Fractional coordinates¶
Crystal coordinates expressing a point as coefficients of the unit-cell lattice basis.
Core Idea¶
A fractional coordinate triplet (u,v,w) represents r=u·a+v·b+w·c relative to a chosen cell origin and lattice vectors, with coordinates equivalent modulo integers in a periodic crystal. Changing the unit cell or basis transforms coefficients contragrediently while periodic translation preserves the represented crystallographic site. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Fractional coordinates belongs to crystallography and is useful where the analyst can specify the typed crystallography carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, then evaluate origin, lattice basis, cell convention, and periodic equivalence are stated and the coordinate reconstructs the same Cartesian point modulo lattice vectors. The scope is broad within that domain but bounded by the need for origin, lattice basis, cell convention, and periodic equivalence are stated and the coordinate reconstructs the same Cartesian point modulo lattice vectors. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making origin, lattice basis, cell convention, and periodic equivalence are stated and the coordinate reconstructs the same Cartesian point modulo lattice vectors the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Fractional coordinates can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Fractional coordinates. Fractional coordinates compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed crystallography carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express origin, lattice basis, cell convention, and periodic equivalence are stated and the coordinate reconstructs the same Cartesian point modulo lattice vectors independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of crystallography because they reuse the typed crystallography carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, Changing the unit cell or basis transforms coefficients contragrediently while periodic translation preserves the represented crystallographic site., and type the carrier, state every parameter and convention in the definition, test that origin, lattice basis, cell convention, and periodic equivalence are stated and the coordinate reconstructs the same Cartesian point modulo lattice vectors, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Fractional coordinates Domain-specific
Parents (1) — more general patterns this builds on
-
Fractional coordinates is a kind of Frame of Reference Prime
The proposed strict upward parent is
prime:frame_of_reference.
Hierarchy path (1) — routes to 1 parentless root
- Fractional coordinates → Frame of Reference → Viewpoint
Neighborhood in Abstraction Space¶
Fractional coordinates sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Crystallographic Coordinates & Symmetry (5 abstractions)
Nearest neighbors
- Rod group — 0.92
- Line group — 0.90
- Crystal structure prediction — 0.89
- Dual lattice — 0.89
- Quincunx matrix — 0.89
Computed from structural-signature embeddings · 2026-09-08