explore | Grilled Cheese

Photographic portrait of Marie-France Vignéras, and a quote : "To solve a problem with a proof is exciting and rewarding because it is true forever."

Triumphant Achilles dragging Hector’s body around Troy, from a panoramic fresco of the Achilleion. by Franz Matsch.

Achilles rides a chariot pulled by black horses, dragging Hector's body around the walls of Troy, while soldiers and onlookers watch from the ramparts.

$Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Archimedes’ Equiareal Map Theorem : Let S be the unit sphere defined relative to the x, y and z axes by x^2 + y^2 + z^2 = 1 and let S′ be the bounding cylinder defined by x^2 + y^2 = 1. Let f be the map f : (x, y, z) → (x / √(x^2 + y^2) , y / √(x^2 + y^2) , z) from S \{(0, 0, 1), (0, 0, −1)} to S ′ which takes a point P on the sphere having height zP, |zP| < 1, to the nearest point on the cylinder which has height zP and which is collinear with P and (0, 0, zP). Then f is an equiareal diffeomorphism.$