#### Alicia Boole Stott’s models of sections of polytopes

Irene Polo Blanco
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I. Polo Blanco (&) Department Mathematics
, Statistics, Computation,
Faculty of Science, University of Cantabria
, Avda de los Castros s/n, 39005 Santander,
Spain
Alicia Boole Stott (1860-1940) was an amateur mathematician who worked on four-dimensional geometry. She is remembered for finding all three dimensional sections of the four- dimensional polytopes (that is, the analogues of the three-dimensional Platonic solids), and for the discovery of many of the semi-regular polytopes. In this text we give a short biography of her and explain her method to calculate the three-dimensional sections of the four-dimensional polytopes. We illustrate her results by showing pictures of her original models and drawings.
1 Introduction
The extraordinary amateur mathematician Alicia Boole
Stott is remembered above all for her contribution to
fourdimensional geometry (Fig. 1). Born in the second half of
the nineteenth century, her opportunities for formal
education were quite limited. She spent the greater part of her
adult life as a full-time wife and mother. In spite of this,
however, she achieved important results in mathematics,
thanks to her surprising capacity to visualise the fourth
dimension. Alicia Boole Stott calculated the
three-dimensional sections of regular four-dimensional polytopes (that
is, the analogies to the Platonic Solids in four dimensions)
and discovered many of the four-dimensional semi-regular
polytopes. In the course of her lifetime, she met two
important geometers of the period, P. H. Schoute and
H. S. M. Coxeter, with whom she collaborated on various
aspects of four-dimensional geometry.
2 A brief biography
Alicia Boole Stott, born in 1860 in Castle Road, Ireland,
was the third of five children of the famous logician George
Boole and his wife, Mary Everest Boole. George Boole
died in 1864 at the early age of 49, when Alicia was only
four. His wife was left with the five children, and very little
means for maintaining them. For this reason she was forced
to move to London with Alicias four sisters, while Alica
was left to live in Cork with another member of the family.
At the age of eleven she moved to London to live with her
mother and sisters. The four sisters also became important
figures of the day for various reasons. For a detailed
account of the Boole family we refer the reader to [6].
Figure 2 shows Alicia with her sisters, her mother and
several offspring.
It should be noted that the English universities of the
time did not offer degrees to women, who could only aspire
to study some of the classics of literature and other arts.
Alicias formal scientific knowledge consisted of only the
first two books of Euclid. How was it then possible for her
to obtain such surprising mathematical results during her
life? One of the reasons is undoubtedly due to the unique
atmosphere in which she grew up and the special education
she received from her mother. Mary Everest Boole was
known in her day for her peculiar ideas about education.
She wrote several books on mathematics learning, and
Fig. 1 Alicia Boole Stott (18601940). Photo
Special Collections. Reproduced by permission
Fig. 2 From left to right, top to
bottom: Margaret Stott Taylor,
Ethel L. Voynich, Alicia Boole
Stott, Lucy E. Boole, Mary E.
Hinton, Julian Taylor, Mary
Stott, Mary Everest Boole,
George Hinton, Geoffrey
Ingram Taylor, Leonard Stott.
Photo University of Bristol
Special Collections.
Reproduced by permission
believed strongly in the importance of early stimulation of
children for an effective learning of geometry and other
aspects of mathematics. During the years she lived in
London, Mary Everest Boole received numerous visitors at
home, among whom was the amateur mathematician
Howard Hinton. Hinton was a mathematics teacher, and
enormously interested in the fourth dimension. He became
famous with his book The Fourth Dimension [5], in which
the subject is treated from a philosophical point of view.
During his visits to the Boole family, Hinton used to pile
up groups of wooden blocks to try to allow the five
daughters to visualize the four-dimensional hypercube.
This greatly inspired Alice in her future work, and she soon
began surprise Hinton with her ability to visualize the
fourth dimension. Alicia contributed to writing part of the
book [4].
In 1890 Alicia married the actuary Walter Stott, with
whom she had two children, Mary and Leonard. Inspired
by Howard Hinton, Alicia Boole Stott began investigating
four-dimensional polytopes in her free time as the children
grew. At that time, Boole Stott worked completely
independently, without any contact with the scientific world,
and proved the existence of the six regular
four-dimensional polytopes. These polytopes were first listed by
Ludwig Schlafli in 1850 (published after his death in 1901
in [10]), and are four-dimensional analogues of the
threedimensional Platonic solids. The six regular
four-dimensional polytopes are the hypercube, hypertetrahedron,
hyperoctahedron, 24-cell, 120-cell and 600-cell. In addition
Fig. 3 Central section of the 600-cell. Left the model by Boole Stott; right the drawing by Schoute in [11]
to proving the existence of these polytopes, Boole Stott
calculated their three-dimensional sections and constructed
models of them in coloured cardboard.
In 1894 the Dutch geometer Pieter Hendrik Schoute
published an article [11] in which he calculated analytically the
central sections of the six regular four-dimensional
polytopes. According to Coxeter [3] Boole Stott learned about this
publication from her husband. After verifying that Schoutes
results coincided with her own, Boole Stott sent pictures of
models illustrating not only the central section of each
polytope calculated by Schoute, but the entire series. Figure 3
shows that the central sections of Boole Stott and Schoute
effectively match for the case of the 600-cell.
