US Flag Day

[lpetrich]

 Flag Day (United States)

In the United States, Flag Day is celebrated on June 14. It commemorates the adoption of the flag of the United States on June 14, 1777, by resolution of the Second Continental Congress.[1] The Flag Resolution, passed on June 14, 1777, stated: "Resolved, That the flag of the thirteen United States be thirteen stripes, alternate red and white; that the union be thirteen stars, white in a blue field, representing a new constellation."[2][3]

 Flag of the United States and  List of flags of the United States have depictions of past and present and possible future flags.

The first flag design was Francis Hopkinson's design, with interleaved 3*3 and 2*2 patterns of stars -- and six-pointed stars, like the Star of David. The stars were soon simplified to fixe-pointed ones, pentagrams.

The circle of 13 stars is the Betsy Ross version, and there is a variant with a circle of 12 stars and the 13th one in the center, the Cowpens version.

There is a fourth 13-star version, the 4-5-4 version: 4 stars in the top row, 5 in the middle row, and 4 in the bottom row.

I'm posting on this day because I like the mathematics of US flag patterns.

 

[lpetrich]

For a rectangular grid of r rows and c columns, there are r*c stars.

Alternating between r rows of c stars and r of c-1 stars gives a total of r*(2c-1)

Interleaving (r,c) and (r-1,c-1) gives r*c + (r-1)*(c-1) stars, like in the Hopkinson flag - (3,3)(2,2) - and in the present-day flag - (5,6)(4,5)

The original 13 states: Connecticut, Delaware, Georgia, Maryland, Massachusetts, New Hampshire, New Jersey, New York, North Carolina, Pennsylvania, Rhode Island, South Carolina, Virginia.


When Vermont and Kentucky were admitted in 1795, the flag went from 13 stars and 13 stripes to 15 stars and 15 stars. The 15 stars were in a (3,3)(2,3) pattern.

More states meant more stars and more stripes, and while the stars were no problem, the stripes would either make the flag longer or else they would get thinner.

So when Tennessee, Ohio, Louisiana, Indiana, and Mississippi were admitted in 1818, the flag went to 20 stars but only 13 stripes to make the stripes easier to sew. The stars were in a (4,5) pattern.

When Illinois was admitted in 1819, the stars went up to 21, and were in a rather ugly 5-4-6-6 pattern. They could have been in a flipped (3,4)(3,3) pattern. There was an alternative version, with the 20 stars on the edges and corners of a pentagram.

When Alabama and Maine were admitted in 1820, the stars went to 23, in a 6-5-6-6 design, also not very pretty.

When Missouri was admitted in 1822, the stars went to 24, in a (4,6) pattern, 6-6-6-6.

When Arkansas was admitted in 1836, the stars went to 25, with 3 patterns: 6-5-7-7, (5,5), and (3,5)(2,5).

When Michigan was admitted in 1837, the stars went to 26, with 2 patterns, 7-6-6-7, and the 26 stars filling in an inverted pentagram. They could have been in a (2,7)(2,6) pattern.

When Florida was admitted in 1845, the stars went to 27, with 7-6-7-7. They could have been in a flipped (3,5)(3,4) pattern.

When Texas was admitted in 1846, the stars went to 28, with (4,7).

When Iowa was admitted in 1847, the stars went to 29, with 8-7-6-8, and also a diamond shape filled with 25 stars with a star on each star-region corner. The diamond is tilted (4,4)(3,3).

When Wisconsin was admitted in 1848, the stars went to 30, with (5,6).

 

[lpetrich]

I'll continue.

When California was admitted in 1851, the stars went to 31, with 7-5-6-6-7.

When Minnesota was admitted in 1858, the stars went to 32, with 7-6-6-6-7. It could have been (4,4)(4,4)

When Oregon was admitted in 1859, the stars went to 33, with 7-7-5-7-7, a diamond like for Kansas but with 2 pairs of stars on each corner, the stars filling a pentagram, and that diamond with the extra stars moved to near the left and right corners.

When Kansas was admitted in 1861, the stars went to 34, with 7-7-6-7-7, and also a pattern with a star in the center, 2 circles with 11 and 18 stars each, and a star in each corner.

When West Virginia was admitted in 1863, the stars went to 35, with (5,7), and also 2 circles with 12 and 19 stars, and a star in each corner.

When Nevada was admitted in 1865, the stars went to 36, with 8-6-8-6-8, and also a pattern with a star in the center with 3 circles with 5, 10, and 20 stars each. It could have been (3,6)(3,6).

When Nebraska was admitted in 1867, the stars went to 37, with 7-8-7-8-7, and also 8-7-7-7-8 and also a pattern with 2 circles with 13 and 24 stars each. The stars are not interleaved in the official version; they should have been interleaved: (3,7)(2,8).

