tag:blogger.com,1999:blog-6464949251805178422024-03-13T20:12:34.609-07:00The Bliss Science MuseumMath, Science and Technology tidbits written for the average person.Daniel Edwinhttp://www.blogger.com/profile/09343146036341064899noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-646494925180517842.post-82150655391986012382015-09-04T14:24:00.000-07:002015-09-04T14:47:49.219-07:00Geometry Puzzle - Grid and Circle - SolutionWinner: Nanpeng! <br />
<br />
Solution:<br />
r = √12.5 = 5/2 * √2<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkviWtDvPw3j8KHOoPGHL85-R7gF96WaGlAvfIqoGAiuk840cJYmxw_PLrthQSZhfv-56DuP3ypJFBo31vLbypfC7PSUXLzKFGXjoT2xFvL4MYr4CtRTQwsUkcV240SBYY5kJXkRkWx_E/s1600/circle5overradical2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkviWtDvPw3j8KHOoPGHL85-R7gF96WaGlAvfIqoGAiuk840cJYmxw_PLrthQSZhfv-56DuP3ypJFBo31vLbypfC7PSUXLzKFGXjoT2xFvL4MYr4CtRTQwsUkcV240SBYY5kJXkRkWx_E/s1600/circle5overradical2.png" /></a></div>
<br />
The first trick to finding this solution is to realize that the circle
does not need to be centered on a grid intersection. <br />
<br />
To understand how you might solve problems like this, you can start by understanding this circle:<br /><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnhRQ9EFPpQulby9CmtakgbPWRepIXkTGbEMvzvWt_GjPvhV1uHIhs0h580-Q1zOJ6H6JzPUpwq9ZfEaI98gSiI_8wNhP5BNT0my8WL_Yuo-SIuYdDlLcXdT9v76f4beXGpVtXnmsOFBo/s1600/circle5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnhRQ9EFPpQulby9CmtakgbPWRepIXkTGbEMvzvWt_GjPvhV1uHIhs0h580-Q1zOJ6H6JzPUpwq9ZfEaI98gSiI_8wNhP5BNT0my8WL_Yuo-SIuYdDlLcXdT9v76f4beXGpVtXnmsOFBo/s1600/circle5.png" /></a></div>
<br />
To me, this is the most obvious potential solution. The circle passes through 12 grid intersections, and so is a candidate for the solution to the problem. This circle is based on the well known 3-4-5 triangle. Because 3² + 4² = 5², the distance from the center to any of the 8 points near the corners is 5, the same as the distance to the 4 points near the edges.<br />
<br />
The solution is actually just a rotated and shrunken version of this. Observe: <br />
<br />
<br />
<br /><div class="separator" style="clear: both; text-align: center;">
</div>
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</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA-4JMROxwhrfX777_409vQoHxTZzQpOb3kV-vTRVLv63IUYV8p0Z6GPBjaym52I2keAQZghBUO1R5G0FcfP4qrnJYdE3X_baS6VGx_L5_ftZzQkrUpXIzKb_pIoFn0ttbh0yDgjeUXN4/s1600/how2solve.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA-4JMROxwhrfX777_409vQoHxTZzQpOb3kV-vTRVLv63IUYV8p0Z6GPBjaym52I2keAQZghBUO1R5G0FcfP4qrnJYdE3X_baS6VGx_L5_ftZzQkrUpXIzKb_pIoFn0ttbh0yDgjeUXN4/s1600/how2solve.png" /></a></div>
This is the r=5 circle from above. I have overlaid a diagonal grid in
green. You can see that if you look only at the green grid, this circle
is identical to the solution.<br />
<br />
There are other ways you could find the actual solution. One of them is to realize that 7² + 1² = 5² + 5². The points near the corners are 5 half-units from center of the circle horizontally, and 5 half-units vertically as well. The points near the edges are 7 half-units one way and 1 half-unit the other. Therefore, they are all the same distance from the center. This is actually the way I did it when I invented the puzzle.<br />
<br />
<br />
<br />
<span style="font-size: large;">Additional Information</span><br />
<br />
I used my mad programming skills to generate a list of circles with notable point counts (with radii less than 10000). The data is below. I considered only circles centered on grid intersections, in the middle of grid squares, or in the middle of grid edges (not pictured above). Each entry in the list looks like this:<br />
<br />
<span style="font-family: "Courier New",Courier,monospace;"> point count: radius</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> x1, y1</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> x2, y2</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> ...</span><br />
(Mirror-image coordinates are not listed. For instance, the circle with a point count of 4 and radius of 1 actually has these points: (0,1), (0,-1), (1,0) and (-1,0). I only listed one of these since the rest can be easily inferred.) <br />
(for circles centered on squares or edges, the coordinates should be divided by two to get actual non-integer coordinates)<br />
<br />
A circle is listed if and only if it has a higher point count than all smaller circles in that category.<br />
<br />
A couple interesting things are visible in the data:<br />
<ol>
<li>For every circle centered on a grid intersection, there is a corresponding circle centered in the middle of a square. (they are shown side-by-side in the table) This circle has the same point count but the radius is smaller by a factor of the square root of two.</li>
<ul>
<li>The points in these circles are related on a sum-and-difference basis. So, for the 3-4-5 circle from above, the points are 4,3 and 5,0. The sum of 4 and 3 is 7, and the difference of 4 and 3 is 1. So, 7,1 is a point in the corresponding circle. Similarly, the sum of 0 and 5 is 5 and the difference of 0 and 5 is 5. So, 5,5 is a point in the corresponding circle. </li>
</ul>
<li>For every circle centered on a grid intersection, there is a corresponding circle centered on the middle of an edge. This circle has half the point count and half the radius.</li>
<li>The point counts do not increase by 4 each time. The pattern of increasing point counts appears to be irregular. </li>
</ol>
The bottom line is that circles centered on grid squares will always win in puzzles like this one. <br />
<br />
<br />
<span style="font-family: "Courier New", Courier, monospace;">Circles centered on Circles centered on Circles centered on<br />
grid intersections: grid squares: grid edges:<br />
4: 1.00000 4: 0.70711 2: 0.50000<br />
1, 0 1, 1 1, 0<br />
8: 2.23607 8: 1.58114 4: 1.11803<br />
2, 1 3, 1 2, 1<br />
12: 5.00000 12: 3.53553 6: 2.50000<br />
4, 3 5, 5 4, 3<br />
5, 0 7, 1 5, 0<br />
16: 8.06226 16: 5.70088 8: 4.03113<br />
7, 4 9, 7 7, 4<br />
8, 1 11, 3 8, 1<br />
24: 18.02776 24: 12.74755 12: 9.01388<br />
15, 10 19, 17 15, 10<br />
17, 6 23, 11 17, 6<br />
18, 1 25, 5 18, 1<br />
32: 33.24154 32: 23.50532 16: 16.62077<br />
24, 23 37, 29 24, 23<br />
31, 12 41, 23 31, 12<br />
32, 9 43, 19 32, 9<br />
33, 4 47, 1 33, 4<br />
36: 65.00000 36: 45.96194 18: 32.50000<br />
52, 39 65, 65 52, 39<br />
56, 33 79, 47 56, 33<br />
60, 25 85, 35 60, 25<br />
63, 16 89, 23 63, 16<br />
65, 0 91, 13 65, 0<br />
48: 74.33034 48: 52.55949 24: 37.16517<br />
55, 50 81, 67 55, 50<br />
62, 41 87, 59 62, 41<br />
70, 25 93, 49 70, 25<br />
71, 22 95, 45 71, 22<br />
