In nature and in art, geometric symmetry is everywhere. It has also progressively found its way into the basic ideas and techniques of mathematics, such as the continuous and discrete groups of transformations and their uses. The impression of beauty and aesthetic pleasure, which are linked to symmetric patterns, is another significant function of symmetry in our lives.

Geometric symmetry is defined mathematically as being invariant to several types of geometric changes. For example, a circle would stay a circle (be invariant/symmetric) under a translation, rotation, and dilation combination, but not a skew transformation. Numerous studies have shown evidence of geometric invariance in the motor, perceptual, and underlying brain activity of primates.

Cognitive geometric transformations also include topic such as how the human brain perceives the image and how it recognizes objects despite of transformation in the image, does it uses some kind of transformation to recognize or it use some other mechanism. This is an active area of research in the field of cognitive science and computer vision.

The representation of cognitive processes in the brain, according to Felix Polyakov of Bar-Ilan University, is based on geometric variables and transformations that are crucial for bridging the gap between cognitive and mechanistic components of behavior, for integrating information from various sensory modalities, for sensory-motor integration, and for compactly representing learned skills. To this account, he proposes that:

- Such ubiquitous conscious use of the notion of symmetry in art and science may be a byproduct of the subconscious core geometric machinery, including geometric intuition, built-in into information processing systems of the brain.
- Seemingly separate components of a cognitive performance, e.g. different geometric shapes being perceived or drawn in sequence, may be interrelated via geometric transformations. Then, sequences of geometric transformations can be applied to a small number of templates to describe a complex performance. A mechanistic incentive for representation with sequences of geometric transformations could be to provide a more compact/instrumentally efficient neural representation while in the course of learning/practice neural system seeks to achieve complexity reduction in processing a complex task.
- The brain adjusts the neural substrate of various cognitive tasks to representation in terms of geometric variables to allow processing by its built-in structures responsible for geometric intuition and sensitive to geometric symmetry. Produced behaviors only partially follow predictions obtained with mechanistic optimality principles.
- Various cognitive tasks implemented by the brain, for example voluntary movements, counting, processing of sequences, use of language, rely on brain’s built-in geometric machinery or share with it common mechanisms on higher levels of control.

Polyakov proposes that geometry constitutes a core set of intuitions present in all humans The proposed ideas may be applied to enhance procedures of learning cognitive tasks, for example by structuring the learned material to exploit geometric redundancy (symmetry/invariance) of the task and expose the learner to corresponding classes of geometric transformations, what may be of interest to educators and sports coaches.

*Are Cognitive Processes Encoded Through Sequences of Geometric Transformations? *Felix Polyakov

Published: November, 2019

DOI: 10.2139/ssrn.3479636