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

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, Victorinus and Fortunatianus, describe an Archimedean toy composed
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 P J. Draw BL, and produce it to cut DC in R. Draw
F L 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 commercial 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 considerably 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) edition, now presented as the joint work of Ball and Coxeter [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 interest”.
F
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B
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L K
J
E
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M
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A
(i) After Suter
(ii) Ausonian elephant ?
Figure 1: Stomachion
The changes over this comparatively short time frame may increase our appreciation, 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 Euclid (?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 ringing 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 technology 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
2
Ball, a lawyer as well as a Fellow of Trinity College, Cambridge — had already
footnoted this view of the Stomachion in 1884 in his 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
of a coat of arms on a shield — in the terminology favoured in heraldry 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
of arms, a unique identity for the bearer of those arms. Naturally, the elements in any
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.
F
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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
4
(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 and avoid turning pieces over altogether. Although Tangram has the mystique
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 seaborne 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 fairly natural to wonder how many ways there are of 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
5
(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
ρ2 = σ 4 = 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
6
(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 π άλη. 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
7
(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]. EvelynWhite 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.
But research can be a tricky business, faute de mieux . A prepared mind — les esprits
préparés — is some advantage, as Pasteur appreciated, but an overly trained, or
perhaps strained, mind is apt to prove a hindrance. Hence, Paul Erdős (1913–1996)
improves on Pasteur with his dictum that research favours the open brain that still
allows observations to speak to it — as Erdős 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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F
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S
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M
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A
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(i) double square
(ii) single square
Figure 9: Stomachion revised
leaving it to Röntgen to discover X-rays (compare [29]). As recounted in [30, pp. 151–
152], Erdős took this story to heart, personally holding it against himself that he had
“missed ” discovering Extremal Graph Theory, but also because in his view Röntgen’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 Figure 9(i), rather than Suter’s square. Oldham had been
researching the history of the Rhône 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
9
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 available commercially, and a write-up in The 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] of work on Archimedes after Dijksterhuis 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
on the same page of his original article [8, (a), p. 69], today we 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 P L, 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
(i)
(ii)
(iii)
Figure 10: More irregular hexagons
with pairs of parallel sides of three different lengths, not two. Since the Stomachion
pieces from the double square board are, like the Tangram pieces, more well-rounded,
there is not the same objection to turning over pieces anyway. So, the conclusion is
that on the double square board turning over pieces would likely be allowed. As it
happens, Oldham’s version of an elephant reproduced in Figure 11(i) requires just
one piece to be turned over, the pentagon, all of whose angles are right angles or
larger. If this piece is not turned over, the result is to give the elephant a higher
tail, as in Figure 11(ii), where Oldham’s modified construction of the Stomachion
board has also been repealed.
(i) original
(ii) restored
Figure 11: Oldham’s elephant
A more serious casualty may be the combinatorial conjecture. For, again as with the
Tangram pieces in Figure 4, the double square board allows us the choice of filling in
that board or, with a nod to Socrates’s dialogue with the slave in Plato’s Meno, the
single square in Figure 9(ii). To be sure, both give rise to enumeration problems of
theoretical interest. It is just that now the recognition of how the pieces fit together
into four congruent isosceles right triangles might seem the more pressing task.
11
4
Weighing in
In this final section, we turn to an Archimedean theme that seems to have been
neglected in discussions of the Stomachion, perhaps because the texts, fragmented
as we have them, do not provide a steer in this direction. But, whichever board
the Stomachion was set out on, there is no mistaking the curious abundance of
triangle centroids, a feature absent in contrast in the Tangram dissections in Figure
4. We see in Figures 12(i) and (ii) that S is the centroid of △HCF and that
correspondingly Q is the centroid of △AF B. Moreover, S is also the centroid of
△BDC, echoed in △DBA by M (see Figure 12(iii)). Once we discover this property
of the Stomachion board, we might also notice that there are virtual centroids, too.
For the erased construction lines AQ and BL intersect BN and F H respectively in
the centroids of triangles △ABM and △BCH. It might also be remarked, for what
it is worth, that the lines on the board defining H ensure that it is the centroid of
△BKB1 , where B1 is such that A is the midpoint of BB1 ; and then, in turn, M is
the centroid of △BCB1 .
