geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, Hesse's principle of transfer (german: Übertragungsprinzip) states that if the points of the

projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...

P¹ are depicted by a

rational normal curve In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the ...

in P^''n'', then the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

of the

projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ...

s of P^''n'' that preserve the curve is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

to the group of the projective transformations of P¹ (this is a generalization of the original Hesse's principle, in a form suggested by

Wilhelm Franz Meyer Friedrich Wilhelm Franz Meyer (1856–1934) was a German mathematician and one of the main editors of the '' Encyclopädie der Mathematischen Wissenschaften''. Life and work Meyer studied in the universities of Leipzig and Munich. In 1878, he ...

). It was originally introduced by

Otto Hesse Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Kingdom of Prussia, Prussia, and died in Munich, Kingdom of Bavaria, Bavaria. He worked mainly on algebraic invariants, and geome ...

in 1866, in a more restricted form. It influenced

Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

in the development of the

Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...

. Since its original conception, it was generalized by many mathematicians, including Klein,

Fano Fano is a town and ''comune'' of the province of Pesaro and Urbino in the Marche region of Italy. It is a beach resort southeast of Pesaro, located where the ''Via Flaminia'' reaches the Adriatic Sea. It is the third city in the region by popula ...

, and Cartan.

References

Original reference

*Hesse, L. O. (1866). "Ein Uebertragungsprinzip", ''

Crelle's Journal ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Augus ...

''. Projective geometry Invariant theory Group theory Symmetry Birational geometry {{geometry-stub

See also

Further reading

References

Original reference