Stomach Punch: Boxing with Archimedes 1 It's a knockout

20 Pages·2009·1.23 MB·English

Stomach Punch: Boxing with Archimedes
ABC
1 It’s a knockout
Dissection Problems The above paradoxes naturally suggest the consideration
of dissection problems. An excellent typical example is to cut a square into 20
equal triangles, and conversely to construct a square of 20 such triangles.
There is an interesting historical example of such a problem. Two late Latin
writers, Victorinusand Fortunatianus, describe an Archimedean toycomposed
of 14 ivory polygons which fitted exactly into a square box, and they suggest
that the puzzle was to fit the pieces into the box. A recent discovery [1899
in [33]] has shown that its association with the name of Archimedes is due
to the fact that he gave a construction for dividing a square into 14 such
pieces (namely, 11 triangles, 2 scalene quadrilaterals, and one pentagon) so
that the area of each piece is a rational fraction of the area of the square. His
construction is as follows: let ABCD be the square, and E, F, G, H, the
midpoints of the sides AB, BC, CD, DA. Draw HB, HF, HC, and let J,
K, L be the mid-points of these lines; draw AKC cutting HB in M, and let
N be the mid-point of AM, and P the mid-point of BF. Draw BN. Draw AP
cutting HB in Q. Draw PJ. Draw BL, and produce it to cut DC in R. Draw
FL cutting AC in S. Draw LG. Rub out AQ and BL. The remaining lines
will give a division as required [see Figure 1(i)], each figure being an integral
multiple of 1/48th of the square. Why Archimedes propounded so peculiar a
division it is impossible to guess, but no doubt the problem has a history of
which we are ignorant.
W. W. Rouse Ball [3, p. 54]
A mathematical formula is eternal. But a winning formula, in order to stay a com-
mercial proposition, remains forever new and improved, changing with changing
tastes. Mathematical books also have their market share. So the book we know
today as Mathematical Recreations and Essays [3], by Walter William Rouse Ball
(1850–1925), has enjoyed continuing popularity for over a century, making the grade
as a landmark [31] of early modern mathematics. But Ball himself reworked it con-
siderably through frequent editions, and a more abrupt and thorough-going change
occurred when Harold Scott MacDonald Coxeter (1907–2003) took it over for the
1
11th edition in 1939. Indeed, a retrospective review [4, (b)] of the 12th (1974) edi-
tion, nowpresented asthejointworkofBallandCoxeter [4, (a)],cautioned“changes
have been so great, one should not discard earlier editions”, although being of the
view that later editions contained “much more material of real mathematical inter-
est”.
C F P B
S
Q
G L K J E
R
M
N
D H A
(i) After Suter (ii) Ausonian elephant ?
Figure 1: Stomachion
The changes over this comparatively short time frame may increase our apprecia-
tion, not only for what remains to us of the works of Archimedes (?287–212), but
also that the acknowledged difficulty of these works makes it likely that they have
been less altered at the hands of transcribers and editors than the Elements of Eu-
clid (?325–?265). The historical irony here is that presumably Ball’s account of the
Stomachion — Ostomachion or loculus of Archimedes — was considered in 1939
of lesser mathematical interest and so dropped by Coxeter. Ball’s comment that
“no doubt the problem has a history of which we are ignorant” is hardly a ring-
ing endorsement and, for all that he describes the research [33] of Heinrich Suter
(1848–1922) published in 1899 as “recent”, he did not include it in Mathematical
Recreations and Essays at his first opportunity, the 4th edition in 1905. While
saving space by not reproducing Suter’s diagram, Ball did revise the labels in the
passage quoted above, a departure in which we follow him, beginning in Figure 1(i).
Clearly Coxeter cannot be expected to have known that progress in imaging tech-
nology would make it possible to read more of the palimpsest now familiarly known
as the Archimedes Codex than had been managed when Ball wrote, and certainly
he was correct in that regard during his own long life. It was only in 2004 that
news [23] broke of a novel proposal that the Stomachion was an early exercise in
combinatorial mathematics, an interpretation since set out more accessibly in [24,
Chap. 10].
