User:PoliTopo/John Flinders Petrie

John Flinders Petrie (1907-1972) was an English mathematician who showed a remarkable geometrical aptitude in his youth. As a schoolboy, he was acquainted with the great geometer H. S. M. Coxeter, beginning a lifelong friendship. They collaborated in the discovery of the infinite skew polyhedra and the (finite) skew polyhedra in the fourth dimension, analogous to these. Besides being the first to realize the importance of the skew polygon that now bears his name, his skills as a draughtsman are still appreciated.

Biography
Petrie, born 26 April 1907, in Hampstead, London, was the only male son of Sir William Flinders Petrie, the renowned Egyptologist. As a schoolboy showed remarkable promise of mathematical ability. In a boarding school in England, he was put in bed next to H. S. M. Coxeter, recovering from some trivial illness in the sanatorium. They became firm lifelong friends. Looking at a geometry textbook with an appendix on the five Platonic solids they wondered why there were only five, and tried to extend them. Petrie said, what about putting four squares around a corner? Of course, they had fall flat, giving a pattern of squares filling the plane. Being inventive with words, he called it a “tessarohedron”; the similar arrangements of triangles, a “trigonohedron”.

Regular skew polyhedra
One day in 1926, Petrie told Coxeter, with much excitement, that he had discovered two new regular polyhedra; infinite, but free from “false vertices” (points other than vertices, where three of more faces meet, like those on regular star polyhedra): one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex, forming a dual polyhedron pair. To the usual protest that there is no room for more than four squares round a vertex, he revealed the trick: to let the faces go up and down in a kind of zigzag formation. When Coxeter understood this, he pointed out a third possibility: hexagons, six at each vertex, being self dual.

Coxeter offered a modified Schläfli symbol {l, m|n} for these figures, with {l, m} implying the vertex figure, m l–gons around a vertex, and n–gonal holes. It then occurred to them that, although these new polyhedra are infinite, they might find analogous finite polyhedra by going into the fourth dimension. Petrie cited one consisting of n2 squares, four at each vertex. They called such figures regular skew polyhedra. Afterwards, Coxeter elaborated further into the subject.

College years and job
Because his father belonged to University College London, Petrie went to study there, where he did quite fine. When the wwii came, he enlisted as an officer and was taken prisoner by the Germans. He organized a choir there. After the war ended and he was released, he went to Darlington Hall, a school in southwest England. He had a rather trivial job there and worked for many years as a school master, and never seemed to fulfill his early promise. He became a tutor who looked after children who were not doing well in school.

Petrie polygon
Petrie still corresponded with Coxeter, and it was the former who noticed that among the edges of a regular polyhedron, we easily pick out a skew polygon or zigzag, in which the first and second are sides of one face, the second and third are sides of another face, and so on. This zigzag is known as a Petrie polygon, and has many applications. Each finite regular polyhedron can be orthogonally projected on to a plane in such way that one Petrie polygon becomes a regular polygon, with the rest of the projection inside it.

Draughtsmanship
His skills as a draughtsman can be seen in the exquisite set of drawings of stellated icosahedra, which provide much of the fascination of the much talked–about book. Later, in order to explain icosahedral symmetry, Coxeter showed an orthogonal projection, representing 10 of the 15 great circles as ellipses. The difficult task of draughtsmanship was done by Petrie in 1932. It now stands out in the cover of a popular recreational mathematics book, with some color added. In periods of intense concentration, he could answer questions about complicated four dimensional figures by “visualizing” them.

Later years
Petrie married and had a daughter. Then, late in the year 1972, his wife suddenly got a heart attack and died. He was so distraught, missed her so terribly that he walked into a motorway near his home and was killed by a car. He died in Surrey, aged 64, just two weeks after his wife.