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How to arrange any number of stars on the U.S. flag
Last modified: 2016-04-14 by rick wyatt
Keywords: united states | stars |
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[U.S.] image by Joe McMillan, 6 May 2003
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See also:
* Stars on U.S. flags
* United States of America
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How can the US national flag canton be patterned for a given number
of stars? Of course we know that both legally and aesthetically you
can chose whichever pattern you like, but usually this question
expects an answer restricted to the regular patterns of ordered rows
as in official use since 1912.
There are four different resulting patterns. For any given final
number of stars (and considering only vexillographically realistic
parameters - e.g. not things like one row of 51 stars should PR, or
DC, or LI, or WE become a state), either several of these four
patterns can be used, or only one, or even none. These four patterns
are:
* The plain pattern (q[?]), made of a number of rows with identical
number of stars each - usually these show in a rank-and-file
arrangement, as in the famous 48-star flag (1912-1959), with six
rows with eight stars on each. To make room for bigger stars,
however, each other row may be shifted sideways, resulting in a
zigzag pattern (and making the whole pattern to look "pointy" and
"rounded" at opposite corners) without changing anything about
the numbers; such is the case of the official version of the
49-star flag (1959-1960), with seven zigzag rows of seven stars
each.
* The staggered pattern with even rows (e[?]), made of an even number
of rows in which every other row has one star less than (or more)
the previous row (the whole will look either "pointy" above and
"rounded" at the bottom, or the opposite - numbers don't change);
such is the 45-star flag (1896-1908) we show at with six rows and
eight stars on the top row.
* The staggered pattern with odd rows and "pointy" corners (p[?]),
made of an odd number of rows in which every other row has one
star less than the previous row and the top row have one star
more than the second row (or the bottom row has one star more
than the next-to-last row), as in the current flag, with nine
rows and six stars on the top one. U.S. flag (1960-): p[?](6;5) =
50 stars
* The staggered pattern with odd rows and "round" corners (r[?]),
made of an odd number of rows in which every other row has one
star more than the previous row and the top row have one star
less than the second row (or the bottom row has one star less
than the next-to-last row), as in the 13-star John Shaw flag,
with three rows and four stars on the top one.
To put these into formulae that will crunch our data and spit out
results, lets consider two numbers, which we'll call w and h: The
number of stars in the longer rows (1st or 2nd row, depending if the
pattern is "pointy" or "rounded") is w, while h is the number of
longer rows (that's the number of rows for the simple rank-and-file
pattern).
The given examples can be expressed thus:
Official 1912-1959 flag : q[?]s(8;6) = 48 stars
Official 1959-1960 flag : q[?](7;7) = 49 stars
Reported 1896-1908 flag : e[?](8;3) = 45 stars
John Shaw flag (1776) : r[?](5;1) = 13 stars
The formulae for each of these functions follows:
q[?](w;h) = wh
e[?](w;h) = wh+h(w-1)
p[?](w;h) = wh+(w-1)(h-1)
r[?](w;h) = wh+(w-1)(h+1)
Note that I used above q[?]s and q[?] to add to the star and row count
also a description of the pattern arrangement (simple and zigzag),
but they share the same algebraic definition - simple multiplication.
Note also that I refer to rows and not to columns because the US flag
has a horizontally oblong canton - should you need to describe a
vertical flag in the same terms, just swap w and h around.
As for the answer to our question - how to design the canton of a
flag with 51, 53, 58, 69 and whatever number of stars? (a prominent
matter in many science fiction settings...) - it is simple to come up
with a list of numbers, by just iterating the formulae above for
integers. However, as said, some arrangements would not fit a
sensible design for an U.S. flag, and I tried to weed them out: I
limited the iterations to values of w between h and 5h (which still
allows for really too oblong arrangements for higher values...) and
listed only totals up to 200 stars, allowing for really aggressive
Manifest Destiny scenarios and/or Balkanization within the current
borders.