This development would surely have appealed to both Ball and Coxeter, with their
strong inclinations towards the mathematics of counting. But re-acquaintance with
Ball’s text reminds us that it had been suggested that the puzzle was to fit the
pieces back into their box long before the combinatorial considerations of [23, 24]
entered the picture. Indeed, Ball’s contempory, James Gow (1854–1923) — like
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Ball, a lawyer as well as a Fellow of Trinity College, Cambridge — had already
footnotedthis view of theStomachion in1884 inhis standard Short History of Greek
Mathematics [13, p. 243, n. 3]. By contrast, Thomas Little Heath (1861–1940), their
junior who made more of a speciality studying Greek mathematics, only speculated
rather vaguely, in his study [15, p. xxii] of the works of Archimedes, that the phrase
“loculus Archimedius” perhaps meant little more than it was cleverly made, in much
the way Archimedes’ name had become a byword for any taxing problem. However,
this comment predates Suter’s publication [33] by a couple of years — from our later
perspective, including the first publication of material from Archimedes Codex by
Johan Ludvig Heiberg (1854–1928), not to mention the latest tranche in [23, 24], it
is instructive to see how these earlier authors recycle their phraseology in expressing
a shared ignorance. Instead, when Heath came to review [17, (b), p. 51] Heiberg’s
volume [17] containing material on the Stomachion, he drew the analogy with “a
sort of Chinese puzzle”. By about the same time, Fritz Kliem (1887-1947?) was
able to add more information about the Stomachion to the German edition [16, (a)]
of Heath’s book.
(i) pieces turned over (ii) no turning over
Figure 2: The irregular hexagon
However, if it had been a “common game to put [the pieces of the Stomachion]
together again into the original square [of ivory from which they had been cut]”,
as Gow suggests, even the Archimedean difficulty of this task could hardly have
prevented players from noticing that the pieces fit together into a square in more
than one way. So, a combinatorial interpretation brings with it a puzzle of its own:
why had no one thought to mention it before? After all, it had been a commonplace
to remark, with mixed amazement and frustration, on the myriad free-form figures
that it is possible to shape with the pieces (compare Figures 1(ii) and 2). Could
it have been that there was some restriction on how the pieces were to be played?
Another traditional explanation of the Stomachion is that, although it could foster
such creativity, yet it had the more serious purpose of helping children to strengthen
their powers of memory and, more specifically, those of pattern recognition. But
unless the idea was to learn all 17,152 solutions by heart, this in turn might suggest
homing in on relatively few, perhaps just one, of those many theoretically possible
ways of completing this square.
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2 Out for the count
The principal halvings of the Stomachion board lend it something of the appearance
ofa coatofarms ona shield —intheterminologyfavoured inheraldry party per pale
for the vertical division; party per bend for the main diagonal; and party per chevron
reversed for the v-shaped cut (see Figure 6). The purpose of these divisions in
heraldry is to achieve, through marshalling and differencing the elements of the coat
ofarms,auniqueidentityforthebearerofthosearms. Naturally,theelementsinany
heraldic composition are displayed face outwards. If the pieces of the Stomachion
were fashioned in such a way as to allow obverse and reverse to be distinguished,
then presumably they too would uniformly face outwards on being placed back in
their box.
C F
S
L K
H
(i) (ii) (iii)
Figure 3: Changing orientation
As it happens, the pieces of the Stomachion board in Figure 1(i) can be slid into
position on the plane of the board to form the Ausonian elephant in Figure 1(ii)
without any being turned over. To put this observation another way, if the elephant
is to march to the right rather than to the left, all the pieces have to be turned
over. Of course, it may be that a stronger sense of pattern can be imparted to a
Stomachion dissection if pieces are allowed to be turned over than if not, as in the
case of the irregular hexagon in Figure 2 — here and in subsequent figures, pieces
that have been turned over are marked with a black spot. With an essentially
empirical observation of this kind, the argument cuts both ways. For, if some
classically described shape is found to require that some pieces be turned over in
order to achieve it in a Stomachion dissection, then that is evidence that turning
over pieces was countenanced at least in this constructive play. Still, it helps to
avoid turning over pieces in creating shapes that one of the four congruent right
triangles into which the board is partitioned is cut so that the pieces can also form
a congruent right triangle having opposite orientation (see Figures 3(i) and (ii);
other ways to reattach the pieces so that edges match are possible, too, as in Figure
3(iii)). Moreover, the sharpness of the angles of some of the pieces perhaps gives
an incentive not to turn pieces over. There may be difficulty enough to work those
angles in such materials as ebony, horn or ivory. But sliding the pieces would help
minimise wear and tear on both pieces and players.
