Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# A Phase-Field Perspective on Mereotopology

Version 1 : Received: 4 August 2021 / Approved: 5 August 2021 / Online: 5 August 2021 (08:39:00 CEST)

A peer-reviewed article of this Preprint also exists.

Schmitz, G.J. A Phase-Field Perspective on Mereotopology. AppliedMath 2022, 2, 54-104. https://doi.org/10.3390/appliedmath2010004 Schmitz, G.J. A Phase-Field Perspective on Mereotopology. AppliedMath 2022, 2, 54-104. https://doi.org/10.3390/appliedmath2010004

## Abstract

Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things like x isConnected y (topology) or x isPartOf y (mereology) by first order logic and Boolean algebra, the phase-field concept describes the geometric shape of things and its dynamic evolution by drawing on a scalar field. The geometric shape of any thing is defined by its boundaries to one or more neighboring things. The notion and description of boundaries thus provides a bridge between mereotopology and the phase-field concept. The present article aims to relate phase-field expressions describing boundaries and especially triple junctions to their Boolean counterparts in mereotopology and contact algebra. An introductory overview on mereotopology is followed by an introduction to the phase-field concept already indicating first relations to mereo- topology. Mereotopological axioms and definitions are then discussed in detail from a phase-field perspective. A dedicated section introduces and discusses further notions of the isConnected relation emerging from the phase-field perspective like isSpatiallyConnected, isTemporallyConnected, isPhysicallyConnected, isPathConnected and wasConnected. Such relations introduce dynamics and thus physics into mereotopology as transitions from isDisconnected to isPartOf can be described.

## Keywords

Region based theory of space; RBTS; Contact algebra; Dyadic and Triadic relations; sequent algebra; boundaries; triple junctions; mereotopology; 4D mereotopology; mereophysics; Region Connect Calculus RCC; invariant spacetime interval; Falaco solitons; phase-field method; intuitionistic logic

## Subject

Physical Sciences, Mathematical Physics

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