Order, disorder and in-between

Pattern formation · Emergent order · Hyperuniformity

Order, disorder and in-between

How does order emerge from disorder? And what can we learn from the fascinating space in-between?

Overview

Nature rarely presents us with perfect order or complete randomness. Snowflakes and crystals are highly ordered; gases and crowds may appear almost random. Yet many of the most intriguing systems inhabit the rich territory between these extremes, where simple local interactions give rise to complex collective organisation. Understanding this interplay offers insights not only into physical materials but also into biological, social, and technological systems that continually organise and reorganise themselves.

My research in this area began with geometry-driven hyperuniformity, where I investigated how simple geometric interactions among neighbouring points spontaneously generate large-scale order without any central coordination. This work revealed how topological defects, orientational domains, and scale-free correlations emerge during ordering transitions towards hyperuniformity, leading to unusual structural and mechanical properties.

Today, these ideas continue to motivate my broader interest in emergence, phase transitions, and universality across complex systems. By combining geometry, topology, computation, and dynamical modelling, I seek common organising principles that transcend individual disciplines — from condensed matter and biological tissues to collective behaviour and adaptive systems.

“What is order? Order was usually considered as a wonderful building, a loss of uncertainty. Typically it means that if a system is so constructed that if you know the location or the property of one element, you can make conclusions about the other elements. So order is essentially the arrival of redundancy in a system, a reduction of possibilities.”

— Heinz von Foerster, Understanding Understanding (2003)

Relevant publications

Below is a list of publications connected to my work on geometry-driven hyperuniformity, pattern formation, local relaxation, and emergent order.

Scale-free correlations in two-dimensional geometry-driven hyperuniformity

Authors: Sungyeon Hong, Mohammad Saadatfar
Year: 2025
Publisher: Data-Driven Modelling: Unravelling Complexity in the Sea of Data

This work investigates how large-scale order can emerge from simple local interactions. Using a computational process known as Lloyd's algorithm, we show that a disordered collection of points can self-organise into a hyperuniform state characterised by orientational domains whose sizes grow with the system itself. The absence of a characteristic length scale suggests a form of scale-free organisation, a hallmark of systems capable of coordinated collective responses across many scales.

Topological mechanical states in geometry-driven hyperuniform materials

Authors: Sungyeon Hong, Can Nerse, Sebastian Oberst, Mohammad Saadatfar
Year: 2024
Publisher: PNAS Nexus

Can disorder be engineered to produce useful physical behaviour? In this study, we reveal how topological defects that emerge during the formation of hyperuniform structures give rise to distinct mechanical states capable of localising vibrational energy. By connecting geometry, topology, and mechanics, this work demonstrates how robust functional properties can arise from self-organising disordered materials.

Dynamical arrest of topological defects in 2D hyperuniform disk packings

Authors: Sungyeon Hong, Michael A. Klatt, Gerd Schröder-Turk, Nicolas François, and Mohammad Saadatfar
Year: 2021
Conference: Powders & Grains 2021 – 9th International Conference on Micromechanics on Granular Media

This paper explores how topological defects evolve during the formation of hyperuniform structures. We show that as the system becomes increasingly ordered, defects slow down and eventually become trapped, leaving a characteristic structural signature. The results provide insight into how local topological processes shape the emergence of large-scale order in complex systems.

Interactive demo

Would you like to see how local interaction rules based on simple geometric relations between neighbouring points result in the emergence of spatial order? The interactive demo provides an intuition-building way to explore the local relaxation dynamics and emergent structures.

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