It is instructive to compare the situation with that of Tangram, as most likely the
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(i) double square
(iii) single square
(ii) two squares
Figure 4: Tangram
“Chinese puzzle” Heath had in mind. For a start, the pieces in Tangram are all
much more well-rounded, with no sharp angles. But, in any case, only one, the
rhomboid, lacks mirror symmetry. Consequently, in order to change the orientation
of a Tangram design, it is enough to turn over this piece, while sliding the others
into place. Alternatively, we might convene a standard orientation for our Tangram
pieces andavoid turning pieces over altogether. AlthoughTangramhasthemystique
of antiquity, there seems to be little or no evidence of it before the opening in the
early 1800s of more extensive contacts between China and Europe through sea-
borne export trade (see [32]). There is some suggestion of a much older Chinese
tradition of tiles coloured only on one side. This might have helped reinforce a
seeming preference on the part of Liu Hui (220–280) to take right triangles paired
as rectangles rather than singly when commenting on the Jiu Zhang Suan Shu, a
compilation of problems already a classic in his day. For the transformation of a
single right triangle into an L-shaped trysquare (gnomon or carpenter’s square) of
equal area illustrated in Figure 5 requires pieces to be turned over, which would be
conspicuous if they were coloured only on one side, whereas this can be avoided by
working instead with right triangles paired into rectangles. This could account in
turn for using the diameter of the inscribed circle of such right triangles, not the
radius.
So, it is fairlynatural towonder how many ways there areof putting the Stomachion
pieces back together again as a square without turning any over, especially as this
is not mentioned in [23, 24]. Here we adopt the naive approach of our beginning
heraldry lesson, coupled with a little elementary group theory concerning cyclic
groups Cn of order n = 2,4 and the dihedral group D4 of order 8. We might notice
that the four congruent right triangles into which the shield is divided are marked
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(i) right triangle (ii) trysquare
Figure 5: Right triangle and trysquare
by distinctive fractions of the main diagonal, but a touch of colour adds realism.
Each rectangular half of our shield can be given a half turn in the plane of the
board independenly of the other, producing the pictorial representations in Figures
6(i)–(iv), in effect a realisation of the Klein four-group or Vierergruppe, C2 C2.
×
But the two rectangular halves may also be exchanged, to yield a second set of four
designs in Figures 6(v)–(viii), which are, however, the first set turned upside-down
in the plane of the board in reverse order, so that the whole group of eight looks
the same either way up. Thus, although each of the operations of rotating the left
rectangle (λ), rotating the right rectangle (ρ) and exchanging the two rectangles
(η) is an involution, yet the group they generate has elements of order 4, so is not
isomorphic to C2 C2 C2. Indeed, with σ = λη, as ρη = ηλ, we find that
× ×
2 4
ρ = σ = ǫ,
where ǫ is the identity operation leaving the shield unchanged, while
ρσρ−1 = σ−1.
We can therefore take ρ and σ as the generators of this group, thereby identifying
it as D4.
For that matter, any design can be rotated in this plane so that the chevron is
couched dexter or sinister, as well as upright and reversed, giving in total four
variants — the results in the case of Figure 6(i) appear in Figures 7(i)–(iv) and
provide an illustration of the cyclic group of order four, C4. Finally, there are two
pairs of congruent pieces that can be interchanged independently of one another
without turning any over, producing a further set of four variants, as illustrated in
Figures 8(i)–(iv) on starting with Figure 6(i), in what is yet another instance of
the Klein four-group. It might be noted here that, while CLG and JBP are
△ △
congruent, they have different orientations, so they can only be swapped if turning
pieces over is allowed. Each of the four patterns in Figure 7 can be developed in four
variants as in the first row of Figure 6; and then each of these 4 4 = 16 patterns
×
admits four further variants as in Figure 8. All told, there are 4 4 4 = 64 designs
× ×
that can be obtained by a combination of these ways of rearranging the elements of
the shield.
Naturally enough, in these 64 designs, the division into two rectangles is invariant.
But it is also noticeable that the v-shaped cut remains intact. However, all three
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(i) (ii) (iii) (iv)
(v) (vi) (vii) (viii)
Figure 6: An instance of D4
(i) (ii) (iii) (iv)
Figure 7: An instance of C4
principal halvings of the board are preserved only in the quartet in Figures 7(i)–
(iv). Therefore, played this way, the Stomachion puzzle might well help strengthen
our ability to reconstruct distinctive features in a pattern. Yet, even under this
comparatively restrictive regime, it is readily apparent that there is a muliplicity of
ways in which the Stomachion pieces can go back in their box.
