Old Glory: a Practical Investigation Into Pattern

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Even in the United Kingdom, we are all familiar with the flag of the United States, known as the "Stars and Stripes" or "Old Glory". The US flag has a long and interesting history which goes back to colonial days before the declaration of independence, and some of this history may be found in Smith (1975). In this article I suggest that there is sufficient interest in the changing arrangements of stars in the flag over the years to spark off a mathematical investigation into the various possible patterns.

First a brief history of the origins of the flag of the United States. It must be remembered that in early days, flags were not so well defined as they are today, and variations in design and shape frequently existed. Up to 1707, the flag of the Honourable East India Company consisted of red and white stripes of equal width, usually thirteen in number, and with the red cross on a white ground of St George in a rectangular canton (the canton is the upper comer of the flag next to the staff, usually the upper left hand corner in pictures). The depth of this canton seems to have been the top five stripes. From 1676, following complaints by the famous diarist Samuel Pepys, the Com- pany's ships were only allowed to display this flag beyond the island of St Helena in the South Atlantic and in eastern waters; nearer home they were obliged to use the red ensign as prescribed in a proclamation of 1674. In 1707, following the union with Scotland, the cross of St George was replaced by the then new union flag which included the white saltire (diagonal cross) of St Andrew on a blue ground, but which did not then contain the red saltire of a St Patrick for Ireland. The union flag in the canton then seems to have usually had a depth equal to the first six of the thirteen stripes. In 1801, after the union with Ireland, the red saltire of St Patrick was added to the union flag to give it its present form and the canton of the East India Company's flag was altered accordingly (Wilson, 1986). By a curious coincidence, an almost identical flag to that of the East India Company was adopted unofficially by the American colonies in 1775, with the then current form of a the union flag indicating their status as British colonies (Fig l(a)). The one difference was that the canton, while still containing the union flag, now occupied the first seven of the thirteen stripes. There seems to be no particular a reason for the thirteen stripes in the East India Company's flag (and other numbers of stripes are found), but in the new flag, the stripes symbolised the thirteen colonies. The fact that there were two virtually identical flags during the years 1775 to 1777 seems to be just a curious coincidence. Since the East India flag was confined to practical distant waters, there was unlikely to be much confusion at first, but with the development of American trade around Cape Horn it could have been confusing in later years if both had remained the same. However the declaration of independence in 1776 meant that the British flag was no longer acceptable to the American states as part of their M: own, and the result was that in 1777 the union flag in the canton was replaced by thirteen white stars on a blue ground (Fig l(b)). The reason for the thirteen stars, as for the thirteen stripes, was to symbolise the thirteen states in 1aa the original union. Problems soon arose with the elevation to statehood of Vermont (1791) and Kentucky (1792). The flag was not in fact changed until 1795, when one with fifteen stars and fifteen stripes was substituted, the canton now occupying the first eight stripes (Fig l(c)). In spite of the creation of five more states (in 1796, 1803, 1812, 1816 and 1817), no 111t0 further official changes were made until a flag with 20 stars but only thirteen stripes to commemorate the original colonies was instituted from 4 July 1818 (Fig 4). One is by Keith Selkirk, tempted to speculate that the return to thirteen stripes University of Nottingham might have been because American seamstresses objected to sewing yet more stripes! Since then extra stars have

42 Mathematics in School, March 1992 (a)1775-1777 (b) 1777-1795

Fig. 1 Flags of the United States: (Wilson, p. 74)

been added on the 4 July following the creation of each has been found with staggered rows of 3/2/3/2/3 (the new state, but the stripes have remained at the original commonest arrangement, Fig 1(b) or 4/5/4, or as twelve thirteen. stars in a circle with the thirteenth at the centre or as The mathematical interest in all this lies in the arrange- thirteen stars in a circle (Fig. 2(b)), as well as at least two ment of stars to produce a pattern. There are various other arrangements (Figs 2(a) and 2(c)). (The circular possibilities for symmetrical designs, but in most cases the arrangement has also been adopted by the European Com- stars have been arranged in rows, which is the most obvious munity and by Malaysia, which has a very similar flag to possibility when their number becomes large. These rows the American one). can either be in rectangular formation or can be staggered, Those of us who are older might remember the forty- that is in triangular formation. The 13 star flag, for example eight star flag which was in use until 3 July 1959 had six

