John Flinders Petrie

Last updated • 4 min readFrom Wikipedia, The Free Encyclopedia

John Flinders Petrie
Died1972
NationalityUnited Kingdom
Known for Petrie polygon

John Flinders Petrie (26 April 1907 – 1972) was an English mathematician. He met the 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, he was also skilled as a draftsperson.

Contents

Biography

Petrie was born on 26 April 1907, in Hampstead, London. He was the only son of the renowned Egyptologists Sir William Matthew Flinders Petrie and Hilda Petrie. [1] 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. [2] 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. He called this arrangement a "tesserohedron", reaching the similar structure of triangles a "trigonohedron."

Polyhedral regular tilings

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. When Coxeter understood this, he mentioned a third possibility: hexagons, six around a vertex, its dual.

Coxeter suggested a modified Schläfli symbol, {l, m | n} for these figures, with the emblem {l, m} implying the vertex figure, m l-gons around a vertex and n-gonal holes. Then it occurred to them that, although the new polyhedra are infinite, they could find analogous finite polyhedra by delving into the fourth dimension. Petrie cited one consisting of n2 squares, four at each vertex. They called these figures "regular skew polyhedra". Later, Coxeter would delve deeper into the subject.

University and work

Because his father belonged to University College London, Petrie enrolled in this institution, where he successfully completed his studies. When the Second World War broke out, he enlisted as an officer and was captured as a prisoner by the Germans, organizing a choir during his captivity. After the war ended and he was released, he went to Darlington Hall, a school in southwest England. He worked many years as a schoolteacher. He was one of the tutors who attended to the children doing poorly in school.

The Petrie polygon

Petrie continued to correspond with Coxeter and was the first to notice that, among the edges of a regular polyhedron, a skew polygon forming a zigzag can be distinguished, in which the first and second are the edges of one face, the second and third are the edges of another face and so on, successively. This zigzag is known as the "Petrie polygon" and has many applications. The Petrie polygon of a regular polyhedron can be defined as the skew polygon (whose vertices do not all lie in the same plane) such that every two consecutive sides (but not three) belong to one of the faces of the polyhedron.

Each finite regular polyhedron can be orthogonally projected onto a plane so that the Petrie polygon becomes a regular polygon, with the rest of the projection inside. These polygons and their projected graphs help visualize the symmetrical structure of regular polytopes of higher dimensions, which are difficult to conceive or imagine without this aid.

His skills as a draftsman are shown in an exquisite set of drawings of the stellated icosahedron, which provides much of the fascination of the much-discussed book he illustrates. On another occasion, to explain the symmetry of the icosahedron, Coxeter showed an orthogonal projection, representing 10 of the 15 great circles as ellipses. The difficult task of drawing was performed by Petrie around 1932. It now prominently features on the cover of a popular recreational mathematics book garnished with a touch of colour. It is reported that, in periods of intense concentration, he could answer questions about complex figures of the fourth dimension by "visualizing" them.

Final years

Petrie got married and had a daughter. In late 1972, his wife suffered a sudden heart attack and passed away. He missed her so much and was so distracted that one day he walked onto a highway near his home and was hit by a car while trying to cross it running. He died in Surrey, at 64, just two weeks after his wife.[ citation needed ]

See also

Related Research Articles

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi-) triangular faces and twelve (dodeca-) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

<span class="mw-page-title-main">Kepler–Poinsot polyhedron</span> Any of 4 regular star polyhedra

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

<span class="mw-page-title-main">Polyhedron</span> Three-dimensional shape with flat faces, straight edges, and sharp corners

In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.

In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Stellation</span> Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

<span class="mw-page-title-main">Regular polytope</span> Polytope with highest degree of symmetry

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension jn.

In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.

<span class="mw-page-title-main">Vertex figure</span> Shape made by slicing off a corner of a polytope

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general n-polytope is sliced off.

<span class="mw-page-title-main">Final stellation of the icosahedron</span> Outermost stellation of the icosahedron

In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron.

<span class="mw-page-title-main">Apeirogon</span> Polygon with an infinite number of sides

In geometry, an apeirogon or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.

<span class="mw-page-title-main">Skew polygon</span> Polygonal chain whose vertices are not all coplanar

In geometry, a skew polygon is a closed polygonal chain in Euclidean space. It is a figure similar to a polygon except its vertices are not all coplanar. While a polygon is ordinarily defined as a plane figure, the edges and vertices of a skew polygon form a space curve. Skew polygons must have at least four vertices. The interior surface and corresponding area measure of such a polygon is not uniquely defined.

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.

<span class="mw-page-title-main">Petrie polygon</span> Skew polygon derived from a polytope

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.

<span class="mw-page-title-main">Density (polytope)</span> Number of windings of a polytope around its center of symmetry

In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets.

<span class="mw-page-title-main">Regular skew apeirohedron</span> Infinite regular skew polyhedron

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron. They have either skew regular faces or skew regular vertex figures.

References

  1. W. H. Auden – ‘Family Ghosts’ «John Flinders Petrie».
  2. A large part of what is known about Petrie is due to Coxeter. See also: Hargittai (2005). "H. S. M. (Donald) Coxeter". Candid science., pág. 5 et seq.