3 Squaring off
At this stage a more pervasive doubt might creep in concerning what we know
about the Stomachion — and by no means just because the Greeks’ enthusiasm for
wrestling might leave boxing beyond the πα´λη. To begin with, when we turn to look
up the allusion Decimus Magnus Ausonius (c. 310–395) makes to the Stomachion,
we find in an appendix [11, (a), pp. 395–397] that the elephant in Figure 1(ii) is
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(i) (ii) (iii) (iv)
Figure 8: An instance of C2 C2
×
the production of the translator and editor, Hugh Gerard Evelyn-White (1884–
1924), fashioned expressly in emulation of the Tangram figures displayed in a then
recently published popular book of mathematical amusements [9, p. 43]. Evelyn-
White was a classically educated archaeologist who had excavated in Egypt (his
early death, aged forty, elicited poignant obituary notices [11, (b,c)]). So, it may
be more significant that he notes [11, (a), p. 395, n. 3] that the Arabic text setting
out the Stomachion board is unpointed, saying at first, by way of cautious gloss,
only that the board is a parallelogram and then, after the mention of right angles,
that it is a rectangle, although he continues to represent the board diagramatically
as the square in Figure 1(i). Reverting to Suter’s article from two decades earlier,
we find that Suter also allows [33, p. 494, n. 6] that the construction he presents
works for any parallelogram and that the presence of right angles is made clear only
in the course of the Arabic text. But it is clear that there is a problem with this
footnote — a copying or typesetting error, if nothing else. However, Suter may,
more seriously, have confused “twice” for “equals”: he volunteers [33, p. 498, n. 30]
that the two expressions are sufficiently close for confusion; and an unpointed text
exacerbates just such uncertainty.
For all that the text from the Archimedes Codex is more fragmentary than what is
available in Arabic, yet the evidence from the first proposition is much more clear
cut, always assuming it to apply to the Stomachion board. As Heiberg [17, pp. 416–
424] sets it up, this first proposition tackles the angles at M in the representation
of the board in Figure 1(i). Beginning with a square, the construction doubles one
side, in effect creating a double square. Eduard Jan Dijksterhuis (1892–1965), in
coming to tidy away the “varia” [8, (a), pp. 66–70, Figs. 168, 169] remaining from
his extended study of Archimedes, concurred in this construction some thirty years
later, although he does not fill out the double square as completely as Heiberg.
Butresearchcanbeatrickybusiness, faute de mieux. Apreparedmind—les esprits
pr´epar´es — is some advantage, as Pasteur appreciated, but an overly trained, or
perhaps strained, mind is apt to prove a hindrance. Hence, Paul Erd˝os (1913–1996)
improves on Pasteur with his dictum that research favours the open brain that still
allows observations to speak to it — as Erd˝os never tired in pointing out, Crookes,
although a gifted experimentalist who invented the cathode ray tube, noticed only
that these tubes were hazardous for photographic film stored in their proximity,
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C F P B
S
Q
G L E
K J
R
M
N
D H A
(i) double square (ii) single square
Figure 9: Stomachion revised
leavingittoR¨ontgentodiscover X-rays(compare[29]). Asrecountedin[30,pp.151–
152], Erd˝ostookthis storytoheart, personally holding itagainst himself thathehad
“missed”discoveringExtremalGraphTheory,butalsobecauseinhisviewR¨ontgen’s
discovery changed the whole direction of Physics, towards the development of the
Atomic bomb. So, too, if less momentously, Heiberg, and later Dijksterhuis, saw a
disparity between the content of their Greek and Arabic sources, but did not put
one and one together. In between, however, an outsider, the distinguished geologist,
Richard Dixon Oldham, FRS (1858–1936), to whom is due the inference in 1906
that the Earth has a core of determinable radius, recognised that the two sources
were brought into agreement on supposing that the Stomachion was set out on a
double square board, as in Figure9(i), rather than Suter’s square. Oldham had been
researching the history of the Rhˆone valley, when he came upon references to the
loculus of Archimedes. His letter [25, (b)] to Nature in March, 1926 brings the same
practical commonsense to this new topic as he already showed in what seems his
first contribution to the columns of Nature, another letter [25, (a)], in May, 1884,
accounting for the presence of double-storied houses and concave roofs in the area
of the Himalayas where his work with the Indian Geological Survey had taken him.
