern that seems to emerge is that simple display function activity is correlated with the inferior region (IPL) of the parietal lobe, whereas motion-related activity, such as mental rotation or sensorimotor integration) is correlated with the superior region (SPL).

For references to a discussion of the work of these researchers, and a diagram of how they connect, see the following PDF file link to a 12-slide excerpt, Arnold Trehub and Related Researchers, from my presentation on a draft of The Culture of Quaternions.

Rivers Flowing Into Music in Our Mind

Music is an unexpected alliance of “rivers” in our head and body of very ancient and very modern forces that reflect the way we operate inside and the social/cultural milieu we swim in and share.

Music reflects motion and travel, and obstacles giving us resistance, and the sight of objects promising us rewards.

Music reflects a sense of the effort it takes to do these things — where to turn, when to wait, how hard to work to scale a steep terrain, how to proceed carefully. The underlying meaning of music comes from how to do these things, not how to describe a scene but how to react emotionally to interaction with a scene. This traces back to our animal nature, but the process is superimposed on our experience of aesthetic emotion, a product of our civilization and humanity, and our spatial sense of traversal and transformation.

So we have at least several forces at work in our experience of music:

  • Animal emotion and its tie to effort and life-preserving traverse. (Kevin Behan basing his ideas on animal emotion researcher, Jaak Panksepp).
  • Experience of inner tension and release as projected by a spectrum of harmonic relationships in a melody. (Carol Krumhansl and others)
  • Aesthetic emotion, a refinement and enjoyment of complex and transforming relationships between artistic elements. (Virtually spatial; rhythmic; textural)

Melody, I and others believe, is experiencing the effects of such a path.

Path Mathematics:  Mathematically, the trace of  line intervals resulting from moving and traversing along from significant point to significant point, is called a graph. The theory of constructing these and analyzing them is called graph theory. The points are called nodes and the connecting intervals are called edges. The theory is  applied to choices of paths, often along the edges of a solid 3D figure such as a tetrahedron (4 equal sides) or more complex shapes, such as a dodecahedron (12 sides).

Somewhere within our cognitive apparatus, I believe, may be a capacity to carry on such activities — to construct, store, remember, and transform such graph paths. Our “river” components catalyze it — travel, effort, animal emotion, and aesthetic emotion.

Gilles Baroin, in his dissertation at the University of Toulouse (2011), shows how to construct such graph paths to represent melodies, using quaternion objects and computer representation to create a picture of visual turning and observation of a melody graph. It does not directly incorporate perceptual experimentation involving music (psychophysics), but does show how a path system can be built around quaternion objects and relationships.

For several quotes and references from researchers, go to page Rivers and Paths

Welcome to the Culture of Quaternions – Past, Present, Future

By Herb Klitzner
Copyright 2015

Can a brilliant but controversial math system virtually sink from sight? Isn’t math permanent, fixed, self-evident? No, it’s not.

Quaternion math is such a system — celebrated around the 1870s and 1880s at some of the finest universities such as Harvard — this system was almost forgotten in the 20th Century after the theory of vector spaces was extracted and fashioned from parts of quaternion elements and thinking, but was then re-encountered in 1985-1995 because it solved many new problems occurring in the fields of aerospace, computer graphics, and signal/image processing, especially those involving the processes of rotation, orientation-determining, and filtering.

Ask three scientists at random if they have ever heard of quaternions — the chances are you will get “no” as the answer from all three. The exception is the mathematical physics community. But even there, ask them to name several new examples of their use after 1985, or even 1935. For some it will be difficult.

In the future, I personally believe, quaternions will be used in a variety of ways (they are already used in well over 25 different applications) — researchers will especially will leverage the quaternions’ 4D structure to analyze a variety of topics in human cognition and in the neuroscience of music.

A modest number of great researchers in cognitive psychology and AI, over the last hundred years, have already used quaternions and their mathematical family, the hypercomplex numbers, to investigate the nature of thought, motion, and personality  — these scientists include Jean Piaget, Karl Pribram, Ben Goertzel, and others.

This diverse site and exploring blog — including history, culture, math design, future speculation, and new research interpreting and highlighting — will help you explore and fill in this remarkable story for yourself.

I provide generous lists of references, links, and commentary in each major cultural/mathematical topic and sub-topic, and provide interesting and informative quotes from many of these papers to give you the real flavor and essence of their value.

Topics and techniques I may highlight in future blogs include:

  • briefly explaining math object constructions based on the unit hypersphere in a 4D space, creating objects such as melodic 4D directed paths (Gilles Baroin), stereographic projections of dodecahedrons (A. Ocneanu), etc.
  • exploring the possible 4D nature of melody and melodic rotation in the brain
  • the role of Ralph Waldo Emerson in advocating the strong development of pure math in young America
  • possible knowledge storage and transfer based on the projective Fano plane within octonion integrated “collections” of quaternions
  • also, in neuroscience, I would like to explore  and sketch the relevance of 4D processes and quaternions to various possible conceptual functions of, and connections between, the parietal lobe (object assembly and enhancement), the prefrontal cortex (planning/control), the occipital lobe (vision), and the thalamus, as suggested by some researchers, such as Jerath & Crawford, Arnold Trehub, Marina Korsakova-Kreyn, Daniel Wolpert, Daniela Dentico, and Ben Goertzel.

It will be interesting to see where this leads. I thought about many of these questions in graduate school 40 years ago, and recently decided to return to these questions after pursuing a long career in computer science and educational research. Your companionship and ideas are appreciated.