In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation
Each curve in the family is determined by a pair of parameter values () (not both zero) and consists of the points in the plane whose homogeneous coordinates satisfy the equation for those parameters. Multiplying both and by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either or (but not both) to be set to zero.
Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving only nine points. The family of cubics through these nine points forms the Hesse pencil. More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points.
The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the Hesse configuration.
Every elliptic curve is birationally equivalent to a curve of the Hesse pencil; this is the Hessian form of an elliptic curve. However, the parameters () of the Hessian form may belong to an extension field of the field of definition of the original curve.
In mathematics, the syzygetic pencil or Hessepencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the...
Hessian matrix, the Hesse normal form, the Hesse configuration, the Hessian group, Hessian pairs, Hesse's theorem, Hessepencil, and the Hesse transfer principle...
SL2(3) of order 24. It also acts on the Hessepencil of elliptic curves, and forms the automorphism group of the Hesse configuration of the 9 inflection points...
curve in a pencil of curves generated by C and by the Hessian curve of C, the Hessepencil. The Hessian polyhedron is a representation of the Hesse configuration...
the Hesse group and the Hessepencil. Grove, Charles Clayton (1906), The syzygetic pencil of cubics with a new geometrical development of its Hesse Group...
Eva Hesse (January 11, 1936 – May 29, 1970) was a German-born American sculptor known for her pioneering work in materials such as latex, fiberglass,...
crown"), a wide silk grosgrain hatband ribbon, a flat brim shaped in a "pencil curl", and a ribbon-bound trim about the edge of the brim. It is traditionally...
In mathematics, a Hessian pair or Hessian duad, named for Otto Hesse, is a pair of points of the projective line canonically associated with a set of...
Duke Ferdinand of Brunswick-Wolfenbüttel and their host Prince Charles of Hesse-Kassel, and the Martinists, under Jean-Baptiste Willermoz. The only dissenting...
Julien Thompson September 30, 2012 (2012-09-30) February 27, 2015 As a pencil with a magical drawing ability has just fallen onto Oggy's house yard, the...
others, stated their admiration for Walser's prose, and authors like Hermann Hesse and Franz Kafka counted him among their favorite writers. Walser published...
Hauser, who, on his deathbed, muttered incoherently about "writing with pencil". Although Hauser was eager that the purse be found, he did not ask for...
the Grand Duchy of Hesse to the South (Province of Starkenburg) and to the North (Province Upper Hesse), the Electorate of Hesse (District of Hanau)...
cachet crashed when his name appeared on everything from key chains to pencil holders". However, the Cardin name was still very profitable, although the...
LeWitt associated. LeWitt also became friends with Hanne Darboven, Eva Hesse, and Robert Smithson. LeWitt taught at several New York schools, including...
first words were "piz, piz", a shortening of lápiz, the Spanish word for "pencil". From the age of seven, Picasso received formal artistic training from...
rise to a projective mapping π {\displaystyle \pi } from pencil S {\displaystyle S} onto the pencil Π y {\displaystyle \Pi _{y}} . The intersection of the...
Tower (1991) in Cleveland, Ohio. Frankfurters often call it Bleistift ("pencil") due to its shape. The construction of the building's foundation set a...
costume. A ball in Zürich is featured in the novel Steppenwolf by Hermann Hesse. A masquerade ball is central to the plot of Mikhail Lermontov's 1835 play...
boy, he carved a tiny mannequin into the end of the wooden ruler from his pencil case and placed it inside the case. He added a stone, which he had painted...
overall grid of rectangles. In this painting the rectangles were drawn in pencil over uneven washes of gray translucent paint. In 1973, she returned to art...
Global Information