F
C
F
P
B
S
C
S
Q
K
L
B
J
M
A
H
(i)
D
(ii)
A
(iii)
Figure 12: Centroids on the Stomachion board
Now, it is sometimes said that, while Archimedes knew that the three medians of
a triangle are concurrent at the point of common trisection, no complete, strictly
geometrical proof has come down to us from the Greeks (see, for example, [1, p. 23,
n. 60a; p. 54, n. 109]. This is all the more remarkable because in Equilibrium of
Planes I.15 , the final, climactic proposition of that work, Archimedes [15, pp. 189–
190, esp. pp. 201–202] gives a stronger result locating the centre of gravity of a
uniform plane trapezial lamina. This proposition implies that the centre of gravity
of a uniform plane triangular lamina lies at the common trisection point of the
medians of the triangle, except that, in the course of his proof for the trapezium,
Archimedes [15, p. 202] makes tacit uses of the result for triangles. Moreover, after
Archimedes a certain vagueness sets in, with Heron (?10–?75), and even a writer as
late as Jordanus de Nemore (1225–1260), speaking of a point from which lines to
the vertices of a triangle trisect the triangle’s area. Archimedes’ understanding of
the way the medians trisect one another is regained by Leonardo Pisano (Fibonacci,
1170–1250). But the concurrency of the medians of a triangle can still excite interest
12
(see [14] for a recent instance and [5] for a characterisation of the medians in terms
of the triangle they form).
1
1
1
2
1
1
1
1
1
2
(i) three medians
(ii) two medians
Figure 13: Dissections of triangles by medians
Of course, the presence of several centroids on the Stomachion board might be
considered merely an artefact of the definition, phrased as it is in terms of midpoints.
But midpoints are conspicuous in Figure 4 without producing triangle centroids.
Perhaps more instructively S, as the centroid of △HCF , is defined by one pair
of medians, but looking over to the congruent triangle △AF B we find Q is the
intersection of a different pair of medians, as though to remind us of the concurrency
of the three medians. However, the most singular benefit these centroids confer
is to make short shrift to the determination discussed in the Arabic text of the
fourteen fractions into which the Stomachion board is divided, both centroids and
fractions being preserved under a lateral stretch, as from Figure 1(i) to Figure 9(i).
In the first place, the medians of a triangle divide it into six parts of equal area,
as in Figure 13(i). Next, suppressing any one of these medians leaves four regions
dividing the triangle in the proportions indicated in Figure 13(ii). Since △HCF
occupies a quarter of the Stomachion board, an immediate application of Figure
13(ii) tells us that △CSL and △F KS each account for one twentyfourth of the
board, while △CF S and the quadrilateral HLSK each take up double that. As
noted in Section 2, there are two pairs of congruent pieces with the same orientation
— △HKM with △F KS and △AHM with △CF S. So, now we know how much
of the board these two further triangles represent. A second application in △AF B
reveals that △P BQ is another twentyfourth of the board. After this, the fractions
for the remaining seven pieces fall out quickly.
Moreover, if we restore the erased construction lines for the Stomachion board, we
obtain a dissection in twenty one pieces, rather than fourteen. But the determination of their proportional areas is eased with the aid of the analogue of Figure 13(ii)
shown in Figure 14(i). A closely related observation is that the central triangle in
the dissection in Figure 14(ii) occupies one seventh of the containing triangle. This
observation is sometimes associated with Richard Phillips Feynman (1918–1988),
leading to speculation as to its origin (see [6, esp. (e)]; compare [7, p. 211], publicised in [12]). So, it may be worth remarking that Thomas Rudd (d. 1656) includes
13
1
35
5
25
25
3
27
10
18
18
10
5
1
5
42
1
(i) Stomachion ?
(ii) Feynman ?
Figure 14: Further dissections of triangles
a problem of this sort among the hundred problems appended to his Practical Geometry [28] of 1650. Rudd borrowed heavily from a Dutch source published in the
early 1600s, but his Question 67 is not to be found there (which may or may not
account for the misprint of CD for ED). However, in the way that it uses interlocking (5, 12, 13) and (8, 15, 17) Pythagorean triangles with common right angle and
helpfully scaled by a convenient factor of 7, it is not untypical of how problems of
the period were constructed.
Quest. 67.
There is a Triangle ADC, whose sides are known, AC is 84, AD 105, DC is
√18081: In this Triangle is drawn two lines out of the angles C and D to the
opposite sides, as EC and DB, these crosse one another in the Triangle in the
point F , and BD is 119, and doth divide AC into two unequall parts, whereof
AB is 56, and BC 28, In like manner doth CE 91 divide AD in unequall
parts, AE 35, and CD [sic] 70.
The Question is: How far the point F is distant from B or E?
