isometry
C2Technical/Academic
Definition
Meaning
A transformation or mapping that preserves distances between points.
In mathematics, a function that preserves distances (also called a rigid transformation or congruence). In cartography, it can refer to a map projection preserving distances along certain lines or from a point. In geometry, two figures are isometric if one can be transformed into the other by an isometry.
Linguistics
Semantic Notes
Primarily a mathematical term. It denotes a strict form of preservation of structure (distance), stronger than a homeomorphism (which preserves continuity) or an isomorphism (which preserves algebraic structure).
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling conventions follow the standard UK/US patterns (e.g., 'rigour' vs. 'rigor' in surrounding text).
Connotations
Identical, purely technical.
Frequency
Equally low frequency and confined to specialised mathematical/geographical discourse in both varieties.
Vocabulary
Collocations
Grammar
Valency Patterns
An isometry of [geometric object, e.g., the plane]An isometry between [X] and [Y]To be an isometry, a map must...Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Core term in geometry, metric spaces, functional analysis, and cartography.
Everyday
Extremely rare, would not be understood by the general public.
Technical
Standard term in mathematics, physics (relating to symmetries), and certain engineering fields.
Examples
By Part of Speech
verb
British English
- The two spaces isometrically embed into a third.
- We need to isometrically map one figure onto the other.
American English
- The function isometrically maps the metric space.
- We can isometrically identify the two structures.
adverb
British English
- The two shapes are isometrically equivalent.
- The spaces map isometrically onto each other.
American English
- The function acts isometrically on the given set.
- The figures were arranged isometrically in the model.
adjective
British English
- The transformation was strictly isometric.
- They studied isometric properties of the surface.
American English
- An isometric embedding preserves distances exactly.
- The isometric view in the diagram helped visualise the object.
Examples
By CEFR Level
- In geometry, a rotation is an example of an isometry.
- The map showed the distances from the capital isometrically.
- The theorem proves that every isometry of the Euclidean plane is either a rotation, translation, reflection, or glide reflection.
- A local isometry between Riemannian manifolds preserves the metric tensor.
Learning
Memory Aids
Mnemonic
Think: 'ISO-METRY' = 'SAME-MEASURE'. An isometry keeps the measure (distance) the same.
Conceptual Metaphor
A perfect, rigid copy; a movement that does not warp or stretch the object.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Not to be confused with 'изометрия' (which is the direct equivalent). Beware of false friends with 'изомерия' (isomerism in chemistry). The Russian term is identical but the conceptual domain is strictly mathematical/geographical.
Common Mistakes
- Using 'isometry' to mean 'isomorphism' in algebra. Confusing it with 'isometric contraction' in physiology (muscle tension without movement). Pronouncing it as /ˈaɪ.səʊ.mɛ.tri/ (stress on first syllable).
Practice
Quiz
In which field is the term 'isometry' LEAST likely to be used?
FAQ
Frequently Asked Questions
By definition in most mathematical contexts, an isometry is a distance-preserving map between metric spaces. It is necessarily injective but need not be surjective unless specified as an 'isometric isomorphism' or 'global isometry'.
In Euclidean geometry, they are essentially synonymous. 'Congruence' often refers to the relationship between two figures, while 'isometry' often refers to the transformation/mapping itself that establishes that relationship.
Yes. Isometries are classified as either 'direct' (preserving orientation, like rotations and translations) or 'opposite' (reversing orientation, like reflections and glide reflections).
Only etymologically. 'Isometric' in physiology (e.g., pushing against a wall) comes from 'iso-' (equal) and '-metric' (measure), meaning 'of equal measure/length', referring to constant muscle length. The mathematical term shares the 'equal measure' root but refers specifically to distance.