73, 14 103, 21 73, 14<br />
74, 7 105, 5 74, 7<br />
64: 166.20770 64: 117.52659 32: 83.10385<br />
120, 115 185, 145 120, 115<br />
132, 101 191, 137 132, 101<br />
141, 88 205, 115 141, 88<br />
144, 83 215, 95 144, 83<br />
155, 60 227, 61 155, 60<br />
160, 45 229, 53 160, 45<br />
164, 27 233, 31 164, 27<br />
165, 20 235, 5 165, 20<br />
72: 268.00187 72: 189.50594 36: 134.00093<br />
191, 188 269, 267 191, 188<br />
208, 169 305, 225 208, 169<br />
215, 160 325, 195 215, 160<br />
236, 127 333, 181 236, 127<br />
247, 104 351, 143 247, 104<br />
257, 76 363, 109 257, 76<br />
260, 65 375, 55 260, 65<br />
265, 40 377, 39 265, 40<br />
268, 1 379, 3 268, 1<br />
80: 371.65172 80: 262.79745 40: 185.82586<br />
275, 250 393, 349 275, 250<br />
301, 218 405, 335 301, 218<br />
310, 205 435, 295 310, 205<br />
317, 194 465, 245 317, 194<br />
334, 163 475, 225 334, 163<br />
350, 125 497, 171 350, 125<br />
355, 110 511, 123 355, 110<br />
365, 70 515, 105 365, 70<br />
370, 35 519, 83 370, 35<br />
371, 22 525, 25 371, 22<br />
96: 400.28115 96: 283.04152 48: 200.14058<br />
300, 265 415, 385 300, 265<br />
311, 252 431, 367 311, 252<br />
329, 228 469, 317 329, 228<br />
337, 216 473, 311 337, 216<br />
356, 183 497, 271 356, 183<br />
360, 175 515, 235 360, 175<br />
375, 140 535, 185 375, 140<br />
384, 113 539, 173 384, 113<br />
392, 81 553, 121 392, 81<br />
393, 76 557, 101 393, 76<br />
399, 32 563, 59 399, 32<br />
400, 15 565, 35 400, 15<br />
128: 895.05586 128: 632.90007 64: 447.52793<br />
655, 610 905, 885 655, 610<br />
703, 554 985, 795 703, 554<br />
710, 545 1013, 759 710, 545<br />
722, 529 1039, 723 722, 529<br />
766, 463 1067, 681 766, 463<br />
769, 458 1095, 635 769, 458<br />
785, 430 1103, 621 785, 430<br />
815, 370 1165, 495 815, 370<br />
830, 335 1185, 445 830, 335<br />
862, 241 1215, 355 862, 241<br />
865, 230 1227, 311 865, 230<br />
874, 193 1229, 303 874, 193<br />
881, 158 1251, 193 881, 158<br />
886, 127 1255, 165 886, 127<br />
890, 95 1257, 149 890, 95<br />
895, 10 1265, 45 895, 10<br />
144: 1443.23422 144: 1020.52070 72: 721.61711<br />
1027, 1014 1469, 1417 1027, 1014<br />
1107, 926 1571, 1303 1107, 926<br />
1133, 894 1597, 1271 1133, 894<br />
1170, 845 1625, 1235 1170, 845<br />
1230, 755 1735, 1075 1230, 755<br />
1245, 730 1765, 1025 1245, 730<br />
1261, 702 1807, 949 1261, 702<br />
1322, 579 1873, 811 1322, 579<br />
1331, 558 1879, 797 1331, 558<br />
1338, 541 1889, 773 1338, 541<br />
1342, 531 1901, 743 1342, 531<br />
1378, 429 1963, 559 1378, 429<br />
1395, 370 1975, 515 1395, 370<br />
1405, 330 1985, 475 1405, 330<br />
1430, 195 2015, 325 1430, 195<br />
1434, 163 2027, 239 1434, 163<br />
1437, 134 2033, 181 1437, 134<br />
1443, 26 2041, 13 1443, 26<br />
160: 2001.40576 160: 1415.20758 80: 1000.70288<br />
1483, 1344 2075, 1925 1483, 1344<br />
1500, 1325 2143, 1849 1500, 1325<br />
1555, 1260 2155, 1835 1555, 1260<br />
1604, 1197 2309, 1637 1604, 1197<br />
1645, 1140 2345, 1585 1645, 1140<br />
1685, 1080 2365, 1555 1685, 1080<br />
1692, 1069 2429, 1453 1692, 1069<br />
1780, 915 2485, 1355 1780, 915<br />
1800, 875 2531, 1267 1800, 875<br />
1811, 852 2575, 1175 1811, 852<br />
1875, 700 2663, 959 1875, 700<br />
1899, 632 2675, 925 1899, 632<br />
1920, 565 2695, 865 1920, 565<br />
1941, 488 2761, 623 1941, 488<br />
1960, 405 2765, 605 1960, 405<br />
1965, 380 2785, 505 1965, 380<br />
1973, 336 2801, 407 1973, 336<br />
1995, 160 2815, 295 1995, 160<br />
1996, 147 2825, 175 1996, 147<br />
2000, 75 2827, 139 2000, 75<br />
192: 2434.81519 192: 1721.67433 96: 1217.40759<br />
1746, 1697 2497, 2371 1746, 1697<br />
1806, 1633 2527, 2339 1806, 1633<br />
1823, 1614 2633, 2219 1823, 1614<br />
1890, 1535 2711, 2123 1890, 1535<br />
1953, 1454 2725, 2105 1953, 1454<br />
1985, 1410 2855, 1925 1985, 1410<br />
2065, 1290 2875, 1895 2065, 1290<br />
2110, 1215 2953, 1771 2110, 1215<br />
2118, 1201 3025, 1645 2118, 1201<br />
2191, 1062 3061, 1577 2191, 1062<br />
2202, 1039 3131, 1433 2202, 1039<br />
2238, 959 3149, 1393 2238, 959<br />
2271, 878 3197, 1279 2271, 878<br />
2282, 849 3241, 1163 2282, 849<br />
2319, 742 3253, 1129 2319, 742<br />
2335, 690 3319, 917 2335, 690<br />
2362, 591 3325, 895 2362, 591<br />
2385, 490 3355, 775 2385, 490<br />
2390, 465 3395, 575 2390, 465<br />
2415, 310 3407, 499 2415, 310<br />
2417, 294 3425, 355 2417, 294<br />
2426, 207 3437, 209 2426, 207<br />
2433, 94 3439, 173 2433, 94<br />
2434, 63 3443, 49 2434, 63<br />
256: 5444.41227 256: 3849.78084 128: 2722.20614<br />
3896, 3803 5511, 5377 3896, 3803<br />
3980, 3715 5567, 5319 3980, 3715<br />
4027, 3664 5721, 5153 4027, 3664<br />
4133, 3544 5755, 5115 4133, 3544<br />
4156, 3517 5935, 4905 4156, 3517<br />
4280, 3365 5997, 4829 4280, 3365<br />
4315, 3320 6215, 4545 4315, 3320<br />
4520, 3035 6315, 4405 4520, 3035<br />
4540, 3005 6495, 4135 4540, 3005<br />
4645, 2840 6665, 3855 4645, 2840<br />
4772, 2621 6705, 3785 4772, 2621<br />
4805, 2560 6837, 3541 4805, 2560<br />
4861, 2452 6935, 3345 4861, 2452<br />
4931, 2308 6957, 3299 4931, 2308<br />
4960, 2245 7051, 3093 4960, 2245<br />
5051, 2032 7071, 3047 5051, 2032<br />
5059, 2012 7083, 3019 5059, 2012<br />
5072, 1979 7205, 2715 5072, 1979<br />
5128, 1829 7239, 2623 5128, 1829<br />
5140, 1795 7313, 2409 5140, 1795<br />
5189, 1648 7365, 2245 5189, 1648<br />
5245, 1460 7393, 2151 5245, 1460<br />
5260, 1405 7485, 1805 5260, 1405<br />
5315, 1180 7545, 1535 5315, 1180<br />
5360, 955 7555, 1485 5360, 955<br />
5380, 835 7635, 995 5380, 835<br />
5413, 584 7645, 915 5413, 584<br />
5420, 515 7673, 639 5420, 515<br />
5435, 320 7677, 589 5435, 320<br />
5437, 284 7691, 363 5437, 284<br />
5443, 124 7695, 265 5443, 124<br />
5444, 67 7699, 93 5444, 67<br />
288: 8778.85101 288: 6207.58508 144: 4389.42550<br />
6240, 6175 9035, 8515 6240, 6175<br />
6497, 5904 9223, 8311 6497, 5904<br />
6625, 5760 9335, 8185 6625, 5760<br />
6663, 5716 9389, 8123 6663, 5716<br />
6740, 5625 9565, 7915 6740, 5625<br />
6864, 5473 9893, 7501 6864, 5473<br />
7111, 5148 9971, 7397 7111, 5148<br />
7176, 5057 10231, 7033 7176, 5057<br />
7353, 4796 10427, 6739 7353, 4796<br />
7428, 4679 10477, 6661 7428, 4679<br />
7487, 4584 10561, 6527 7487, 4584<br />
7521, 4528 10663, 6359 7521, 4528<br />
7568, 4449 10729, 6247 7568, 4449<br />
7692, 4231 10853, 6029 7692, 4231<br />
7839, 3952 10985, 5785 7839, 3952<br />
7865, 3900 11219, 5317 7865, 3900<br />
8000, 3615 11335, 5065 8000, 3615<br />
8100, 3385 11435, 4835 8100, 3385<br />
8135, 3300 11485, 4715 8135, 3300<br />
8200, 3135 11615, 4385 8200, 3135<br />
8268, 2951 11765, 3965 8268, 2951<br />
8385, 2600 11791, 3887 8385, 2600<br />
