Author: Rene PuaKoala

From Euclid to Tensors

Scope of PuaKoala’s work are symbolic systems and the emergence of meaning from perception. This can be studied with topics that are beyond everyday language while dealing with seemingly simple processes like the translation and rotation of objects.

From Euclid to Descartes numbers begin to describe geometry. Scalars, vectors, matrices, tensors – linear algebra – what drives scientists like Gauss, Hamilton, Minkowski, Ricci, Lie to explore these structures. What makes it possible that abstract symbols and their arrangement provide the same meaning to different people and how did this meaning develop? There are some interesting works that might illuminate these questions:

Die Entwicklung des räumlichen Denkens beim Kinde, Jean Piaget

Die Entwicklung des Tensorkalküls: Vom absoluten Differentialkalkül zur Relativitätstheorie, Karin Reich

Die Grundlage der allgemeinen Relativitätstheorie, Albert Einstein

Space in Language and Cognition, Explorations in Cognitive Diversity, Stephen C. Levinson

Computation and Gestalts

It is incredible how complex and sometimes organic particle systems can appear. Mathematically these can be described as a set of numerical solutions of differential equations of 3D space and time. How a particle system changes its appearance in dependence of initial values or parameters can be hard to predict. This is already the case with quite simple differential equations. The concept of causality in these deterministic systems is questioned by the huge space of possible solutions and the 3 dimensional shapes often reveal their essence as determined by Gestalt laws interacting with the perspective projection.

Having in mind that most natural processes and also human social and economic behaviour can be modelled by complex differential equation systems, there are interesting questions about which cross modal Gestalts project to the human mind.

Objects in the audio domain

Let’s say the term ‘object’ may be characterised as a cognitive process that is coupled to properties like persistence over time, countability or forgeground background differentiation. An engineering approach of the digital era to these properties is connected to the notions of sampling and resolution. Roughly speaking the required temporal sampling in digital multimedia systems for the human visual domain is below 100 Hz, for the human audio domain it is in the range of 40.000Hz.

In response to a pure sound at a frequency below about 300 Hz, an auditory-nerve fiber fires action potentials at every cycle of stimulation and at a fixed phase [3,5]. Above 300 Hz the axon starts to skip cycles, but action potentials still occur at a preferred phase of the stimulus. […] Temporal information about the stimulus frequency is [] greatest for frequencies below 800 Hz, declines from 800 Hz to 4 kHz, and vanishes for still greater frequencies. In some species, such as the barn owl, phase locking can continue up to 10 kHz [8].

Source: Frequency decoding of periodically timed action potentials through distinct activity patterns in a random neural network, Tobias Reichenbach and A. J. Hudspeth

Ignoring the different underlying physiological and perceptual concepts the following questions arise. 

Emerging from its higher temporal resolution – are there object like percepts in the audio domain that do not have an equivalent in the visual domain ? Or is this resolution only projected to spectral percepts like timbre, pitch, and spatial localisation?

Can objects emerge, persist and defocus on time scales smaller than 50ms ?