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A Galaxy of New Constellations

When the "stars and stripes" design of the flag was made official in 1777, it was merely stated that the "union be thirteen stars, white in a blue field, representing a new constellation." However, until the Twentieth Century, there was no official law proclaiming exactly how those stars should be arranged. This led to a century and a half of diverse and creative patterns for the American flag. Some of the more interesting examples appear here.

This flag from the 1780s has the 13 stars representing the original colonies; however, instead of the familiar circle pattern of the "Betsy Ross flag," the stars are arranged into a "great star" pattern.
"Great star" patterns were popular throughout the first century of the nation. The patterns themselves grew more complex as the number of stars increased. This 26 star pattern from the early 1840s celebrates the arrival of Michigan as the 26th state with a large star in the center.
The "medallion" flag of 37 stars celebrates the 13 original colonies with an inner ring of 13 stars. The outer ring of the double wreath is a circle of 24 stars, one for each that had been admitted into statehood. This flag probably dates from the late 1860s.
Interestingly, although the union of this flag dates from the celebration of American’s First Centennial, it also pays tribute to the "Stars and Bars" flag of the Confederacy.
This union of this flag of 48 stars symbolizes the history of America. The great star is patterned after the Great Seal of the United States and also contains 13 stars, representing the 13 colonies. The stars of the inner ring represents the number of states admitted between 1776 and 1876, and the outer ring symbolizes the states admitted since the Centennial.
The makers of this flag of 33 stars from about 1860 used different sized stars to accentuate its unique design. The inner ring also contains a star for each of the 13 original colonies.
While not entirely accurate in the number of stars (81), the union of this dazzling flag celebrating the Centennial of the USA has one of the most imaginative designs ever put to Old Glory.

Source: The Stars And The Stripes, The American Flag as Art and as History from the Birth of the Republic to the Present by Boleslaw and Marie-Louise D’Otrange Mastai.