Quite surprised by Boole Stotts results, Schoute replied
to her immediately, proposing a collaboration that would
last for almost 20 years, until Schoutes death in 1913.
During this period, Schoute travelled to England during the
summer holidays and worked with Boole Stott on various
topics regarding the fourth dimension. Their collaboration
combined Boole Stotts extraordinary capacity to visualise
the fourth dimension with Schoutes analytical method.
The work of Boole Stott culminated in an honorary
doctorate being awarded to her by the University of Groningen
in 1914, in recognition of her contribution to
four-dimensional geometry.
After Schoutes death, Boole Stott set her mathematical
investigations aside to devote herself exclusively to
domestic life. In 1930 she resumed her work when her
nephew, the famous physicist and applied mathematician
Geoffrey Ingram Taylor, introduced her to the geometer H.
S. M. Coxeter. Although Coxeter was only 23 years old
and Boole Stott 60, they developed a close friendship and
worked together on various aspects of four-dimensional
geometry. There are no joint publications, but the
contributions of Boole Stott are known thanks to numerous
references to them in the work of Coxeter. His book Regular
Polytopes [3] also contains numerous facts about Boole
Stotts life, and together with [6] is the main source of
information about her biography. A more detailed account
of Boole Stotts life can also be found in [7].
Boole Stott published her principal mathematical results
in two articles: [1] in 1900 and [2] in 1910 (see [9] and [8]
for a detailed description of both articles respectively). In
what follows we will focus on the first of the publications
related to the sections of polytopes and the drawings and
models that Boole Stott made of these sections.
Fig. 5 Drawings of the parallel sections of the 120-cell, conserved at the University of Groningen, The Netherlands. Photo author, reproduced
by permission
Fig. 6 Models of the perpendicular sections of the 120-cell, conserved at the University of Groningen, The Netherlands. Photo author,
reproduced by permission
3 Three-dimensional sections of four-dimensional
polytopes
Boole Stotts publication of 1900, On certain series of
sections of the regular four-dimensional hypersolids
[1] is an exhaustive study of the parallel
three-dimensional sections of the six regular polytopes. These
sections are the result of intersecting three-dimensional
space with the polytope, the three-dimensional space
being parallel to one of the three-dimensional faces of
the polytope.
With regard to the method used by Boole Stott in her
article, several points deserve mention. Just as a Platonic
solid can be unfolded in a plane, a four-dimensional
polytope can be broken down into three dimensions. Once
that is done, the calculations of the sections are
considerable simplified, greatly facilitating visualisation.
Figure 4 shows one of her drawings representing part of
the unfolded hypercube (note that in the unfolding, several
of the vertices, edges, etc. appear more than once, and must
be identified in order to reconstruct the original hypercube).
A detailed description of this method from a modern point
of view can be found in [9].
Boole Stott also studied perpendicular sections of these
polytopes, which are characterised by the three-dimensional
spaces being considered perpendicular to the segment
connecting a vertex with the centre of the polytope. In particular,
she made drawings and cardboard models of sections of the
two most complex polytopes: the 120-cell and 600-cell.
Regarding the sections of the 120-cell (the polytope
composed of 120 dodecahedra), we find as many original
drawings of the parallel sections as cardboard models of
perpendicular sections made by Boole Stott. The drawings in
Fig. 5 depict the parallel sections of the 120-cell.
Fig. 7 Drawings of the unfolded planes of the perpendicular sections of the 600-cell, conserved at the University of Groningen, The
Netherlands. Photo author, reproduced by permission
Fig. 8 Models of the perpendicular sections of the 600-cell,
conserved at the University of Groningen, The Netherlands. Photo
author, reproduced by permission
Furthermore, Boole Stott built models of the perpendicular
sections of the polytope. Figure 6 shows these models, today
held in the museum of the University of Groningen.
Boole Stott also studied perpendicular sections of the
600-cell and made drawings and cardboard models of these
sections. This is the most complex polytope and consists of
600 tetrahedra. At the University of Groningen are found
Boole Stotts original drawings corresponding to the
unfolded planes of these sections (Fig. 7). We also find
models of these perpendicular sections in the museum of
the University of Groningen (Fig. 8), and at the University
of Cambridge (Fig. 9).
4 Conclusion
Alicia Boole Stott is an exceptional example of an amateur
mathematician born in the nineteenth century. Isolated
from the mathematical community, many of her
discoveries were never published. However, this isolation may have
helped Boole Stott to develop an intuitive ability to work
with the fourth dimension that was very different from the
analytical method used at the time, leading her to her
discoveries. Boole Stott is remembered today for her
Fig. 9 Models of the perpendicular sections of the 600-cell, conserved at the University of Cambridge, UK. Photo author, reproduced by
permission
outstanding contribution to the four-dimensional geometry.
Her models and drawings reflect the complexity and beauty
of her results.
Translated from the Spanish by Kim Williams.