Euler's Theorem, that every non-trivial rotation $R$ of 3-space has a unique axis. It really just days that $R$ acting on the space of lines through the origin has a unique fixed point.

(Added April 11, 2013) I just received my copy of the latest issue of The Journal of Fixed Point Theory and its Applications (Vol.12, Nos. 1--2) and starting on page 27 there is an article with the title "Chasles' fixed point theorem for Euclidean motions". Chasles' theorem is a generalization of Euler's Theorem; it says that every orientation preserving Euclidean motion of 3-space that is not a pure translation is a "twist" or "screw motion", that is, a rotation about some unique line (NOT necessarily through the origin) called the axis followed by a translation that is parallel to the axis. I really should have given this as my example rather than Euler's Theorem, since as I said it is more general. And I have no excuse for not recalling it since the authors of that paper are myself and my son Bob.

Euler's Theorem, that every non-trivial rotation $R$ of 3-space has a unique axis. It really just says that $R$ acting on the space of lines through the origin has a unique fixed point.

(Added April 11, 2013) I just received my copy of the latest issue of The Journal of Fixed Point Theory and its Applications (Vol.12, Nos. 1--2) and starting on page 27 there is an article with the title "Chasles' fixed point theorem for Euclidean motions". Chasles' theorem is a generalization of Euler's Theorem; it says that every orientation preserving Euclidean motion of 3-space that is not a pure translation is a "twist" or "screw motion", that is, a rotation about some unique line (NOT necessarily through the origin) called the axis followed by a translation that is parallel to the axis. I really should have given this as my example rather than Euler's Theorem, since as I said it is more general. And I have no excuse for not recalling it since the authors of that paper are myself and my son Bob.

Euler's Theorem, that every non-trivial rotation $R$ of 3-space has a unique axis. It really just days that $R$ acting on the space of lines through the origin has a unique fixed point.

Euler's Theorem, that every non-trivial rotation $R$ of 3-space has a unique axis. It really just days that $R$ acting on the space of lines through the origin has a unique fixed point.

(Added April 11, 2013) I just received my copy of the latest issue of The Journal of Fixed Point Theory and its Applications (Vol.12, Nos. 1--2) and starting on page 27 there is an article with the title "Chasles' fixed point theorem for Euclidean motions". Chasles' theorem is a generalization of Euler's Theorem; it says that every orientation preserving Euclidean motion of 3-space that is not a pure translation is a "twist" or "screw motion", that is, a rotation about some unique line (NOT necessarily through the origin) called the axis followed by a translation that is parallel to the axis. I really should have given this as my example rather than Euler's Theorem, since as I said it is more general. And I have no excuse for not recalling it since the authors of that paper are myself and my son Bob.