(a) unofficial variant 1777 (b) infrequent variant

Fig. 2 Var. of. the. thirteen star flag (smith pp. 41, 41, 32

Mathematics in School, March 1992 43 It is interesting to note that the designers of the curren, flag preferred to use a staggered pattern of stars rather than have five rows of ten stars which does not fit so well into the required space. Perhaps the maximum difference between the numbers of stars in the rows and in the columns might be three, or possibly the number of columns should not exceed the number of rows by more than 50%. Such numbers could be called "near-square" numbers, a near-square number being one with two factors which do not differ by more than three or with two factors of which one is not 50%/ more than the other, according to which of the two choices is made. This constraint is an example of the need in design to overcome both mathematical and artistic difficulties. With staggered rows there are rather more possibilities, Fig. 3 Vermont flag of 1804-1837 with 17 stars and 17 stripes (recon- structed). (Smith; p. 225) and for odd numbers of rows they come in pairs according to whether the first row is a long one or a short one, for example 32 can be produced by rows of 6/7/6/7/6 and 33 by 7/6/7/6/7 (Compare figs 3 and 6). The numbers which occur in these patterns are of the type pq + (p+ 1)(q- 1) or rows of eight stars placed in a rectangular pattern. Similar pq+(p-1)(q-1), where p and q are sufficiently large arrangements occurred several times in earlier years positive integers and have certain artistic constraints. (Figs 4(b) and 5(a). For one year there were then forty- Where there are odd numbers of rows, these patterns have nine stars, and from 4 July 1960 the flag has had fifty stars the advantage of both horizontal and vertical reflective in staggered rows of 6/5/6/5/6/5/6/5/6 (Fig 6). symmetry. Between 1818 and 1959 there have been many variations There is also, of course the possibility of staggering with on these themes. It would serve little purpose to detail all the same number in each row, for example 30 could be the patterns through the years, even if I was able to find illustrations of them, but there have at various times been 6/6/6/6/6 (Figs l(c) and 4(a)). This design lacks vertical reflective symmetry (Fig l(c)), and for even numbers of flags with the numbers of stars as follows, each from the rows horizontal reflective symmetry as well (Fig 4(a)), relevant fourth of July: although the latter case has rotational symmetry of order 21 (1819) 30 (1848) 38 (1877) 2. It is really a variation on the rectangular pattern, but it 23 (1820) 31 (1851) 43 (1890) does allow greater flexibility to the overall rectangular 24 (1822) 32 (1858) 44 (1891) shape of the block of stars. 25 (1836) 33 (1859) 45 (1896) It is difficult to find suitable arrangements for some 26 (1837) 34 (1861) 46 (1908) numbers, for example between 20 and 60 the really difficult 27 (1845) 35 (1863) 48 (1912) ones are 21, 29, 34, 47 and 57. Some ingenuity will be 28 (1846) 36 (1865) 49 (1959) needed to devise suitable arrangements for these numbers. 29 (1847) 37 (1867) 50 (1960) Some other numbers such as 31 require rather more rows than looks correct, in this case 4/5/4/5/4/5/4. Curiously the It is much more interesting to set a project to design only one of the really difficult numbers which must have suitable patterns for these numbers of stars, and this could been solved in practice is 21 (though this might have been easily be linked with a history project on the development done by three rows of seven). I have not seen any illus- of the United States. There are a number of general trations of how this flag was arranged in practice and I observations which can be made. shall leave other possible solutions to readers (or their If rectangular patterns only are investigated, then the pupils). If the worst comes to the worst, then other patterns prime numbers can only be done as single rows, thus can be tried, (Figs 2(a), 2(c) and 5(b), and obviously various emphasising their particular properties. Two rows of stars designers have felt the need to show their originality in are unlikely to be acceptable as a suitable pattern, and even these three examples. three or four rows would look wrong with larger numbers The reference above to the difficulties of seamstresses is of stars, so that long, narrow rectangles are not acceptable. not entirely without relevance, since the costs of continually

(a) staggered rows (4 Jul to 10 Sep 1818) (Smith p. 38) (b) rectangular pattern (from Sep 1818) (Wilson p. 51)

Fig. 4 Two varieties of the 20 star flag

44 Mathematics in School, March 1992 (a) Florida flag 1846-47

jrgo

(b) Cavalry guidon (1863-65)

Fig. 5 Later patterns: (Smith, pp. 85, 38, 105)

producing new flags with such a complicated design must If you have any boxes of surplus gummed good-conduct have been considerable, as flags were sewn from different stars, your classes could have a fine time making designs coloured cloths rather than printed. It is likely, therefore and sticking the stars on, but most of us will not wish to that many users, for example shipowners, only replaced be so profligate with our use of materials. Nor will we old flags when they were worn out and continued to use want pupils to spend a lot of time on drawing the thirteen out-of-date designs for several years after the introduction stripes in Old Glory, or carefully drawing large numbers, of each new one. Alternatively, non-regular patterns were of five-pointed stars. But this does not stop us from working used by inserting odd home-made stars where there was out possible patterns, and to do this the easiest way is space for them. Wilson (1986) has a good example which probably to use spotty paper of both square and triangular gives evidence of the latter. On pages 70 and 71 there is a types. Probably isosceles triangle paper is slightly prefer- reproduction of a flag chart which was published by R. H. able to equilateral triangle. The spots used in any given Laurie in 1842. Such charts were used by mariners, for pattern can then be emphasised, possibly with a felt-tipped example, for recognition purposes. Here the flag is shown pen, the border being inserted round the spots. with 26 stars, which at the time of publication would have If you want to find out more of the history of the United been correct from 1837. There are four rectangular rows States flag, then you should read Smith, which also has of six stars as would presumably have been correct for the many fine colour illustrations. Apart from this I hardly flag from 1822 to 1836, and two stars have clearly been need to give any tips about how to set about organising added to the left of this rectangle between the first and your next project. second rows and between the third and fourth rows to With the emphasis in the National Curriculum on cross- account for the additional stars which were added in 1836 curricular work, this could be an ideal project for a class and 1837 respectively. Clearly the publishers wished to which is studying the growth and development of the avoid expensive alterations to the original printing plates. United States as part of their history programme, as well as having useful links with geography. There are oppor- tunities for the preparation of colourful joint displays, the mathematics contribution might include a section on pattern, with the each individual flag illustrating it drawn by a different group of pupils in order to avoid too much tedious work, this could then be accompanied by the relevant maps for the various periods. As you may have noticed there are also simple mathematical developments into early algebra, symmetry, prime numbers and factors which can act as suitable reminders of the work contained in these topics. If you are still short of ideas, what about the problem of how to draw a five- pointed star accurately?

References Smith, Whitney (1975). The Flag Book of the United States. Second Edition. William Morrow, New York. I Fig. 6 Current US flag. (Smith, frontispiece) Wilson, Timothy (1986). Flags at Sea. HMSO, London.

Mathematics in School, March 1992 45