Are Cognitive Processes Encoded Through Sequences of Geometric Transformations?

Posted: 12 Nov 2019 Last revised: 26 Dec 2019

See all articles by Felix Polyakov

Felix Polyakov

Bar-Ilan University - Department of Mathematics

Date Written: November 2, 2019


Geometric symmetry is abundant in nature, in art and has gradually been inscribed into the fundamental mathematical concepts and tools; for example, continuous and discrete groups of transformations and their applications. Another important role of symmetry in our life comes from experiencing beauty and aesthetic pleasure as they are connected to the perception of symmetric patterns. Mathematically, geometric symmetry is characterized by invariance under classes of geometric transformations. Say, circle remains circle (is invariant/symmetric) under a combination of dilation, rotation and translation but not under skew transformations. A number of works reported representation of geometric invariance in primate motor performance and perception and in underlying brain activity.

According to recent research, geometry constitutes a core set of intuitions present in all humans. Moreover, several studies show that different cognitive processes, from learning spatial sequences to linguistic comprehension, interact with geometric variables. While, for decades, numerous works postulate mechanistic optimality in brain's controlled behavior, they do not account for the cognitive component. To my view, representation of cognitive processes in the brain is based on geometric variables and transformations that are essential for bridging between cognitive and mechanistic components of behavior, for sensory-motor integration, integration of information from different sensory modalities, and for compact representation of acquired skills. I propose that:

1. 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.

2. 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.

3. 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.

4. 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.

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 sport coaches.

This note discusses how empirical evidence from different studies led me to state and to put together the above propositions about the role of geometric representation in brain control of our activity, and formulates research questions aimed to deepen our understanding of the topic.

Keywords: geometric intuition, geometric transformation, compact representation, geometric symmetry, geometric primitive, cognitive processes, learning

Suggested Citation

Polyakov, Felix, Are Cognitive Processes Encoded Through Sequences of Geometric Transformations? (November 2, 2019). Available at SSRN: or

Felix Polyakov (Contact Author)

Bar-Ilan University - Department of Mathematics ( email )

Ramat Gan, 5290002

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