The Princess of Polytopia: Alicia Boole Stott and the 120-cell
Alicia Boole Stott's first article, On certain series of sections of the regular four-dimensional hypersolids, published in 1900, is the basis for this month's feature column...
Tony Phillips
Stony Brook University
tony at math.sunysb.edu
Fig. 1.
A reconstruction of seven of Alicia Boole Stott's models showing
3-dimensional slices of the 120-cell. These slices go from the bottom
to the midpoint; then the first six would repeat in reverse order.
Photograph by Tony Scarlatos, Stony Brook.
A paradoxical childhood and an unusual hobby
Alicia Boole (1860-1940) grew up in the highest
intellectual circles of Victorian England, and in deep penury.
Her geneaology was excellent: Boolean Algebra and Mount Everest
are named, one for her father George Boole and the other for her
maternal uncle, Sir George Everest. But Boole married late and died
young, leaving his wife Mary with five daughters, the eldest eight,
the youngest six months old, and no money. Alicia, four at the time,
was sent to live with relatives. When she rejoined her family
they were living "in a poor, dark, dirty, uncomfortable lodging
in Marylebone" according to H. S. M. Coxeter, who collaborated
with her much later and who contributed a short biography to
"Women of Mathematics."
In London the girls, especially the younger, received little or
no formal education.
Alicia is said to have learned the first two books of Euclid, and
no more. But Mary Boole, who corresponded
with Darwin and was friends with H. G. Wells,
received many visitors, guests who could keep the
family acquainted with developments in science and society
(or at least those on the mystical fringes of both:
Coxeter characterizes them as "a continual stream of cranks").
One of the cranks had a son named Howard Hinton, a brilliant
polymath, who courted and married the eldest of the Boole daughters.
Hinton was fascinated by four-dimensional geometry.
He brought over an elaborate system of wooden cubes he had designed
to make it comprehensible, and set his
fiancée's younger sisters to work on them.
Alicia, then around 18, "gradually began to surpass him in her
ability to understand what happens in hyperspace" (Coxeter). She
worked out by herself the six regular polytopes (the 4D analogues of our
regular polyhedra) and built cardboard models of their sections
by parallel 3-planes.
Alicia was building her models just before Edwin A. Abbott's book "Flatland"
appeared (1880) and before W. I. Stringham published Regular
figures in n-dimensional space in the American Journal of
Mathematics (also 1880), but she considered her research her
own private hobby. Only towards the end of the century, when her
husband (she had married Walter Stott in 1890) showed her that figures
published by the Dutch geometer Pieter Schoute closely
resembled her own constructions, did she contact a professional
mathematician. Schoute became her enthusiastic supporter, and
arranged for the publication of her diagrams in the proceedings
of the Royal (Netherlands) Academy of Science.
Alicia Boole Stott's first article,
On certain series of sections of the regular four-dimensional
hypersolids, published in 1900,
is
the basis for this month's feature column. We will focus on her
analysis of one of the "hypersolids," the 120-cell, with
boundary consisting of 120 dodecahedra. In particular we will show how to use
her diagrams as templates to reconstruct seven of the polyhedral models
she had built some twenty years before.
Alicia left Schoute with a complete set of models for the 120-cell
and its dual, the 600-cell; these are still
on display at the University of Groningen, where Schoute taught,
and can be seen as part of Groningen's online display of
Mathematical models of surfaces.
Another set
of her models is on exhibit at The Faulkes Institute for Geometry,
Centre for Mathematical Sciences,
Cambridge University. The
Cambridge set duplicates most of the two series
of 600-cell sections at Groningen (some with different
color-coding), but comes with
a different set of drawings. Cambridge does not
seem to have any models of the 120-cell sections.
A Flatlander's views of the dodecahedron
A 120-cell is sitting in
x, y, z, w-space
with
one of its dodecahedral faces in the 3-plane {w = 0},
and the rest of it in the "upper" half-space where w > 0.
Alicia Stott examined its sections by 3-planes {w = c}, parallel
to the x, y, z-plane, as c increases from 0 to the
height of the 120-cell, approximately 6 units if the dodecahedra have
edge length 1. These sections are what would appear if
a 120-cell traveled "downwards" on a path intersecting
the space we live in.