8441, 2412 11923, 3461 8441, 2412<br />
8488, 2241 12017, 3119 8488, 2241<br />
8511, 2152 12049, 2993 8511, 2152<br />
8544, 2017 12071, 2903 8544, 2017<br />
8569, 1908 12107, 2749 8569, 1908<br />
8583, 1844 12149, 2557 8583, 1844<br />
8632, 1599 12233, 2119 8632, 1599<br />
8684, 1287 12259, 1963 8684, 1287<br />
8697, 1196 12337, 1391 8697, 1196<br />
8740, 825 12365, 1115 8740, 825<br />
8756, 633 12379, 947 8756, 633<br />
8760, 575 12385, 865 8760, 575<br />
8767, 456 12401, 593 8767, 456<br />
8775, 260 12415, 65 8775, 260<br />
320: 8608.37165 160: 6087.03797<br />
12217, 12131 8730, 8485 <br />
12485, 11855 8786, 8427 <br />
12635, 11695 9030, 8165 <br />
13121, 11147 9083, 8106 <br />
13165, 11095 9115, 8070 <br />
13237, 11009 9450, 7675 <br />
13555, 10615 9547, 7554 <br />
13625, 10525 9765, 7270 <br />
13919, 10133 9925, 7050 <br />
14275, 9625 10014, 6923<br />
14375, 9475 10242, 6581<br />
14707, 8951 10325, 6450<br />
14765, 8855 10550, 6075<br />
15019, 8417 10590, 6005<br />
15125, 8225 10762, 5691<br />
15305, 7885 10821, 5578<br />
15491, 7513 10955, 5310<br />
15655, 7165 11010, 5195<br />
15745, 6965 11158, 4869<br />
15943, 6499 11190, 4795<br />
15985, 6395 11221, 4722<br />
16027, 6289 11355, 4390<br />
16205, 5815 11410, 4245<br />
16265, 5645 11502, 3989<br />
16399, 5243 11595, 3710<br />
16453, 5071 11675, 3450<br />
16595, 4585 11718, 3301<br />
16625, 4475 11810, 2955<br />
16775, 3875 11829, 2878<br />
16823, 3661 11925, 2450<br />
16937, 3091 11950, 2325<br />
16975, 2875 12026, 1893<br />
17035, 2495 12075, 1550<br />
17101, 1993 12085, 1470<br />
17125, 1775 12123, 1114<br />
17185, 1045 12130, 1035<br />
17189, 977 12134, 987<br />
17195, 865 12165, 470<br />
17213, 359 12170, 315<br />
17215, 245 12174, 43<br />
192: 7795.21207<br />
11138, 10909<br />
11210, 10835<br />
11302, 10739<br />
11789, 10202<br />
11885, 10090<br />
11918, 10051<br />
11990, 9965<br />
12403, 9446<br />
12422, 9421<br />
12541, 9262<br />
12790, 8915<br />
12958, 8669<br />
12986, 8627<br />
13261, 8198<br />
13315, 8110<br />
13373, 8014<br />
13474, 7843<br />
13747, 7354<br />
13810, 7235<br />
13885, 7090<br />
14003, 6854<br />
14074, 6707<br />
14174, 6493<br />
14435, 5890<br />
14477, 5786<br />
14515, 5690<br />
14563, 5566<br />
14710, 5165<br />
14806, 4883<br />
14867, 4694<br />
14990, 4285<br />
15026, 4157<br />
15082, 3949<br />
15166, 3613<br />
15203, 3454<br />
15235, 3310<br />
15362, 2659<br />
15389, 2498<br />
15394, 2467<br />
15410, 2365<br />
15469, 1942<br />
15485, 1810<br />
15518, 1501<br />
15562, 941<br />
15565, 890<br />
15571, 778<br />
15581, 542<br />
15590, 115</span><br />
<br />Daniel Edwinhttp://www.blogger.com/profile/09343146036341064899noreply@blogger.com0tag:blogger.com,1999:blog-646494925180517842.post-58661678232213255912015-09-02T20:01:00.001-07:002015-09-07T08:43:06.963-07:00Geometry Puzzle - Grid and CircleYou have a sheet of graph paper and a compass. Draw a circle that
passes through more than 8 grid intersections. Specifically, draw the
smallest possible such circle.<br />
<br />
For example, the following circle only passes through 8 grid intersections, and so is not a solution. <br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieoi9KNrgFh17KVDMJAIEiUFCHXzfVX6QlGXkJI53hGwPc0ZZxWKAvz_FrBXpr8e1lhKDefH1WDImCiVclMfe5lHx0fodk0WAxXPB8q_khIs3y6ObhRsTHCpvUg2cl5_7lcF9Zw-PSW9w/s1600/circle9.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieoi9KNrgFh17KVDMJAIEiUFCHXzfVX6QlGXkJI53hGwPc0ZZxWKAvz_FrBXpr8e1lhKDefH1WDImCiVclMfe5lHx0fodk0WAxXPB8q_khIs3y6ObhRsTHCpvUg2cl5_7lcF9Zw-PSW9w/s1600/circle9.png" /></a></div>
<br />
To post your answer, just post a comment with the
radius of the circle (please do not post an image). For example, if you were posting the example above as an answer, post "r=square root of 5" or similar text. I will post the correct answer (and winner) as another blog post.<br />
<br />
(FYI, this has also been posted on Facebook. Facebook comments will also be in the running to win.)<br />
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Whitespace complete.Daniel Edwinhttp://www.blogger.com/profile/09343146036341064899noreply@blogger.com4tag:blogger.com,1999:blog-646494925180517842.post-24514681489633709572012-11-10T12:36:00.001-08:002012-11-15T11:36:43.403-08:0051 Star Flag (continued)A previous post on this blog discussed the mathematics of grid and checkerboard flags. These are suitable for most numbers of stars, including 51. Occasionally, however, these techniques are not sufficient. For instance, there are no suitable grid/checkerboard flags for 62, 79 or 89 stars. You may also just get bored of checkerboard and grid flags and want something different. As it turns out, there are plenty of options. Without further ado:<br />
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<div style="text-align: center;">
<span style="font-size: large;">Alternative Flag Styles</span></div>
<div style="text-align: center;">
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUx2rBwRXTClHMTvRObTwKKPHeSNltQfKNwNdexd2-Dep1sOB9XPwJOUG40WwPYQdzJIVJSIVZ66ivwrIS0vwGh2OaVNSkmucnwJZPOQaxK_3hyphenhyphenKIxx2AbR_fEQuEkuuR8MZthLOrMLE4/s1600/stateflagstyles.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUx2rBwRXTClHMTvRObTwKKPHeSNltQfKNwNdexd2-Dep1sOB9XPwJOUG40WwPYQdzJIVJSIVZ66ivwrIS0vwGh2OaVNSkmucnwJZPOQaxK_3hyphenhyphenKIxx2AbR_fEQuEkuuR8MZthLOrMLE4/s1600/stateflagstyles.png" /></a></div>
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The first five styles were actually used for US flags. They are named after the state that whose joining caused that flag. The other styles have never been used, but are valuable mathematically.<br />
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Michigan style is a simple grid flag with an extra star in the top and bottom rows.<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = rows x columns + 2</b></span><br />
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Oregon style is a simple grid flag with two stars removed from the center row. (though it only looks good if the number of rows is odd, so that the the modified row can be centered)<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = rows x columns - 2</b></span><br />
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Kansas is like Oregon except that it only has one star removed.<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = rows x columns - 1</b></span><br />
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</div>
