Roman surface

ruzuzu commented on the word Roman surface

See the Century's tenth definition of surface, which states as follows:

"The surface often originally, and better, called the Roman surface discovered by Jacob Steiner (1796-1863), undoubtedly the greatest of all geometricians, being a quartic surface of the third class, having three double lines. In its symmetrical form its appearance is thus described: Take a tetrahedron, and inscribe in each face a circle. There will be, of course, two circles touching at the mid-point of each edge of the tetrahedron; each circle will contain, on its circumference, at angular distances of 120°, three mid-points; and the lines joining these with the center of the tetrahedron, produced beyond the center, meet the opposite edges … joining the mid-points. … Now truncate the tetrahedron by planes parallel to the faces, so as to reduce the altitudes, each to three fourths of the original value; and from the center of each new face round off symmetrically up to the adjacent three circles; and within each circle scoop down to the center of the tetrahedron, the bounding surface of the excavation passing through that is, containing the three right lines, and the sections by planes parallel to the face being in the neighborhood of the face nearly circular, but, as they approach the center, assuming a trigoidal form, and being close to the center an indefinitely small equilateral triangle. We have thus the surface, consisting of four lobes united only by the lines through the mid-points of opposite edges—these lines being consequently nodal lines, the mid-points being pinch-points of the surface, and the faces singular planes, each touching the surface along the inscribed circle. (Cayley, Proceedings London Math. Soc., V. 14.)"

August 2, 2011