"Flatland" explains this process by analogy with
a 2-dimensional being's perception of a 3-dimensional solid. All the
Flatlander can see is the sequence of planar figures produced as
the solid drops through Flatland. It is a useful exercise to imagine
how a Flatlander would see a dodecahedron passing face-first, vertex-first
or edge-first, because the three families of polygons
produced are exactly those we will encounter in Alicia's models.
Fig. 2. Three Flatland perspectives on the dodecahedron.
- Slices parallel
to a face. The vertices are encountered in four groups of 5.
As the slices move up, the pentagonal intersection grows.
Between the two middle sets of vertices, the slices are decagons
(not shown), with a regular decagon at the midpoint. Then another
family of pentagons, rotated with respect to the first, decreasing
to the top face.
- Slices perpendicular to a diameter joining opposite vertices.
The vertices are encountered, in groups of 1, 3, 6, 6, 3, 1,
at six levels. Between level 1 and level 2, the sections are
equilateral triangles. Between level 2 and level 3, the sections
are hexagons, with a regular hexagon (not shown) occurring along the
way. Between the middle levels, the hexagons change shape, with
another, larger, regular hexagon at the very center. The process
reverses as we slice towards the top: hexagons and then triangles.
- Slices perpendicular to the bisector of two opposite edges.
Now the vertices are encountered 2, 4, 2, 4, 2, 4, 2 at seven
levels. The first slices are very thin rectangles, becoming
a perfect square at level 2. Between levels 2 and 3 the slices
are octagons (not shown). Then hexagons between levels 3 and 5,
with a regular hexagon appearing once between 3 and 4, and again
between 4 and 5 (not shown). The first steps then reverse: octagons,
square, rectangles diminishing to a line segment.
The 120-cell sits for its picture
 Fig. 3. The 120 dodecahedra constituting the 3-dimensional boundary
of the 120-cell can be analyzed into 9 layers containing
1, 12, 20, 12, 30, 12, 20, 12, 1 solids respctively. This picture
illustrates how the layers stack up when the polytope is placed with
one solid face in x, y, z-space. One representative is shown
from each layer, with indications of how the layers
are attached. In this
orientation of the polytope the top and bottom cells
have no 4th-dimensional
extent. They appear here as the bottom yellow
pentagon and the top turquoise pentagon. Roman numerals
refer to diagrams in Alicia Boole Stott's 1900 paper.
Up through the layers
The models in this sequence were assembled with the help
of George Hart, Lucienne Pereira and Steven Anderson. They were
photographed by Tony Scarlatos. The images from Stringham's paper
were photographed by Bill Casselman. The templates from which
the models were made were photographed by Robert Sammis
from the fragile fold-out plates attached
to Stott's article. Thanks to all!
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Layer 1. Alicia's diagrams give us a 3-dimensional
perspective on the 120-cell, analogous to the first (yellow)
Flatland picture, where one face is in the inhabited plane.
The first thing the Flatlander sees is that face. The first
thing we see is a single, entire dodecahedron popping into
view. We can think of its skin as formed by the twelve
bottom pentagons of the Layer-2 solids, and color it yellow.
Assembled from Stott's Diagram VIII.
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Layer 2 consists of twelve dodecahedra attached
to the twelve faces of Layer 1. As we move up through
Layer 2, each dodecahedron gives larger and larger pentagons.
This model shows the largest, just before we encounter
the first trace of Layer 3. Assembled from Stott's Diagram
IX.
The image on the right is Fig. 18 from Stringham's 1880
American Journal paper. It shows all of Layer 2, before
the Layer 3 polyhedra have been added. Stringham projected
the entire construction into 3-space, which distorts
the geometry but makes the organization intelligible.
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Layers 2 and 3. This model is assembled from Stott's
Diagram X. One Layer 3 dodecahedron is inserted vertex-first
into each of the 20 dimples shown in Stringham's Fig. 18.
In this section, each of these polyhedra contributes a
triangle.
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The top of Layer 2; Layer 3. From Stott's Diagram
XI. This is the last we will see of Layer 2. Each of
the twelve shows its top face before disappearing. The
Layer 3 polyhedra are now contributing hexagons.
On the right is Stringham's Fig. 19. It shows how the
Layer 3 polyhedra form pentagonal pits. (At the bottom
of each pit is the top face of one of the level-2 dodecahedra)
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Layers 3 and 4. From Stott's Diagram XII. One
(turquoise-colored) dodecahedron is plugged into each
of the 12 pits shown in Stringham's Fig. 19. These new
solids are stacked directly on top of the Layer 2 doecahedra
and will also give pentagonal sections.