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Nevada is the most complicated design so far. The best way to look at it is to mentally combine the left and right columns and make a simple grid flag with one extra star. (But only if the number of rows is odd)<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = rows x columns + 1</b></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzGWno23QB8NUOmNPUGsIK41AKTbmz9DtK98DQbVL9LXkQyad72FemeTomdV3lr84uLDbJn3oGJOMTqYQLd7kKUhQEhrLCiUUU7RQ-XWvixQqpnCzcpJEbiQ5aBlqQZ_3EgHS2IFl6uwk/s1600/nevadamath.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzGWno23QB8NUOmNPUGsIK41AKTbmz9DtK98DQbVL9LXkQyad72FemeTomdV3lr84uLDbJn3oGJOMTqYQLd7kKUhQEhrLCiUUU7RQ-XWvixQqpnCzcpJEbiQ5aBlqQZ_3EgHS2IFl6uwk/s1600/nevadamath.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Mathematical trick for Nevada</td></tr>
</tbody></table>
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Colorado is a simple grid with a star removed from the top and bottom rows. This gives it the same math as Oregon, but with no odd rows restriction.<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = rows x columns - 2</b></span><br />
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No Corners is the heaviest modification so far. It is a grid with four stars removed.<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = rows x columns - 4</b></span><br />
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Modified Checkerboard can be best thought of as a checkerboard flag with an extra star added to the left and right columns. This design only works if it is based on an odd x odd checkerboard pattern with corners black (see previous post for checkerboard flag analysis).<br />
<span style="font-family: Courier New, Courier, monospace;"><b>Number of stars = (rows x columns - 1) / 2 + 2 </b></span><br />
<span style="font-family: Courier New, Courier, monospace;"><b> (Checkerboard with black corners, plus 2)</b></span><br />
<br />
This gives us an army of new formulas with which to do our flag hunting. Solving them all fo<span style="font-family: inherit;">r <b>rows x columns</b> we </span>get:<br />
<br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) - 2 Michigan</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) - 1 Nevada</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) + 1 Kansas</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) + 2 Oregon or Colorado</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) + 4 No Corners</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) x 2 - 3 Modified Checkerboard</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b><br />
</b></span> <span style="font-family: Courier New, Courier, monospace; font-size: x-small;"><b>From the last post:</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) x 2 Even checkerboard flags</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) x 2 - 1 Odd checkerboard, white corners</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) x 2 + 1 Odd checkerboards, black corners</b></span><br />
<span style="font-family: Courier New, Courier, monospace; font-size: small;"><b>rows x columns = (Number of stars) Simple grid flags </b></span><br />
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The procedure for finding a flag with a given number of stars i<span style="font-family: inherit;">s to use these formulas to find possible values of <b>rows x columns</b> for your given number of stars. Then, try to factor <b>rows x columns</b> into appropriate values for rows and columns.</span><br />
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For 62, this gives us:<br />
<span style="font-family: Courier New, Courier, monospace;">Michigan: 62 - 2 = 60 = <span style="background-color: lime;">6 x 10</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Nevada: 62 - 1 = 61 = <span style="color: red;">PRIME</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Kansas: 62 + 1 = 63 = <span style="background-color: lime;">7 x 9</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Oregon/Colorado: 62 + 2 = 64 = <span style="background-color: lime;">8 x 8</span> </span><br />
<span style="font-family: Courier New, Courier, monospace;"> (Oregon requires odd rows, though)</span><br />
<span style="font-family: Courier New, Courier, monospace;">No Corners: 62 + 4 = 66 = <span style="background-color: lime;">6 x 11</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Mod. Check: 62 x 2 - 3 = 121 = <span style="background-color: lime;">11 x 11</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Even Check: 62 x 2 = 124 = 4 x 31 <span style="color: red;">UGLY</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Odd Check, White Corners: 62 x 2 - 1 = 123 = 3 x 41 <span style="color: red;">UGLY</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Odd Check, Black Corners: 62 x 2 + 1 = 125 = 5 x 25 <span style="color: red;">UGLY</span></span><br />
<span style="font-family: Courier New, Courier, monospace;">Simple Grid: 62 = 2 x 31 <span style="color: red;">UGLY</span></span><br />
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This gives us five options for a 62 star flag:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCa2hGGMGSXKFyvn7aNxjq_9DtIIyf01uIy9ao7jpkKbxBZf3FmRklEzUo2roM2M_g9zn1ZtLZJr0cWbuNNldi0G1OPOetSbxXK91On9vsFdd3XO-VeLny8Op6lKUy82rtF21qCisbnAI/s1600/62stars.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCa2hGGMGSXKFyvn7aNxjq_9DtIIyf01uIy9ao7jpkKbxBZf3FmRklEzUo2roM2M_g9zn1ZtLZJr0cWbuNNldi0G1OPOetSbxXK91On9vsFdd3XO-VeLny8Op6lKUy82rtF21qCisbnAI/s1600/62stars.png" /></a></div>
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There are other flag designs that could be imagined, but this collection gives us plenty of formulas to work with, and all of the patterns are aesthetically pleasing.<br />
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The last post mentioned that 62, 79 and 89 are hard to find flag patterns for. 62 has been done just now. Here are suitable patterns for 79 and 89, using the mathematics described above.<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYj3dc0ofnWEQL9jprGomI4p48zUgQcN8-Z_UbyaYf6tLC_NFsk-w6oFb3nbYUllGZmtwXJfB2PjqkoyH88UC6Mx4joj4I_13tZ6QAz_CpXraaEIk3MB1rGVXDEjYhP_Ra0sGP3xd4lhI/s1600/79and89options.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYj3dc0ofnWEQL9jprGomI4p48zUgQcN8-Z_UbyaYf6tLC_NFsk-w6oFb3nbYUllGZmtwXJfB2PjqkoyH88UC6Mx4joj4I_13tZ6QAz_CpXraaEIk3MB1rGVXDEjYhP_Ra0sGP3xd4lhI/s1600/79and89options.png" /></a></div>