Perhaps it was exactly this quality of mind that facilitated his penetrating insights
even where his professional scientific interests were not primarily engaged, as he
always presented himself as a somewhat reluctant seismologist, despite the acclaim
accorded his research. Thus, for Harold Jeffreys (1891–1987), who capped Oldham’s
work on the Earth’s core by arguing that it must be molten, Oldham had been “the
only man I ever met who did first rate work in a subject that disinterested him”
(for further information on Oldham’s professional work, see [25, (d,e,f)]).
For Heath and Evelyn-White, Stomachion had about it something of a “Chinese
puzzle”, and Oldham’s double square makes the resemblance with the tangram
board more marked, and perhaps more marketable — aficionados of board games
might have us recall that rithmomachy was played for at least half a millennium on
a double rectangular board, usually the double chess board, but sometimes an 8 by
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14 square grid (for a brief account of this family of games, see [10, Chap. 17]; further
historical detail is provided in [22]). At all events, Oldham was clearly experienced
in fingering fault lines, and his letter in 1926 triggered a Stomachion jitter, with kits
of Stomachion pieces availablecommercially, and a write-up inThe New York Times
that August. This may have been more than fifteen minutes of fame, but, again,
the historical irony is that it did not register with Coxeter in 1939, still less with
Dijksterhuis in 1943 — indeed, a research item [25, (c)] on puzzle crazes in Nature
early in 1927 makes no mention of Oldham’s letter from only the previous year. The
retrospective postscript [8, (c), p.439]ofworkonArchimedes afterDijksterhuis does
pick up the identification of another literary allusion to the Stomachion, this time in
a more celebrated composition, De Rerum Natura by Titus Lucretius Carus (c. 99
–c. 55), the subject of a learned note [26] where diligent readers can find reference
to Oldham’s letter. But unfortunately this kind of referential memory is not in
general transitive, although Oldham’s letter has been recalled as recently as 2000
in [21, (b,c)], seemingly on account of [26] (the double square Stomachion board
is also mentioned favourably, if only in passing, in [32, (a), p. 11], copyrighted in
2001). Almost needless to say the possibility of any alternative to Figure 1(i) does
not feature in [24, Chap. 10]. Yet, this confronts us more clearly than it did for
Oldham, or even for Dijksterhuis. For, whereas Dijksterhuis printed his figures only
onthesame pageofhis originalarticle[8, (a), p.69], todaywe have them juxtaposed
side-by-side in English translation [8, (c), p. 411, Figs. 169, 170].
On the other hand, whatever Oldham’s many skills, he was not a geometer. Indeed,
he was impatient of the received geometrical details of the Stomachion board in the
interests of enhancing free-form play, revising the construction so as to make LR the
production of PL, rather than of BL as in the Arabic text — in this he reveals the
limitations of his approach, an issue to which we return in the next section. But,
having expressed concern at the practical difficulties inherent in the “very acute
angles” of Suter’s board, Oldham is naturally pleased by the relief to be gained
in this regard by stretching the board laterally into a double square. However, he
misses a more telling geometrical argument in favour of the double square board.
For, whether it be 6 ANB or 6 MNB that is obtuse depends on where N stands in
relation to the foot B′ of the perpendicular from B onto AC. On Suter’s square
board, there is no question about this, as B′ = K, the centre of the board, ensuring
that already 6 AMB is obtuse, so that 6 ANB can only be yet more obtuse. In
contrast, on the double square board in Figure 9(i), not only is B′ not identified,
6 AMB is now acute, meaning that M is on the far side of B′ to A. Thus, the
first proposition of the palimpsest is much more in contention, the question being
whether N is still between A and B′.
Sad to say, Evelyn-White’s Ausonian elephant in Figure 1(ii) does not survive the
lateral stretching of the board, although Oldham, for whom the working elephant
equipped with howdah was no doubt a familiar sight during his time in India and
Burma, musters a more than passable replacement. But there are more subtle
consequences for other constructions. For example, with the pieces from the double
square board, as indicated in Figures 10(i) and(ii), the irregular hexagon resembling
that in Figure 2 is both more elongated and requires the turning over of some pieces
— the best we can do without turning over pieces is the hexagon in Figure 10(iii)
10