SURFER image properties:

brightness = 0, glow = 100

This picture won the SURFR competition in Kassel, hosted by ‹Spektrum der Wissenschaft› und ‹Kasseler Sparkasse› within the category ‹Users outside of Kassel›.

The formula: x² y² + y² z² + x² z² + xyz = 0
describes a shape, which was discovered by Jakob Steiner. While he stayed in Rome in 1844, he studied its geometrical properties - hence it is called Roman surface. However, it was his friend Karl Weierstraß, who first published a paper on the surface and Steiner’s results in 1863, the year of Steiner’s death.

In the picture there are 18 Roman surfaces positioned symmetrically around the origin, each surface only slightly translated off the origin along one or two of the coordinate axes, so they intersect each other. Additionally, only a small portion of the whole arrangement is shown in this picture: everything within a small spherical neighborhood of the origin. Everything outside is simply cut off.

There are four blue ones, four red ones and four yellow ones, each color within one of the three coordinate planes. Their centers lie on the diagonal lines x= +/- y, y = +/- z, and z = +/- x respectively. The centers of the two orange, two green and two purple ones lie on the vertices of a cube, each color on one diagonal of the cube, matching the three coordinate axes.

Even thogh the Roman surface is non-orientable, which means there is no outside or inside - just one side, the SURFER program assigns two colors to it. Here the first color of each surface is picked as described above, and all surfaces have the same second color, which is black. Black is also the background color.

SURFER image properties:

a = 0.47, three surfaces colored red, yellow and blue, transparency = 60%, reflection = 100, various colorful lights :-)

Le plan projectif réel est l’espace des droites passant par l’origine dans l’espace euclidien tri-dimensionnel (R3). Lors d’un séjour à Rome, le mathématicien Jakob Steiner proposa une réalisation du plan projectif réel dans R3. La surface qui en résulte possède des auto-intersections. On l’appelle surface romaine ou surface de Steiner.

L’origine est un point triple pour cette surface et chacun des trois plans de coordonnées est tangent à la surface. Les segments partant de l’origine le long des axes de coordonnées sont des lignes de points doubles qui aboutissent à six points singuliers.

Sur cette image, on admire une surface romaine en jaune, entourée par six morceaux de surface romaine en couleurs, disposés aux points de pincement, soulignant les nombreuses symétries de la surface.

SURFER image properties:

b = 0.38, transparency = 90%, colorful lights

Imagine sewing the edge of a Möbius strip, which is just one closed curve, to the boundary of a disk! The resulting closed surface still contains the Möbius strip and will therefore be non-orientable (you cannot tell what is the outside or the inside - it is all the same). That is why it must have selfintersections in our real three-dimensional space. But can it be done smoothly with no sharp or pointy edges?

The famous mathematician David Hilbert assigned his student Werner Boy to prove that it is impossible. To the surprise of his teacher, Boy constructed such a smooth surface in 1901. He described it by giving the corresponding curves of its insersection with a family of parallel planes.

Now in this picture, you see a slice of two Boy surfaces (one being the mirror image of the other), cut not by planes but by two spheres with the same center and of slightly different radii.