A nice article from Slate (<a href="http://www.slate.com/articles/life/do_the_math/2010/06/13_stripes_and_51_stars.html" target="_blank">www.slate.com</a>) and a fun widget from PopSci (<a href="http://www.popsci.com/science/article/2010-06/how-preserve-symmetry-star-spangled-banner" target="_blank">www.popsci.com</a>) about this topic do not have solutions for 29, 69 and 87 stars. With the mathematics described above, we can find patterns for all of them.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFkB4CEdC6x42wZnnNjVFUu-ke3CUMzXmDpj85HeqJ4u4h55puG1qWJ5xyI0nJ14cZ2LdJpGe5iqszVZdrOyqyp9H4jGDpwpNpwTJPuC4mUA6_I8lDAtrZH2IOzgCIvqCfkiMMbuQkv7E/s1600/difficultflags.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFkB4CEdC6x42wZnnNjVFUu-ke3CUMzXmDpj85HeqJ4u4h55puG1qWJ5xyI0nJ14cZ2LdJpGe5iqszVZdrOyqyp9H4jGDpwpNpwTJPuC4mUA6_I8lDAtrZH2IOzgCIvqCfkiMMbuQkv7E/s1600/difficultflags.png" /></a></div>
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With this I bring you my final set of options for a 51 star flag:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6imu9cr2XMdUeC5hoCmQjIILgcqJCjJtafVCD-sj1D5SWWsrGlNSI6CQwNUCZfMlXCT_LedFsKZMWSEzQOX_BkWlDL-WegU7-whuif2zLWUF37V3BU0u1QRFuIS3dbFJwmAT-3pgHb00/s1600/51starflagoptions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6imu9cr2XMdUeC5hoCmQjIILgcqJCjJtafVCD-sj1D5SWWsrGlNSI6CQwNUCZfMlXCT_LedFsKZMWSEzQOX_BkWlDL-WegU7-whuif2zLWUF37V3BU0u1QRFuIS3dbFJwmAT-3pgHb00/s1600/51starflagoptions.png" /></a></div>
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The mathematics I have so far described gives us Checkerboard, Michigan and Modified Checkerboard styles. The Checkerboard style is the currently accepted design if Puerto Rico should become a state. I add two other styles here. One is the Special Circular. 51 happens to be an excellent number to arrange in a circle. The circular pattern is a popular alternative to the accepted checkerboard and it was not designed by this blog. The 'Special' pattern is my own invention, and it has a lot of interesting mathematics behind it. It is actually a further modification of the idea of a checkerboard flag. But those mathematics are for another post :-)<br />
<br />Daniel Edwinhttp://www.blogger.com/profile/09343146036341064899noreply@blogger.com2tag:blogger.com,1999:blog-646494925180517842.post-3426168188017986012011-11-12T13:41:00.000-08:002011-11-12T13:41:23.537-08:0051 Star FlagThe United States Flag is beautiful. Throughout our country's history, the flag has changed many times. The general concept remained the same, but the number of stars has changed repeatedly as new states were added. This gave the country an interesting mathematical challenge - how to arrange n stars in a rectangle in an aesthetically pleasing way. America may once again be faced with this challenge if Puerto Rico becomes a state. I think that many Americans dread this - not because they dislike Puerto Rico, but because it would mean fitting one more star on the flag, and ruining its symmetry. Well, rest easy, America. Its symmetry is not in jeopardy. There is already a plan for a 51 star flag that is quite symmetrical. But this begs the question: what are the techniques for fitting n points (stars) into a rectangle?<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-k75e5QxlnWWrEKLlvavbyUbhfwZn5sAb26j6eqU8xTHIx1UL5CQb6hC-RA0dxeISj8zbnvPidxnVpcDzlvqTchGIeUhuQrxT1_9fYI6R02_Qf2THLes6TQG0WktlrgXY8xUlpMBQ3L0/s1600/us+flag.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-k75e5QxlnWWrEKLlvavbyUbhfwZn5sAb26j6eqU8xTHIx1UL5CQb6hC-RA0dxeISj8zbnvPidxnVpcDzlvqTchGIeUhuQrxT1_9fYI6R02_Qf2THLes6TQG0WktlrgXY8xUlpMBQ3L0/s1600/us+flag.png" /></a></div>
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Let's start by analyzing the current 50 star flag. It may surprise you to learn that the arrangement of stars on the 50 star flag has nothing to do with the fact that 50 is a multiple of 10. The fact that 5 x 10 = 50 does not matter. The more relevant fact is that (9 x 11 + 1) / 2 = 50, as you will understand in a few minutes. The 50 star flag is what I call a 'checkerboard' flag. Checkerboard flags are when you cover the blue region of the flag in a checkerboard pattern, and you put stars only on the white squares. This is why the 50 star flag seems to be made up of diagonal lines; the diagonal lines are like the lines that a bishop moves along on the chess/checkerboard. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLPd_UfgS76MLN37rNkRGvTSKb455EYvsc_as6sSEDCSXjgwJA_Xyv8G2wCw79WvLiduGsv9kto5HMgrhcDJpkk34dKbj5OOSHWucAsWj2k98WH4VhNuiN6-XDZUR8kqREnxaSjJjmArA/s1600/4x6+flag.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLPd_UfgS76MLN37rNkRGvTSKb455EYvsc_as6sSEDCSXjgwJA_Xyv8G2wCw79WvLiduGsv9kto5HMgrhcDJpkk34dKbj5OOSHWucAsWj2k98WH4VhNuiN6-XDZUR8kqREnxaSjJjmArA/s1600/4x6+flag.png" /></a><br />
If our checkerboard was, say, 4 rows high and 6 columns wide, it would have 4 x 6 = 24 squares on it. Since every other sq<span style="font-size: small;">uare is white, we can count all the white squares by dividing this number by 2. So, a 4 x 6 checkerboard gives us 24 / 2 = 12 white squares. This gives us the formula:</span><br />
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<span style="font-size: small;"><span style="font-family: "Courier New",Courier,monospace;"><b>Number of stars = rows x columns / 2</b></span></span><br />
<br />
<span style="font-size: small;">Also note that we can swap the colors and have a flag that is a mirror image of the original.</span><br />
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It gets more complicated, though, if the number of squares is an odd number. If it is an odd number, then you cannot just divide by 2 to get the number of stars - you would not get an integer. What you do instead is divide it by 2 and then take the integers above and below your answer. For example, if the number of squares is 99, you would divide by 2 to get 49.5. Then you would take the integers immediately above and below this answer: 49 and 50. This means that you can get a 49 star flag or a 50 star flag, depending on which squares you pick to be white. If you pick the corners to be white, you get a 50 star flag, and if you pick the corners to be black, you get a 49 star flag.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0hrD96b55LOG9d9_M3H-s3JsLbcngeqv4jlUgtnn83zYUg-CvRAOEPtdsfBAj7HV0Gu6QeFSvRZs0yJraRlRZqJhtvytrInnxo16z3uislKlhRdulnAayAHK6gjlz7eL1QSHtc4nTvm4/s1600/9x11+flag.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0hrD96b55LOG9d9_M3H-s3JsLbcngeqv4jlUgtnn83zYUg-CvRAOEPtdsfBAj7HV0Gu6QeFSvRZs0yJraRlRZqJhtvytrInnxo16z3uislKlhRdulnAayAHK6gjlz7eL1QSHtc4nTvm4/s1600/9x11+flag.png" /></a></div>
That 9 x 11 pattern with the corners white is the standard pattern for the current United States flag.<br />
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<span style="font-size: small;">This gives us two new equations:</span><br />
<br />
<span style="font-size: small;"><span style="font-family: "Courier New",Courier,monospace;"><b>Number of stars = ( rows x columns + 1 ) / 2 </b> (corners white)<b> </b></span></span><br />