Naturally the area of the central triangle, as indeed that of other such subdivisions,
can be computed more generally in terms of ratios in which lines drawn from the
vertices of the triangle cut opposite sides. Edward John Routh (1831–1907), in his
Treatise on Analytical Statics [27, Chap. IV, Art. 132, p. 89; 2nd ed., p. 82] of 1891,
observed how areas of this sort “often occur” (the citation in [7] leads to the later
dating of 1896 in [12]). Good Cambridge coach that he was, he made up for not
having previously met the general formula for the area of the central triangle by
presenting it in a careful footnote to his discussion of reactions at joints (compare
[7, pp. 211, 219–220] and [6, (b)]). However, about a quarter of a century after
Rudd’s book, in 1678, Giovanni Ceva (1648–1734) published the case where these
lines are concurrent, so that this triangle vanishes, in the theorem that now bears his
name. As it happens, Routh’s more general formula can be derived using a related
theorem of Menelaus of Alexandria (c. 70–140), one that Ceva also proved on his own
account; in fact, the two results, Menelaus’ and Ceva’s, are equivalent. But Routh
14
(i) Title page.
(ii) Question 67
Figure 15: Thomas Rudd’s Practical Geometry, 1650
mentions neither Ceva nor Rudd, still less Menelaus. Strangely enough, the gist of
Ceva’s theorem does not appear to have registered in English until well into the next
century, when it was set as the Prize Question [2, (a)] in the Ladies’ Diary for 1735,
communicated by Samuel Ashby, the author of The Young Analyst’s Exercise (1737;
2nd ed., 1741), a “choice collection of a hundred algebraical problems”, in much the
same expository tradition as Rudd. It seems to have taken another century, in the
1830s, about the time Routh was born, before authors in English notice that others
elsewhere show prior knowledge of it. But then, as pointed out in [18, (a), pp. 9–10],
Abū ‘Āmir Yūsuf ibn Ah.mad ibn Hūd, Al-Mu’taman (d. 1085), in Al-Andalus, had
already proved this theorem some six hundred years ahead of Ceva. Whether or not
Al-Mu’taman would have accepted Ceva’s appeal to levers in proving the result, it
might have brought a smile of recognition to Archimedes, as perhaps also to Routh,
given his applied bent — that such mechanics retains its appeal today is seen in the
neat proof of Routh’s generalisation given in [19].
It is quite another matter whether any of this really has bearing on the Stomachion
board. But it brings us to a final link between the mechanics of centres of gravity
and the angles on the double square Stomachion board. Notwithstanding Oldham’s
inclination to revise the alignment of LR, the construction as it stands meshes well
with the theme of fitting angles together to make right angles or straight lines with
which Archimedes appears concerned in exploring the board. For, on the double
square board, the tangent of 6 GLR is 1/3, and it is a curious fact that angles with
tangents 1/2, of which there are several instances in Figure 9(i), and 1/3 add to π/4.
In [20, (a)], this result was obtained, as suggested in Figure 16(i), by considering the
15
equilibrium of a uniform square lamina ABCD supported from B and loaded at A
with mass equal to that of the lamina, so that a vertical plumb line through B hangs
through the midpoint W of AK. Indeed, regarding Figure 16(ii) in the same spirit,
if a uniform right triangular lamina ABD supported at B carries a mass equal to
that of the lamina at A, the vertical plumb line though B hangs through N, the
midpoint of AM, recalling from Figure 12(iii) that M is the centroid of △BDA.
B
B
C
C
K
K
M
W
N
A
A
H
D
D
(i) square lamina
(ii) right triangular lamina
Figure 16: Weighing up angles
Following up on [20, (a)], there is a connection with Pythagorean triangles that
supplements the mechanics (compare [20, (b)]). For, if q/p is rational with 0 < q < p,
then the right triangle with legs 2pq and p2 − q2 and hypotenuse p2 + q2 has half
angles with tangents 1, q/p and (p − q)/(p + q). Thus, the tangents 1, 1/2 and 1/3
appear conspicuously in this way in that most familiar of all Pythagorean triangles,
the (3, 4, 5) triangle. On the double square Stomachion board in Figure 9(i), angles
with these tangents happen to appear at the corner G. If we look over at corner B on
the double square board, we see angles with tangents 1, 2/5 and 3/7 that also come
together as half angles in the (20, 21, 29) Pythagorean triangle. The appearance of
these almost isosceles Pythagorean triangles might evoke Greek interest in rational
approximations of the square root of two.
Even after all these centuries, sparring partners would surely be disappointed not
to go for another round. In this spirit, we leave readers to ponder a reciprocity
between the two Stomachion boards hinted at in Figure 17 on considering further
the triangle △BKB1 that slipped in at the end of the opening paragraph of this
section.
16
(i)
(ii)
(iii)
Figure 17: Another round?
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