Mathematician:Felix Christian Klein

Mathematician

German mathematician best known for his work establishing the connections between geometry and group theory.

Architect of the Erlangen program, which classifies geometries according to their symmetry groups.

Noted for the Klein bottle and the Klein $4$-group.

Not to be confused with Morris Kline.

Nationality

German

History

• Born: 25 April 1849 in Düsseldorf, Prussia (now Germany)
• 1875: Married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel
• Died: 22 June 1925 in Göttingen, Germany

Theorems and Definitions

• Erlangen Program

• Klein Bottle
• Klein $4$-Group
• Klein Quartic
• Kleinian Group
• Klein Geometry
• Klein Quadric
• Klein's $j$-Invariant

• Beltrami-Klein Model (with Eugenio Beltrami) (also known as the Cayley-Klein Model, with Arthur Cayley, the Klein Model or the Klein Disk Model)

Results named for Felix Christian Klein can be found here.

Definitions of concepts named for Felix Christian Klein can be found here.

Publications

• 1868: Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form
• 1872: Vergleichende Betrachtungen über neuere geometrische Forschungen
• 1882: Ueber Riemann's Theorie der Algebraischen Functionen und ihre Integrale
• 1884: Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade (English translation by G.G. Morrice: Lectures on the Icosahedron; and the Solution of Equations of the Fifth Degree, 1914)
• 1890, 1892: Vorlesungen über die Theorie der elliptischen Modulfunktionen (2 volumes) (with Robert Fricke (2 volumes: 1890 and 1892)
• 1893: Evanston Colloquium (before the Congress of Mathematics, reported and published by Alexander Ziwet, 1894)
• 1894: Über die hypergeometrische Funktion
• 1894: Über lineare Differentialgleichungen der 2. Ordnung
• 1895: Vorträge über ausgewählte Fragen der Elementargeometrie) (English translation by W.W. Beman and D.E. Smith: Famous Problems of Elementary Geometry, 1897)
• 1897 - 1910: Theorie des Kreisels (with Arnold Sommerfeld)
• 1897 - 1912: Vorlesungen über die Theorie der automorphen Funktionen (2 volumes: the second volume was published in three parts) (with Robert Fricke)
• 1897: Mathematical Theory of the Top (Princeton address, New York)
• 1908: Elementarmathematik vom höheren Standpunkte aus
  • 1925: Elementarmathematik vom höheren Standpunkte aus, 3rd ed.
  • 1939: Elementary Mathematics from an Advanced Standpoint: Geometry (translated by Earle Raymond Hedrick and Charles Albert Noble from Elementarmathematik vom höheren Standpunkte aus, 3rd ed.)

• The Felix Klein Protocols

Notable Quotes

Plato said, "God is a geometer." Jacobi changed this to, "God is an arithmetician." Then came Kronecker and fashioned the memorable expression, "God created the natural numbers, and all the rest is the work of man."

-- Quoted as an epigraph to the preface to 1980: David M. Burton: Elementary Number Theory (revised ed.)

Also known as

Felix Christian Klein is also sometimes reported as Christian Felix Klein.

Sources

• John J. O'Connor and Edmund F. Robertson: "Felix Christian Klein": MacTutor History of Mathematics archive

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Preface
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Klein, Christian Felix (1849-1925)
• 1991: David Wells: Curious and Interesting Geometry ... (previous) ... (next): A Chronological List Of Mathematicians
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Klein, Christian Felix (1849-1925)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Klein, Christian Felix (1849-1925)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Klein, (Christian) Felix (1849-1925)

... and while we're about it

A mathematician named Klein

Thought the Möbius band was divine.

He said, "If you glue

The edges of two,

You get a weird bottle like mine."

-- Leo Moser

-- (Quoted by Martin Gardner in one of his Mathematical Games columns in Scientific American, c. $\text {1960}$ – $\text {70}$)