<span style="font-size: small;"><span style="font-family: "Courier New",Courier,monospace;"><b>Number of stars = ( rows x columns - 1 ) / 2 </b> (corners black)<b></b></span></span><br />
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<span style="font-size: small;">There is one more kind of flag we have not talked about. This is a simple grid flag. It is like the checkerboard flag except that you put stars in all of the squares, not just the white ones. The formula for this kind of flag is simple and hardly worth elaborating on:</span><br />
<br />
<span style="font-size: small;"><span style="font-family: "Courier New",Courier,monospace;"><b>Number of stars = ( rows x columns )</b></span></span><br />
<br />
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<span style="font-size: large;">Putting it All Together:</span><br />
<br />
<span style="font-size: small;">The goal, now, is to find a suitable pattern for a given number of stars. So, given 51 stars, what patterns will work? The mathematics make this easy. Just solve all of the above equations for rows x columns, like this:</span><br />
<br />
<div style="font-family: "Courier New",Courier,monospace;">
<span style="font-size: small;"><b>rows x columns = (Number of stars) x 2 </b> even checkerboard flags</span></div>
<div style="font-family: "Courier New",Courier,monospace;">
<span style="font-size: small;"><b>rows x columns = (Number of stars) x 2 - 1 </b> odd checkerboard flags, white corners</span></div>
<div style="font-family: "Courier New",Courier,monospace;">
<span style="font-size: small;"><b>rows x columns = (Number of stars) x 2 + 1 </b> odd checkerboards flags, black corners</span></div>
<span style="font-size: small;"><span style="font-family: "Courier New",Courier,monospace;"><b>rows x columns = (Number of stars) </b> simple grid flags </span></span><br />
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We can use these formulas to find patterns for a 51 star flag. By plugging 51 into the formulas above, we see that rows x columns can equal 102, 101, 103 or 51, respectively.<br />
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Now we need to factor these numbers to see if any are suitable. 101 and 103 are prime numbers, so they will not work. 51 is 17 x 3. But a 17 x 3 grid flag would be ugly, so 51 will not work. 102 is 17 x 6. A 17 x 6 checkerboard flag would be just fine! So there is our answer: a 51 star flag should be a 17 x 6 checkerboard flag:<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4xXz2z_xiGtwT6v4O3kQoWMWG5J8cgHPxg8EiW129bcAZbHXdYH3lsX2XT1IY8WaIBcrVMQEKq132qYNtr-L0Xlp4VAEUbGNxa1RNK8u5nRTaTvf1NgbPu0NFMSHvdscZhVmOc_2nmjU/s1600/51+star+flag.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4xXz2z_xiGtwT6v4O3kQoWMWG5J8cgHPxg8EiW129bcAZbHXdYH3lsX2XT1IY8WaIBcrVMQEKq132qYNtr-L0Xlp4VAEUbGNxa1RNK8u5nRTaTvf1NgbPu0NFMSHvdscZhVmOc_2nmjU/s1600/51+star+flag.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The 51 star flag, a 17 x 6 checkerboard flag. You may notice that the checkerboard is squeezed. It has to fit 17 columns, but only 6 rows. This pushes the stars together horizontally. Despite this, the flag still looks good. Checkerboard flags can handle a lot of squeezing before they start to look funny.</span></td></tr>
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This technique can be used to find suitable flags for all the numbers from 50 to 100 except 62, 79 and 89. Many flags have several suitable arrangements. Here are arrangements for flags 51 to 70:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieZnl7b9yvp3xKcmZadao4dJ0f4_FMFkJkWdzOJEnbOQiof3Yc8LkmYA1MwntwrPb6nZ4Durigo7cBh9TWCL0yRNTXIw1GXshNBtLAeiRycYIhGSr49ZvkkONPiixcUdFLDeIAU7XaMi8/s1600/51+through+70.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieZnl7b9yvp3xKcmZadao4dJ0f4_FMFkJkWdzOJEnbOQiof3Yc8LkmYA1MwntwrPb6nZ4Durigo7cBh9TWCL0yRNTXIw1GXshNBtLAeiRycYIhGSr49ZvkkONPiixcUdFLDeIAU7XaMi8/s1600/51+through+70.png" /></a></div>
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Now you must be wondering what you can do with a 62, 79 or 89 star flag. Actually, there are plenty of options. But that is for another post.<br />
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You will notice that there are only five simple grid flags in the above display: 54, 60, 63, 66 and 70. Good grid flags are quite a bit more scarce than good checkerboard flags. And, frankly, I do not think that grid flags look very good. All of the grid flags shown here could be replaced with checkerboard flags of the same number of stars. I chose to use the grid flags in this display for the sake of diversity, but if it were up to me, they would all be checkerboard style. The only grid flag that I like is the 48 star 6 x 8 grid flag, and that is only because it was used in World War II, and raised in the famous photo at Iwo Jima.<br />
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There are a couple more interesting styles of flag, but they will be left for a later post. If just one reader sleeps more easily knowing that we do not have a 51-star flag crisis looming, my work here is done.Daniel Edwinhttp://www.blogger.com/profile/09343146036341064899noreply@blogger.com10tag:blogger.com,1999:blog-646494925180517842.post-8065171027866766842011-10-07T20:14:00.000-07:002011-10-07T12:57:09.338-07:00Deep Space to Scale<div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"></div><div class="separator" style="clear: both; font-family: Verdana,sans-serif; text-align: center;"></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: small;"><br />
</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;">Celestial objects are bigger than you think. </span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;">Nebulae, galaxies and star clusters – they may be distant, but they are also huge. Many of them are so big, in fact, that they appear larger than a full moon in the night sky. So why can you not see them? They are difficult to see not because of their size, but because they are faint. This is why the purpose of telescopes is not just to magnify the sky but to gather a lot more light than the naked eye can.</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;">Jupiter and Saturn are among the smallest objects in the sky. Most of the pictures of nebulae, galaxies and star clusters that you have seen range in size from the size of Jupiter, to larger than a full moon. Examples of well known pictures that fall into this range are: The Orion Nebula, the Horsehead Nebula, the Andromeda Galaxy, the Whirlpool Galaxy, the Crab Nebula, the Eagle Nebula, and even the Hubble Deep Field images – along with many, many others.</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;">A short discussion of units is important. Apparent size, which is the size that your eye or a camera perceives something to be, is measured in <b>degrees</b></span><span style="font-size: small; font-weight: normal;">. For example, the Moon is about 0.5</span><span style="font-size: small;"><span style="font-weight: normal;">° degrees across. This means that if you drew a line from the left side of the Moon all the way to your eye, and another line from the right side of the Moon all the way to your eye, the angle between the lines is 0.5° degrees. Degrees are divided into 60 </span></span><span style="font-size: small;"><b>arcminutes</b></span><span style="font-size: small;"><span style="font-weight: normal;">. The Moon is, therefore, about 0.5 * 60 = 30 arcminutes across.</span></span><br />