1 star e[?](1;1) p[?](1;1) q[?](1;1) r[?](1;1)
2 stars p[?](1;2) q[?](1;2)
3 stars e[?](1;2) p[?](1;3) q[?](1;3)
4 stars p[?](1;4) q[?](1;4) q[?](2;2) r[?](1;2)
5 stars e[?](1;3) p[?](1;5) p[?](2;2) q[?](1;5)
6 stars e[?](2;2) q[?](2;3)
7 stars e[?](1;4) r[?](1;3) r[?](2;2)
8 stars p[?](2;3) q[?](2;4)
9 stars e[?](1;5) q[?](3;3)
10 stars e[?](2;3) q[?](2;5) r[?](1;4)
11 stars p[?](2;4)
12 stars q[?](2;6) q[?](3;4) r[?](2;3)
13 stars p[?](3;3) r[?](1;5)
14 stars e[?](2;4) p[?](2;5) q[?](2;7)
15 stars e[?](3;3) q[?](3;5)
16 stars q[?](2;8) q[?](4;4)
17 stars p[?](2;6) r[?](2;4) r[?](3;3)
18 stars e[?](2;5) p[?](3;4) q[?](2;9) q[?](3;6)
20 stars p[?](2;7) q[?](2;10) q[?](4;5)
21 stars e[?](3;4) q[?](3;7)
22 stars e[?](2;6) r[?](2;5)
23 stars p[?](2;8) p[?](3;5)
24 stars q[?](3;8) q[?](4;6) r[?](3;4)
25 stars p[?](4;4) q[?](5;5)
26 stars e[?](2;7) p[?](2;9)
27 stars e[?](3;5) q[?](3;9) r[?](2;6)
28 stars e[?](4;4) p[?](3;6) q[?](4;7)
29 stars p[?](2;10)
30 stars e[?](2;8) q[?](3;10) q[?](5;6)
31 stars r[?](3;5) r[?](4;4)
32 stars p[?](4;5) q[?](4;8) r[?](2;7)
33 stars e[?](3;6) p[?](3;7) q[?](3;11)
34 stars e[?](2;9)
35 stars q[?](5;7)
36 stars e[?](4;5) q[?](3;12) q[?](4;9) q[?](6;6)
37 stars r[?](2;8)
38 stars e[?](2;10) p[?](3;8) r[?](3;6)
39 stars e[?](3;7) p[?](4;6) q[?](3;13)
40 stars q[?](4;10) q[?](5;8) r[?](4;5)
41 stars p[?](5;5)
42 stars q[?](3;14) q[?](6;7) r[?](2;9)
43 stars p[?](3;9)
44 stars e[?](4;6) q[?](4;11)
45 stars e[?](3;8) e[?](5;5) q[?](3;15) q[?](5;9) r[?](3;7)
46 stars p[?](4;7)
47 stars r[?](2;10)
48 stars p[?](3;10) q[?](4;12) q[?](6;8)
49 stars q[?](7;7) r[?](4;6) r[?](5;5)
50 stars p[?](5;6) q[?](5;10)
(this is where the fun begins)
51 stars e[?](3;9)
52 stars e[?](4;7) q[?](4;13) r[?](3;8)
53 stars p[?](3;11) p[?](4;8)
54 stars q[?](6;9)
55 stars e[?](5;6) q[?](5;11)
56 stars q[?](4;14) q[?](7;8)
57 stars e[?](3;10)
58 stars p[?](3;12) r[?](4;7)
59 stars p[?](5;7) r[?](3;9)
60 stars e[?](4;8) p[?](4;9) q[?](4;15) q[?](5;12) q[?](6;10) r[?](5;6)
61 stars p[?](6;6)
63 stars e[?](3;11) p[?](3;13) q[?](7;9)
64 stars q[?](4;16) q[?](8;8)
65 stars e[?](5;7) q[?](5;13)
66 stars e[?](6;6) q[?](6;11) r[?](3;10)
67 stars p[?](4;10) r[?](4;8)
68 stars e[?](4;9) p[?](3;14) p[?](5;8) q[?](4;17)
69 stars e[?](3;12)
70 stars q[?](5;14) q[?](7;10)
71 stars r[?](5;7) r[?](6;6)
72 stars p[?](6;7) q[?](4;18) q[?](6;12) q[?](8;9)
73 stars p[?](3;15) r[?](3;11)
74 stars p[?](4;11)
75 stars e[?](3;13) e[?](5;8) q[?](5;15)
76 stars e[?](4;10) q[?](4;19) r[?](4;9)
77 stars p[?](5;9) q[?](7;11)
78 stars e[?](6;7) q[?](6;13)
80 stars q[?](4;20) q[?](5;16) q[?](8;10) r[?](3;12)
81 stars e[?](3;14) p[?](4;12) q[?](9;9)
82 stars r[?](5;8)
83 stars p[?](6;8)
84 stars e[?](4;11) q[?](6;14) q[?](7;12) r[?](6;7)
85 stars e[?](5;9) p[?](7;7) q[?](5;17) r[?](4;10)
86 stars p[?](5;10)
87 stars e[?](3;15) r[?](3;13)
88 stars p[?](4;13) q[?](8;11)
90 stars e[?](6;8) q[?](5;18) q[?](6;15) q[?](9;10)
91 stars e[?](7;7) q[?](7;13)
92 stars e[?](4;12)
93 stars r[?](5;9)
94 stars p[?](6;9) r[?](3;14) r[?](4;11)
95 stars e[?](5;10) p[?](4;14) p[?](5;11) q[?](5;19)
96 stars q[?](6;16) q[?](8;12)
97 stars r[?](6;8) r[?](7;7)
98 stars p[?](7;8) q[?](7;14)
