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MORIN'S SURFACE
Bernard
Morin (1933): French mathematician.
Article of B. Morin and J.P. Petit on turning a sphere inside out: new.lutecium.org/www.jppetit.com/science/maths_f/Retournement_sphere/PLS_79.pdf François Apery: models of the projective plane, page 104. en.wikipedia.org/wiki/Morin_surface www.math.uiuc.edu/~jms/Photos/MathArt/Maubeuge/denner/ mathinfo.unistra.fr/fileadmin/upload/IREM/Publications/L_Ouvert/n094/o_94_3245.pdf http://www.mathart.eu/Documents/pdfs/Cagliari2013/Cagliari_Denner_Springer009.pdf 
François Apery's Cartesian parametrization: where 
Morin's surface is an immersion of the sphere that comes up in the central phase of the process of turning a sphere inside out imagined by B. Morin and J.P. Petit (cf. the above article).
J.P. Petit's drawing opposite shows its topology. The sphere initially had a face colored in grey and a face colored in orange. The swapping of the two faces occurs during the central phase of the process by a simple rotation by . 

For n = 2 et k = 1, the above parametrization provides a model (the case n = 3, k = 1 gives Boy's surface). 

Opposite, a polyhedral version of this surface owed to J.P.
Petit.
Another version is owed to Richard Denner (see the links above).


Apery's parametrization provides a family of surfaces with rotational symmetry of order n; opposite the case n = 5. 

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© Robert FERRÉOL
2017