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<span style="font-size: small;"><span style="font-weight: normal;">Below I have a collection of to-scale images of celestial objects. <b> All of these are scaled relative to how big they actually look in the night sky.</b> If you stand back 22 feet from your computer screen, these images will actually look the same size as they are in the sky. This is true not only of the Moon, but also of ALL the pictures in this post.</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><br />
</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">The Basics - To Scale</span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; font-family: Verdana,sans-serif; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMkxFimPrjlA5j89a8pmx5AtXezG1-6B5VX4qmXlQ921cWCwWcl4KZdDYpLxJR9gIVSKzOgKkUw6e2EDp38lA9bT4oRD_gmgx4lzl_UtSwSld5c8NI6vMQJtcXXuRT-jmwFnzqRRHAGpY/s1600/sun%252Bmoon.jpg" style="margin-left: auto; margin-right: auto;" /></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The most familiar objects in the sky. Use this image as a reference for the other images. All of the images in this post are to scale. </span></td></tr>
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</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;"> The Great Nebulae - To Scale</span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="font-family: Verdana,sans-serif; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDN-Nn-UdqTOHVKgP3nSpSf0utgf9cbNY_yapfxpQ-UOsJAFlsKe9UoAMQAO9OdZVrsSZgFSilZhJG8MXlGcqmMoYH-vdSFP9sE5rBA8vYUepoyyEASki4rLhUZMVMP8Q8m4mn9bSCb7c/s1600/horsehead%252Borion.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDN-Nn-UdqTOHVKgP3nSpSf0utgf9cbNY_yapfxpQ-UOsJAFlsKe9UoAMQAO9OdZVrsSZgFSilZhJG8MXlGcqmMoYH-vdSFP9sE5rBA8vYUepoyyEASki4rLhUZMVMP8Q8m4mn9bSCb7c/s1600/horsehead%252Borion.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">These are probably the two most well known nebulae. Both are larger than a full moon. Notice the full moon partially visible in the lower right corner. This is for convenient size comparison.</span></td></tr>
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</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">The Great Nebulae - To Scale, Continued</span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="font-family: Verdana,sans-serif; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiht61mmCm8oFP8VON_We3PqPstzl3EwJFd4ty-FcY4AocGK_dy-MUepNFHvnbIcnTQAmQHPcK05pB3BoymkaNdNu-A07h3ylk6nWtMZ5Ei7cml3So0vzvNs1ut-R1n8MmQF724Cg1HNog/s1600/eagle%252Bhelix.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiht61mmCm8oFP8VON_We3PqPstzl3EwJFd4ty-FcY4AocGK_dy-MUepNFHvnbIcnTQAmQHPcK05pB3BoymkaNdNu-A07h3ylk6nWtMZ5Ei7cml3So0vzvNs1ut-R1n8MmQF724Cg1HNog/s1600/eagle%252Bhelix.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">Also very popular, these nebulae rival the full moon in size. The Eagle Nebula is home to the famous Hubble picture "Pillars of Creation". As a side note: the colors in these and most other pictures from space are artificial. Nearly all objects in space look white in real life. The artificial coloration is usually based on spectrum data that the human eye cannot perceive normally.</span></td></tr>
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</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">The Great Andromeda Galaxy - To Scale</span></span><br />
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<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTcKqUfk93hYXgo-InqiZlf4MGwTExknL0tDw7dxmGIfIGbHTcCQONyxqN7QEB7GOsayVjRuR-jEOhBYN5cBh4-fWGrkan8N-61VidrXm6-wOwbsL7Swm0Ky9owfIkECMrl29svcBcTgw/s1600/andromeda.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTcKqUfk93hYXgo-InqiZlf4MGwTExknL0tDw7dxmGIfIGbHTcCQONyxqN7QEB7GOsayVjRuR-jEOhBYN5cBh4-fWGrkan8N-61VidrXm6-wOwbsL7Swm0Ky9owfIkECMrl29svcBcTgw/s1600/andromeda.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The Andromeda Galaxy is huge - it is much larger than the full moon in the night sky. Even better, this galaxy is actually bright enough to see with the naked eye on very dark nights. That's right - you can just look up into the sky and see a whole galaxy. Astronomers believe that the Andromeda galaxy is heading toward the Milky Way, and may one day collide with it.</span></td></tr>
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</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">Famous Galaxies - To Scale</span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="font-family: Verdana,sans-serif; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP51HXJ5wydObRz5jVxO8napOd7_kb0bxLCjDDmkqAaOinIXU_POsArMONVqQfL9hhAHfHJ-srz6iDj5AQ7vuhEqZLudm1LDy3vJY2yd2VTrDnny4n-TdCBYNFBkllnspxBa_v42pHKVQ/s1600/triangulum%252Bwhirlpool%252Bsombrero.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP51HXJ5wydObRz5jVxO8napOd7_kb0bxLCjDDmkqAaOinIXU_POsArMONVqQfL9hhAHfHJ-srz6iDj5AQ7vuhEqZLudm1LDy3vJY2yd2VTrDnny4n-TdCBYNFBkllnspxBa_v42pHKVQ/s1600/triangulum%252Bwhirlpool%252Bsombrero.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The Triangulum Galaxy is larger than a full moon. Like the Andromeda Galaxy, this too is visible to the naked eye. The Sombrero and Whirlpool galaxies are just two of many similar galaxies that are scattered about the sky.</span></td></tr>
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<span style="font-size: small;"><span style="font-weight: normal;">The Pleiades - To Scale</span></span><br />