Stringham's Fig. 20, on the right, shows how the tops
of the Layer 4 dodecahedra form 12 plateaus, while the
tops of the Layer 3 dodecahedra appear as 20 triangular
peaks. The sides of the plateaus and the peaks form a
network of 30 four-sided troughs.
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Layers 3, 4 and 5. One (pink) Layer-5 dodecahedron
is inserted edge-first into each trough. In this section,
assembled from Stott's Diagram XIII, it contributes a square.
The Layer-3 dodecahedra make their last appearance as triangles,
while those in Layer-4 continue to contribute pentagons.
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Layers 4 and 5. In this
section, from Stott's Diagram XIV, each Layer-4 dodecahedron
shows its top face, while each of the 30 Layer-5 docecahedra
contributes its central section. This is the half-way point
of the slicing process: we have seen 1 + 12 + 20 + 12=
45 full dodecahedra and half of 30 others. As we contine
to move up thrugh the 120-cell, the sections will now become
smaller. We could use the same models in reverse order,
except yellow and turquoise would need to be interchanged
to follow the color scheme of Fig. 3.
Stringham's Fig. 21 (at right) gives a different perspective
on the end of the process. In his figure the entire Layer
5 has been added. Each trough has been transformed into
a 4-sided ridge; the Layer-5 polyhedra enclose 12 pits
(at the bottom of them you can see the last pentagons
from Layer 4); and three by three they form 20 dimples.
This solid contains 75 dodecahedra, with surface geometry
the exact opposite of the 45-dodecahedron solid shown
in his Fig. 20. Locally it is clear how the two solids
mesh together perfectly: dimple to peak, pit to plateau
and ridge to trough. Globally this matching, which produces
a topological 3-sphere, cannot be carried out in our
world.
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The templates
Alicia Boole Stott's 1900 article in the Royal Netherlands
Proceedings has 14 diagrams illustrating polygonal configurations;
the first seven relate to sections of the 600-cell, and the last seven,
numbered VIII to XIV, relate to the 120-cell. These diagrams can be used
as templates to assemble models exactly like the ones she made
herself and left in Groningen. The only difference is that the
Groningen series includes several intermediate sections for which
she did not publish diagrams. These diagrams are all exactly to
the same scale. They are published in black and white, but I
have followed the example of her actual models in color-coding
by layer.
Diagrams VIII and IX
Diagram X
Diagram XI
Diagram XII
Diagram XIII
Diagram XIV
The assembly of Diagram XIV. Note the George Hart method of
double tabs, Elmer's glue applied with a brush, and clips.
For further reading ...
Coxeter's biography of Alicia Boole Stott appears in - Grinstein, Louise and Paul J. Campbell, eds., Women of
Mathematics: A Biographical Sourcebook, Greenwood Press, Westport CT 1987.
He wrote her a shorter but more personal biographical notice, and
of course everything you need to know about the 120-cell, in
- Coxeter, H.S.M., Regular Polytopes, Dover, New York, 1973.
There is more material about Alicia and her brilliant,
eccentric family in
- MacHale, Desmond, George Boole: His Life and Work, Boole Press, Dublin 1985.
and in her nephew Sir Geoffrey Ingram Taylor's "Scientific Diversions," published on pp. 137-148 in - Higginbotham, S. W., ed., Man, Science, Learning and
Education, Rice University Studies LXIX supplement 2, 1963.
For a more systematic treatment of the problem of slicing convex
bodies in n-space, see Hermann Weyl's The elementary theory
of convex polyhedra in
- Tucker, A. W. and Harold Kuhn, Contributions to the Theory
of Games, Vol. I, Annals of Mathematics Studies #24, Princeton 1950.
Full references for the articles referred to in the text are:
- Stringham, W. I., Regular figures in n-dimensional space,
Amer. Jour. Math. 3 1880 1-14.
- Stott, Alicia Boole, On certain series of sections of the regular four-dimensional hypersolids, Verhandelingen der Koninklijke Akademie
van Wetenschappen te Amsterdam (eerste sectie) 1900.
Tony Phillips
Stony Brook University
tony at math.sunysb.edu
NOTE: Those who can access JSTOR can find some of the
papers mentioned above there. For those with access, the American Mathematical
Society's MathSciNet can be used to get
additional bibliographic information and reviews of some these materials. Some of the
items above can be accessed via the ACM
Portal, which also provides bibliographic services. |
