Stars on The Flag: Introduction

Recently, I became quite fascinated with the arrangement of stars on the US flag, especially possible arrangements on future flags with more stars. I became so fascinated that I wrote a C++ program to generate all of the viable possibilities for flags of 51 stars or more! By viable, I mostly mean a reasonable aspect ratio; not too tall or wide.

The blue part of the flag that contains the stars is called the canton or union. Many possible arrangements of stars in the canton exist, as detailed in two blog posts by someone named Daniel Edwin and an article from Slate.com. All of these arrangements, along with the additional ones I have coded, fall in to three basic categories: Checkerboard, Grid, and Circular.

Our current 50-star flag is a checkerboard flag. Imagine drawing a checkerboard on the canton. Every other square has a star on it. If the total number of squares is odd, the total number of stars is different depending on whether the first square is blank or has a star in it. If the total number of squares is even, both starts give the same number of stars. This gives three patterns.

The most famous grid-based arrangement is the 48 star flag that flew from 1912 to 1959 and was raised over Iwo Jima in World War II. This is the “Simple Grid” flag; all of the variants of the this arrangement subtract stars in some pattern to achieve a different total number in a reasonable aspect ratio.

One can also arrange the stars in concentric circles. The mathematically simplest way to do this is with centered polygonal numbers, which are “each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer.” For example, the centered pentagonal numbers are 1, 6 (1+5), 16 (1+5+10), 31 (1+5+10+15), etc. US flags have historically used variants on the centered pentagonal and hexagonal numbers. Variations include adding stars in the corners or sides of the canton, omitting some of the rings, or slightly altering the number of stars in a ring. For example, the second ring of the centered hexagonal numbers contains 12 stars, but many historical flags increase this to 13 as an homage to the original 13 colonies. The 36-star flag below omits the third ring of the centered hexagonal numbers: 36 = 1 + 5 + 10 + 20.

Mathematician Skip Garibaldi wrote a computer program to figure out all possible combinations for flags of any number of stars. Writer Chris Wilson of Slate used this program made an interactive flag calculator based on the six most common star configurations used in US history for future flags of 51 to 100 stars. I took this a farther by creating a program that draws all viable flags for all 12 patterns in the articles above, many more patterns based on other historical flags, and a few variants of my own design.

You may thing this has been a colossal waste of time. If that is the case, I doubt you have read this far anyway. I disagree. I think it has been fun, has exercised my brain, sharpened my coding skills, and given me lots (in principle an infinite amount) of material for my blog!

Without further ado, the cantons.

Historical Arrangements

These are all patterns that I think have been used in past or present US flags. Since there was no standard arrangement of stars until Executive Order 1637 in 1912, several different arrangements were often used at once. In the table below, I show the most recent canton for each pattern. I give a name for each arrangement with the name from the Slate article, if applicable, in parentheses, along with the years that this arrangement was used. Many arrangements are named after the state whose admittance prompted that design.

My sources for what arrangements have been used historically are a Wikipedia article and US Flag Depot.

From Slate and Daniel Edwin

I started my program by reproducing the designs from the blog posts and articles above.

Name	Basis	Example
Odd CB, “White” Corners (Long)	CB	50 stars (1960-present)
Even CB (Alternate)	CB	49 stars(1959-1960)
Grid (Equal)	Grid	48 stars(1912-1959)
Michigan (Wyoming)	Grid	44 stars(1891-1896)
Colorado	Grid	38 stars(1877-1890)
Nevada	Grid	36 stars(1865-1867)
Circle Pentagonal	Circle	36 stars(1865-1867)
Kansas	Grid	34 stars(1861-1863)
Oregon	Grid	33 stars(1859-1861)

Algorithms I Developed

I have also perused the many other historical designs that have been used and included algorithms to reproduce many of them.

Name	Based on	Example
Oklahoma	Grid	46 stars(1908-1912)
South Dakota	CB	43 stars(1890-1891)
Colorado 2	Grid	38 stars(1877-1890)
Circle Hexagonal	Circle	38 stars(1877-1890)
Nebraska	Grid	37 stars(1867-1877)
Diamond	Grid	33 stars (1859-1861)
California	Grid	31 stars (1851-1858)
Alabama – Maine	Grid	27 stars(1845-1846)
Arkansas	Grid	25 stars(1836-1837)
Illinois	Grid	21 stars(1819-1820)

Expanded Arrangements

From Daniel Edwin

Mr. Edwin also crated three patterns that have not yet been used in real flags.

Name	Based on	Example
Odd CB, “Black” Corners (Short)	CB	49 stars
Modified CB	CB	49 stars
No Corners	Grid	50 stars

My Expanded Arrangements

Nevada – 4, Nevada – 8, and Colorado – 4 simply subtract more stars (4 or 8) from the simple grid design than the Nevada or Colorado patterns.  Even Checkerboard Inverted is just a mirror image of the Even Checkerboard. My main motive for creating these patterns and many of the historical patterns was to allow my algorithm to find viable arrangements for increasingly large numbers of stars. I currently have at least one viable arrangement for all star numbers from 1 to 4167.

Name	Basis	Example
Nevada – 4	Grid	78 stars
Nevada – 8	Grid	74 stars
Colorado – 4	Grid	50 stars
Even CB Inverted	CB	49 stars

Complete Table

Name	Basis	Example
Odd CB, “White” Corners (Long)	CB	50 stars (1960-present)
Odd CB, “Black” Corners (Short)	CB	49 stars
Even CB (Alternate)	CB	49 stars(1959-1960)
Even CB Inverted	CB	49 stars
Modified CB	CB	49 stars
Grid (Equal)	Grid	48 stars(1912-1959)
Oklahoma	Grid	46 stars(1908-1912)
Michigan (Wyoming)	Grid	44 stars(1891-1896)
South Dakota	CB	43 stars(1890-1891)
Colorado	Grid	38 stars(1877-1890)
Colorado – 4	Grid	50 stars
Colorado 2	Grid	38 stars(1877-1890)
No Corners	Grid	50 stars
Circle Hexagonal	Circle	38 stars(1877-1890)
Nebraska	Grid	37 stars(1867-1877)
Nevada	Grid	36 stars(1865-1867)
Nevada – 4	Grid	78 stars
Nevada – 8	Grid	74 stars
Circle Pentagonal	Circle	36 stars(1865-1867)
Kansas	Grid	34 stars(1861-1863)
Oregon	Grid	33 stars(1859-1861)
Diamond	Grid	33 stars (1859-1861)
California	Grid	31 stars (1851-1858)
Alabama – Maine	Grid	27 stars(1845-1846)
Arkansas	Grid	25 stars(1836-1837)
Illinois	Grid	21 stars(1819-1820)