Four-Dimensional Polytopes: Alicia Boole Stott’s Algorithm

Abstract

B etween 1850 and 1852, the Swiss mathematician Ludwig Schlafli developed a theory of geometry in n-dimensions. In Theorie der vielfachen Kontinuitat ([4]), he defined the n-dimensional sphere, introduced the concept of four-dimensional polytopes, which he called polychemes, and proved that there are exactly six regular polytopes in four dimensions but only three in dimensions higher than four. Unfortunately, his work was not accepted for publication, and only fragments were published some years later. The entire manuscript did not appear until 1901. Thus, mathematicians in the second half of the century were unaware of Schlafli’s discoveries. Between 1880 and 1900 the six regular polytopes were independently rediscovered by, among many others, Stringham in 1880 [5], Gosset in 1900 [3] and Boole Stott in 1900 [1]. The work of Stringham and Gosset is well known today, but that of Boole Stott, an amateur mathematician, has remained almost unnoticed. In this note we present her original algorithmic approach, together with her original drawings and models. We emphasize that, as a woman born in the mid-nineteenth century, Boole Stott never received any formal mathematical training. Her discoveries came from an extraordinary capacity to visualize the fourth dimension. Rigorous mathematical proofs can therefore not be expected in herwork, but instead we find a watershed of surprising and original ideas. Alicia Boole was born near Cork (Ireland) in 1860, the third daughter of the famous logician George Boole. He died when Alicia was four years old, and her mother became an innovative educator. The amateur mathematician Howard Hinton, a frequent guest in their home, was deeply interested in the fourth dimension. He taught the children to visualize four-dimensional shapes with small cubes; this may (or may not) have inspired Alicia’s later research. Whatever the inspiriation, Alicia Boole Stott (she married in 1890) rediscovered the six four-dimensional polytopes by computing their three-dimensional sections. In 1895 she was introduced to the Dutch geometer P. H. Schoute. They collaborated for more than 20 years, combining Schoute’s analytical methods with her unusual visualization ability; in 1914, after Schoute’s death (1913), the University of Groningen awarded Boole Stott an honorary doctorate. After that, she was isolated from the mathematical community until about 1930, when her nephew, G. I. Taylor, introduced her to H. S. M. Coxeter. Despite the nearly 50 year difference in their ages, Boole Stott and Coxeter collaborated productively until her death in 1940. (For more details, see [2].) To clarify her approach to four-dimensional polytopes, we first apply Boole Scott’s method to the five regular polyhedra (Figure 1). Constructing the parallel two-dimensional sections of any polyhedron (i.e., the sections parallel to one of its faces) is quite elementary. To compute, for example, the sections of the cube, we intersect the plane containing a given face of the cube with the cube itself. This intersection is, of course, the face of the cube; that is, the parallel section is a square. Translating the plane towards the center of the cube, we see that all parallel sections are isometric squares. Similarly, parallel sections of the tetrahedron are decreasing triangles, triangles and hexagons for the octahedron, pentagons and decagons for the dodecahedron and triangles, hexagons and dodecagons for the icosahedron. Diagonal sections of a regular polyhedron P are sections H \ P, where H is a plane perpendicular to the segment OV joining the center of the polyhedron with a vertex. We can visualize a regular solid by unfolding it to a planar net. Roughly speaking, this means ‘‘cutting’’ certain edges of the polyhedron and mapping it to a two-dimensional space. The well-known net for the cube is shown in Figure 2. Note that to recover the three-dimensional cube from the unfolded version, one must identify certain edges. This allows us to describe the parallel sections of the cube in a very easy way. Namely, one parallel section could be one of the squares in Figure 2, for example, the middle square (call it MS). In order to obtain the other sections (which will be parallel to the square MS after folding the net) one just needs to move the four edges of MS in the unfolded cube parallel towards the remaining squares. In each case, one obtains a square isometric to the square MS (after necessary identification of end points of the edges).