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<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisknTNXXanRsefOpb9XbzrmsDLKKrEc5y10C0A4Y1hFaV1SBa-itcCF62XbRZPaVtpfyLIfMl116HBSY4e247nA4vK-X_OqVjdQxweywuGitVefPcsMENrDcU5wm5p4vKJtK9fslTnJ_U/s1600/pleiades.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisknTNXXanRsefOpb9XbzrmsDLKKrEc5y10C0A4Y1hFaV1SBa-itcCF62XbRZPaVtpfyLIfMl116HBSY4e247nA4vK-X_OqVjdQxweywuGitVefPcsMENrDcU5wm5p4vKJtK9fslTnJ_U/s1600/pleiades.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The Pleiades ('plee-uh-deez) is a prime example of an open star cluster. Open clusters are very beautiful, and are great for amateur astronomers. Clusters are still visible even when light pollution washes out nebulae and galaxies.</span></td></tr>
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</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">The Virgo Cluster - To Scale</span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="font-family: Verdana,sans-serif; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhszmbLVsq92vDDtnWyUfMJCysVxQlxFJEpKQBA8MZZtixCDXltz8xJWwBPzus_Z1w5WKnz1E5XRTnyJuOskxfDcDd9mmaXcNiRfFDcFjTbTu2VO74G1rdN4NdV5LgpAnFTqDYU15BBN40/s1600/virgo.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhszmbLVsq92vDDtnWyUfMJCysVxQlxFJEpKQBA8MZZtixCDXltz8xJWwBPzus_Z1w5WKnz1E5XRTnyJuOskxfDcDd9mmaXcNiRfFDcFjTbTu2VO74G1rdN4NdV5LgpAnFTqDYU15BBN40/s1600/virgo.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The Virgo Cluster is a large cluster of galaxies near the Milky Way (near is a relative term). The Virgo Cluster is the center of the Local Supercluster, of which the Local Group (Milky Way, Andromeda and Triangulum) is an outlying member. The galaxies are about the same apparent size as the dark seas and oceans on the Moon.</span></td></tr>
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</span></div><div style="font-family: Verdana,sans-serif;"><span style="font-size: small;">Assorted Small Objects - To Scale</span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="font-family: Verdana,sans-serif; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNyjNP5p23dFeJFyeI3849WyeaOcw6o0IV8XdIDqDJ94BPh4jgRIxoeJuamYTx1DEfMGqzI43D4y7DslpGCCtxJ8DBtpe6t6gaEpbcyHOCx5vht5W2HnX2r8DABDZvb0PWseP1xVyk9NQ/s1600/crab%252Bdumbbell%252Bring%252Bhdf%252Bplanets.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNyjNP5p23dFeJFyeI3849WyeaOcw6o0IV8XdIDqDJ94BPh4jgRIxoeJuamYTx1DEfMGqzI43D4y7DslpGCCtxJ8DBtpe6t6gaEpbcyHOCx5vht5W2HnX2r8DABDZvb0PWseP1xVyk9NQ/s1600/crab%252Bdumbbell%252Bring%252Bhdf%252Bplanets.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">These are just a few of the many nebulae available to amateur astronomers. The Crab Nebula is an example of a supernova remnant, which is a nebula left behind after a supernova. Notice how small the planets are compared to the other celestial objects. The famous deep field pictures are shown here in outline. Look these up on Wikipedia if you are interested. You may wonder why the ultra deep picture is larger than the deep one. This is because 'deep' refers to sensitivity, not magnification. The Hubble Ultra Deep Field photograph is very sensitive, with a total exposure time of eleven days. </span></td></tr>
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</span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">Human Visual Acuity - To Scale</span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="font-family: Verdana,sans-serif; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRe14Ax7tm3vioSni7ZiQyyqZhaYHvn5gpWN2wCNvccCs6x9rMdmwHoXl1BnfM9fqpsdIQ0vzkSmkNB8hobMU-wcruN0z4I2jMDuO_m-kvLn_919wuB2iY6jMEuGtJVnJcA0B38xFZ6i8/s1600/acuity.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRe14Ax7tm3vioSni7ZiQyyqZhaYHvn5gpWN2wCNvccCs6x9rMdmwHoXl1BnfM9fqpsdIQ0vzkSmkNB8hobMU-wcruN0z4I2jMDuO_m-kvLn_919wuB2iY6jMEuGtJVnJcA0B38xFZ6i8/s1600/acuity.jpg" /></a></span></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">This picture illustrates the limits of human visual acuity. A person with 20/20 vision can make out details down to 1 arcminute across. The pixelated picture of the Moon shows the level of detail visible with 20/20 vision. The sideways E is from a Snellen eye chart on the 20/20 vision row. In theory, a person with 20/20 vision could just barely determine the direction of the E if it was in the sky at that size. Stand back 22 feet from your computer screen to view these images in actual size, and see what you think.</span></td></tr>
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</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">If you would like to experience some of this for yourself, there are lots of ways. One fun way to start is to pick up a pair of binoculars and start looking. You can use the Google Sky app for your smartphone to help you find objects. It makes finding celestial objects so easy, it is sinful. </span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;"><br />
</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">Or, just start scanning the sky with your binoculars. Your are usually bound to find something. This can be rewarding even if you live near a city. City lights are annoying, but usually they are not as bad as they seem at first. Just pick a clear, moonless night, find a suitable location, and start scanning for anything you find interesting. Even if the light pollution is so bad that nebulae and galaxies are washed out, you can still look for star clusters, planets, and the moon. The moon, especially a partial moon (i.e. not full), is very beautiful up close.</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;"><br />
</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">If you are really serious, there are lots of great telescopes in the 300-500 dollar range. Some of them are 'Go-To' style, where you just tell the computer what to look at and it points the scope for you.</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;"><br />
</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">By far the best way, though, is to find an astronomy club (or observatory) that has public viewing nights. In the Washington DC area, The NOrthern Virginia Astronomy Club (NOVAC) has public viewing nights every month. These are where a bunch of club members set up their telescopes in a field, and members of the public are invited to look through them free of charge.</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;"></span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;"><br />
</span></span></div><div style="font-family: Verdana,sans-serif; margin-bottom: 0in;"><span style="font-size: small;"><span style="font-weight: normal;">I hope you have enjoyed this post. All of the original images are from the WikiMedia Foundation. The data I used to size the images correctly is from Google Sky. Most of the other information is from Wikipedia. All of this is made possible by NASA. </span></span></div>Daniel Edwinhttp://www.blogger.com/profile/09343146036341064899noreply@blogger.com1