invariant

mathematics

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June 29, 2026

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In mathematics, an invariant is a property that remains unchanged under specified transformations or operations. It is something constant and unchanging.1

For example, in projective geometry, certain properties of geometric figures are preserved even when they are projected onto another surface. While measurements like distances and angles may change, properties such as collinearity (points lying on the same line) and the cross ratio of four collinear points remain invariant. These invariants help mathematicians understand fundamental characteristics of geometric objects that persist through transformations.2

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projective geometry
Projective invariants
With Desargues’s provision of infinitely distant points for parallels, the reality plane and the projective plane are essentially interchangeable—that is, ignoring distances and directions (angles), which are not preserved in the projection. Other properties are preserved, however. For instance, two different points have a…
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With Desargues’s provision of infinitely distant points for parallels, the reality plane and the projective plane are essentially interchangeable—that is, ignoring distances and directions (angles), which are not preserved in the projection. Other properties are preserved, however. For instance, two different points have a…
Read More

mechanics of solids
Principal stresses
…and are therefore called stress invariants. One may readily verify that they have the same values when evaluated in terms of σ_ij′ above as in terms of σ_ij by using the tensor transformation law and properties noted for the orthogonal transformation matrix.
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…and are therefore called stress invariants. One may readily verify that they have the same values when evaluated in terms of σ_ij′ above as in terms of σ_ij by using the tensor transformation law and properties noted for the orthogonal transformation matrix.
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David Hilbert
…extensively modified the mathematics of invariants—the entities that are not altered during such geometric changes as rotation, dilation, and reflection. Hilbert proved the theorem of invariants—that all invariants can be expressed in terms of a finite number. In his Zahlbericht (“Commentary on Numbers”), a report on algebraic number theory published…
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…extensively modified the mathematics of invariants—the entities that are not altered during such geometric changes as rotation, dilation, and reflection. Hilbert proved the theorem of invariants—that all invariants can be expressed in terms of a finite number. In his Zahlbericht (“Commentary on Numbers”), a report on algebraic number theory published…
Read More