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The patterns of stars on American flags vary greatly, though they can generally be categorized and described based on common traits.  Below are some examples of patterns you may find on the cantons of American flags.
Even Rows.   This early 48 star flag dates to the beginning of the 20th century, and is especially rare because of the beautiful hand embroidering that reads "Nov. 11 1918."  The homemade flag celebrates Armistice Day, also known as Remembrance Day and today celebrated as Veteran's Day. The even rows of 48 stars were legislated by executive order of President Taft in 1912.
Staggered Rows.   The rows on this silk 45 star flag are staggered, as was commonly done on odd star counts.  Note the stars which tilt to 1 o'clock in the 7-star rows (rows 1, 3 and 5) and to the 11 o'clock in the 8-star rows (rows 2, 4 and 6).  This particular silk flag has a beautiful overprint in block painted gilt letters that reads "In Memoriam POST 185, G.A.R. LOWELL, MASS."  It is a funeral flag made for a Civil War veteran by the Grand Army of the Republic, the Union Army's fraternal organization.
Notched Rows.   In times where states were being rapidly added to the Union, flag makers sometimes produced flags in a notched configuration, leaving room for the addition of stars over the life of the flag.  This flag of 38 stars, from the era of the plains expansion, North West territories and the Indian Wars, features a notched configuration.  The flag is the size, shape and style of a Union Infantry Regimental Battle Flag, and most likely served that purpose between 1877 and 1890.
Great Star or Grand Luminary.   This classic Civil War Grand Luminary Flag was produced at the height of the Civil War and is a great reaffirmation of the Union that Americans were fighting to preserve.  Constructed of sewn cotton stars on wool bunting, the flag's colors remain rich and vivid today.
Random.  The stars on this 36 star flag of the Civil War era seem at first to be in rows, but the randomness of their placements leads to no real discernibly consistent pattern.  Here is another great example of a flag with a random pattern of stars.
Spiral.  This beautiful canton on a 45 star flag appears random but in fact is in a well defined spiral pattern, which starts at the center "key" star.  Without locating the key star and seeing the first correct move to begin the spiral, the stars appear random.
Snowflake.  One of the rarest of all patterns, this pattern resembles a "snowflake" with its six arms.  The pattern is known on a very rare few printed parade flags of the Civil War era, and only one known sewn flag, within the Rare Flags collection here.
Wreath.  A wreath consists of a ring of stars arranged in either a circle or oval.  The most famous wreath pattern on an American flag is the Betsy Ross pattern of 13 stars arranged in a single circle.  Shown here is a flag in the 3rd Maryland or Cowpens pattern, which features a wreath of 12 stars surrounding a single central star.
Medallion.  Concentric wreaths of stars are known as the medallion pattern.  This flag of 35 stars from the Civil War era features two wreaths of stars surrounding a single center star, with four outliers, one in each corner.  It is a beautiful symmetrical design.
Great Star in a Wreath.  Another very rare variation, found on printed parade flags of the Civil War era, is the Great Star in a Wreath pattern where some of the stars of an inner Great Star are shared by a circular wreath of stars.  The result is a visually dizzying combination of two great patterns in flag collecting:  the Medallion and the Great Star.  This example of 36 stars contains four outliers and two inliers.
Circle in a Square.  The circle in a square pattern is extremely rare.  The few flags known with this pattern of stars are primarily printed parade flags of the Centennial Era, though some variations from the Civil War period are also known.  This particular example of 38 stars dates to the period of 1876-1890 and is unusual for its teal blue canton and press dyed wool construction.
Phalanx.  The term "phalanx" dates to ancient Greece, and describes a square military formation of soldiers intended to protect the center.  Phalanx pattern flags are very rare.  A handful of phalanx pattern flags with an open center are known.  On this flag of thirty-seven stars, an phalanx with yet to be filled ranks surrounds a large center star representing Nebraska.
Beehive.  The beehive pattern is a rarity in flag collecting.  This pattern is known on a handful of printed parade flags in star counts of 34, 38 and in this shown count of 42.  The resemblance of the pattern to a beehive may be coincidental, but most likely is intentional, given the industriousness that bees and a beehive represent.  Even the smaller skewed corner stars seem to represent bees flying outside the hive. Less than 10 flags in this pattern are known.
Peace Sign.  This flag of began life as a double-applique 50 star flag, but in the tumultuous period of the early 1970s, at the height of America's involvement in the Vietnam War, its stars were removed and repurposed to produce a single-sided 25 Star Peace Flag.  The stars are hand sewn into the internationally recognized symbol for peace, designed in 1958 by British designer Gerald Holtom. Its shape derives from a combination of the semaphore signals for the letters "N" and "D" in support of the British Campaign for Nuclear Disarmament. Period Peace Sign flags in the Stars and Stripes format are scarce and very symbolic of their time.
The Starry Flower.  The name for this style of flag, which is exceptionally rare, was coined by Boleslaw and Mary-Louise D'Otrange Mastai in their famous 1973 book on flag collecting. While some "Great Star" patterns resemble flowers, with their arms bowed outward and resembling petals, this variation of the Starry Flower has clusters of stars in the corners and in the center. It is a beautiful and exuberant pattern, exceedingly rare with two similar, but individually unique, examples known.
The Starry Constellation. Although the pattern may seem random, an exceptionally rare set of unique flags feature recognizable "asterisms" or star constellations on them. I'm aware of just two early American flags that feature constellations or asterisms on their cantons. IAS-00400 is an Abolitionist flag that shows the bowl of the Big Dipper and the North Star; and IAS-00535 shows the Pleiades on the obverse and the horns of Taurus the Bull on the reverse.

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