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Concept
John Flinders Petrie
Summary
John Flinders Petrie (April 26, 1907-1972) was an English mathematician who demonstrated remarkable geometric aptitude in his youth. He met the great geometer Harold Scott MacDonald Coxeter as a student, beginning a lifelong friendship. They collaborated in discovering infinite warped polyhedra and (finite) warped polyhedra in the fourth dimension, analogous to the previous ones. In addition to being the first to realize the importance of the warped polygon that now bears his name, his skills as a draftsperson are still appreciated.
Petrie was born on April 26, 1907, in Hampstead, London. He was the only son of the renowned Egyptologists Sir William Matthew Flinders Petrie and Hilda Petrie. As a student, he showed the remarkable potential of his mathematical abilities, and his incredible visual memory, which his father also showed, allowed him to visualize complex geometries. While studying at a boarding school, he met Coxeter in a sanatorium while recovering from a minor illness, beginning a friendship that would remain throughout their lives. Looking at a geometry textbook with an appendix on Platonic polyhedra, they wondered why there were only five and tried to increase their number. Petrie commented: How about we put four squares around one corner? In practice, they would lie on a plane, forming a pattern of squares covering the plane. Being clever with words, he called this arrangement a "tesserohedron", reaching the similar structure of triangles a "trigonohedron."
One day in 1926, Petrie told Coxeter that he had discovered two new regular polyhedra, infinite but free of "false vertices" (points distinct from the vertices, where three or more faces meet, like those that characterize regular star polyhedra): one consisting of squares, six at each vertex and another consisting of hexagons, four at each vertex, which form a dual or reciprocal pair. To the common objection that there is no room for more than four squares around a vertex, he revealed the trick: allow the faces to be arranged up and down, marking a zigzag.
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