99 stars q[?](9;11)
100 stars e[?](4;13) q[?](10;10) q[?](5;20)
101 stars r[?](3;15)
102 stars e[?](6;9) p[?](4;15) q[?](6;17)
103 stars r[?](4;12)
104 stars p[?](5;12) q[?](8;13) r[?](5;10)
105 stars e[?](5;11) e[?](7;8) p[?](6;10) q[?](5;21) q[?](7;15)
108 stars e[?](4;14) q[?](6;18) q[?](9;12)
109 stars p[?](4;16)
110 stars q[?](10;11) q[?](5;22) r[?](6;9)
111 stars p[?](7;9)
112 stars q[?](7;16) q[?](8;14) r[?](4;13) r[?](7;8)
113 stars p[?](5;13) p[?](8;8)
114 stars e[?](6;10) q[?](6;19)
115 stars e[?](5;12) q[?](5;23) r[?](5;11)
116 stars e[?](4;15) p[?](4;17) p[?](6;11)
117 stars q[?](9;13)
119 stars e[?](7;9) q[?](7;17)
120 stars e[?](8;8) q[?](10;12) q[?](5;24) q[?](6;20) q[?](8;15)
121 stars q[?](11;11) r[?](4;14)
122 stars p[?](5;14)
123 stars p[?](4;18) r[?](6;10)
124 stars e[?](4;16) p[?](7;10)
125 stars e[?](5;13) q[?](5;25)
126 stars e[?](6;11) q[?](6;21) q[?](7;18) q[?](9;14) r[?](5;12)
127 stars p[?](6;12) r[?](7;9) r[?](8;8)
128 stars p[?](8;9) q[?](8;16)
130 stars p[?](4;19) q[?](10;13) r[?](4;15)
131 stars p[?](5;15)
132 stars e[?](4;17) q[?](11;12) q[?](6;22)
133 stars e[?](7;10) q[?](7;19)
135 stars e[?](5;14) q[?](9;15)
136 stars e[?](8;9) q[?](8;17) r[?](6;11)
137 stars p[?](4;20) p[?](7;11) r[?](5;13)
138 stars e[?](6;12) p[?](6;13) q[?](6;23)
139 stars r[?](4;16)
140 stars e[?](4;18) p[?](5;16) q[?](10;14) q[?](7;20)
142 stars r[?](7;10)
143 stars p[?](8;10) q[?](11;13)
144 stars q[?](6;24) q[?](8;18) q[?](9;16) r[?](8;9)
145 stars e[?](5;15) p[?](9;9)
147 stars e[?](7;11) q[?](7;21)
148 stars e[?](4;19) r[?](4;17) r[?](5;14)
149 stars p[?](5;17) p[?](6;14) r[?](6;12)
150 stars e[?](6;13) p[?](7;12) q[?](10;15) q[?](6;25)
152 stars e[?](8;10) q[?](8;19)
153 stars e[?](9;9) q[?](9;17)
154 stars q[?](11;14) q[?](7;22)
155 stars e[?](5;16)
156 stars e[?](4;20) q[?](6;26)
157 stars r[?](4;18) r[?](7;11)
158 stars p[?](5;18) p[?](8;11)
159 stars r[?](5;15)
160 stars p[?](6;15) q[?](10;16) q[?](8;20)
161 stars e[?](7;12) q[?](7;23) r[?](8;10) r[?](9;9)
162 stars e[?](6;14) p[?](9;10) q[?](6;27) q[?](9;18) r[?](6;13)
163 stars p[?](7;13)
165 stars e[?](5;17) q[?](11;15)
166 stars r[?](4;19)
167 stars p[?](5;19)
168 stars e[?](8;11) q[?](6;28) q[?](7;24) q[?](8;21)
170 stars q[?](10;17) r[?](5;16)
171 stars e[?](9;10) p[?](6;16) q[?](9;19)
172 stars r[?](7;12)
173 stars p[?](8;12)
174 stars e[?](6;15) q[?](6;29)
175 stars e[?](5;18) e[?](7;13) q[?](7;25) r[?](4;20) r[?](6;14)
176 stars p[?](5;20) p[?](7;14) q[?](11;16) q[?](8;22)
178 stars r[?](8;11)
179 stars p[?](9;11)
180 stars q[?](10;18) q[?](6;30) q[?](9;20) r[?](9;10)
181 stars p[?](10;10) r[?](5;17)
182 stars p[?](6;17) q[?](7;26)
184 stars e[?](8;12) q[?](8;23)
185 stars e[?](5;19) p[?](5;21)
186 stars e[?](6;16)
187 stars q[?](11;17) r[?](7;13)
188 stars p[?](8;13) r[?](6;15)
189 stars e[?](7;14) e[?](9;11) p[?](7;15) q[?](7;27) q[?](9;21)
190 stars e[?](10;10) q[?](10;19)
192 stars q[?](8;24) r[?](5;18)
193 stars p[?](6;18)
194 stars p[?](5;22)
195 stars e[?](5;20) r[?](8;12)
196 stars p[?](9;12) q[?](7;28)
198 stars e[?](6;17) q[?](11;18) q[?](9;22)
199 stars r[?](10;10) r[?](9;11)
200 stars e[?](8;13) p[?](10;11) q[?](10;20) q[?](8;25)
Antonio Martins-Tuvalkin, 5 July 2012