Methexis - “participation” - is Plato’s name for the relation between particulars and forms: a beautiful thing is beautiful by participating in Beauty, a just act by sharing in Justice. The term does real metaphysical work: it explains how one Form can be present to many things without being divided among them, the way a single melody is fully present in every performance. Methexis is the bridge across Plato’s two-level picture of reality - the joint where the intelligible realm touches the sensible world.
Plato himself pressed on the difficulty in the Parmenides: how exactly does the many share in the one? Is the Form wholly in each particular, or partly? Aristotle sharpened the critique and answered by relocating form into things as Formal Causation - participation recast as immanent structure. Neoplatonism resolved it differently: participation as radiation, each level of the order of Emanation & Descent receiving what it can of the level above, as mirrors receive light without diminishing the sun. In both descendants the core intuition survives: particulars are what they are by relation to pattern, not in self-sufficient isolation.
The word also names something deeper than a logical relation - a mode of being and of knowing. To participate is to be constituted by what one shares in, and the knower rises toward the Forms by contemplation precisely because the soul already participates in what it seeks (anamnesis). This is the ancient root of what the vault elsewhere tracks as participation and participatory knowing: Vervaeke’s participatory knowing, Tarnas’s participatory epistemology, and Owen Barfield’s “original participation” all descend from methexis - the claim that we know reality by sharing in it, not by observing it from outside.
Understanding methexis means seeing that the relation between pattern and instance is the live question hiding in every domain - how the universal is present in the particular, the Archetype in the person, meaning in the moment - and that “participation” was Plato’s wager that the relation is real, intimate, and two-way.
See also form, Plato, participation, participatory, intelligible realm, Formal Causation, Emanation & Descent, anamnesis