When Colorado was admitted in 1877, the stars went to 38, with 7-8-8-8-7, and also a pattern with a star in the center, 2 circles with 13 and 20 stars each, and a star in each corner. The official version could have been two interleaved rectangles: (3,8)(2,7).

When North Dakota, South Dakota, Montana, Washington, and Idaho were admitted in 1890, the stars went to 43, with 8-7-7-7-7-7.

When Wyoming was admitted in 1891, the stars went to 44, with 8-7-7-7-7-8. Though it could have been 7-8-7-7-8-7 or (4,6)(4,5).

When Utah was admitted in 1896, the stars went to 45, with (3,8)(3,7).

When Oklahoma was admitted in 1908, the stars went to 46, with 8-7-8-8-7-8.

When New Mexico and Arizona were admitted in 1912, the stars went to 48, with (6,8)

 

[lpetrich]

When Alaska was admitted in 1959, the stars went to 49, with (4,7)(3,7).

When Hawaii was admitted in 1960, the stars went to 50, with (5,6)(4,5).

-

So the various official and semiofficial flag designs have had varying amounts of symmetry over the last two centuries. They have had mostly rectangular patterns of stars, though some have had circular patterns.

Possible 51-star flags: (3,8)(3,7) and a pattern with a star in the center and 4 circles with 5, 10, 15, and 20 stars each, making a disk of stars.

Possible 52-star flag: (4,7)(4,6).

Possible 53-star flag: (4,8)(3,7).

Possible 54-star flag: (6,9).

Possible 55-star flag: 6-9-8-9-8-9-6.

I've thought of more.

Possible 56-star flag: (7,8) or (4,8)(3,8).

Possible 57-star flag: (3,10)(3,9).

Possible 58-star flag: ?

Possible 59-star flag: (5,7)(4,6).

Possible 60-star flag: (6,10).

I've gone as far as I want to go.

 

[lpetrich]

I've found some sites that feature collections of antique US flags.

ZFC - Home - the Zaricor Flag Collection

Lots of goodies in it, like a 13-star flag with hexagram stars in a filled hexagram pattern - a Star of David - a 1-4-3-4-1 flag.

 Star of David discusses how the Star of David became a Jewish symbol.

Also one with a central star, a ring of 8 stars, and a star on each corner.

Also lots of 4-5-4 flags.

Rare Flags - Antique American Flags, Historic American Flags

Like a flipped 2-3-3-3-2 13-star flag. Flipped meaning that the numbers are for columns instead of rows. A flipped Star of David flag would be 2-1-2-3-2-1-2.

Also a rectangle of 12 stars with a center star.

A lot of the flags do not seem to have very precise sewing, and in some of them, the stars seem randomly placed.

I looked in that site, and it seems to have independently invented the notation that I use for approximately rectangular star patterns: #1-#2-#3-...-#(n-1)-#n where the #k are the number of stars in each row.

The Hopkinson pattern is (3,3)(2,2) or 3-2-3-2-3.

The site has names for some arrangements of stars. Forming a big star is a "Great Star", and concentric circles is a "Medallion".

 

[lpetrich]

Gore Vidal once expressed disdain for what he called the "proto Op Art" appearance of the US flag. I checked on  Op art and it is about artwork that features optical illusions. I don't see anything Op Art in any of the US flag designs.

 

[ZiprHead]

Many don’t know this, but the current design of the American flag was actually created as a school project by Saginaw native, Robert G. Heft. Before 50 states were recognized, the flag was designed differently. Bob’s headstone, which is in Holy Cross Lutheran Church Cemetery on Brockway here in town- is a beautiful tribute to him and a stunning piece of art for a man who, in my opinion, is under appreciated. Bob created the design as a 17 year old student and was given a B- as his grade ?????? His design, along with appx 1500 others, was given to desk of sitting President Dwight D. Eisenhower, and chosen to represent the Free world. Afterwards, his teacher changed his grade to an A. He passed away in Saginaw, by heart attack and rests here as the first tomb on the left as you pull in.
Happy National flag day!

 

[hyzer]

Growing up in Silver Spring, Maryland, right on the north "point" of Washington DC, I have long been fascinated by the geometry of the District. A perfect square, 10 miles by 10 miles, placed on exactly 45 degrees and inset into (well, carved out of) Maryland. So, thinking about adding DC as the 51st state, this is my suggestion of how to add DC, and keep the current pattern.

Also this would allow all existing flags to be retrofitted for the addition of DC . . . . not very capitalistic, but there ya go.

 

[hyzer]

View attachment 34085

Many don’t know this, but the current design of the American flag was actually created as a school project by Saginaw native, Robert G. Heft. Before 50 states were recognized, the flag was designed differently. Bob’s headstone, which is in Holy Cross Lutheran Church Cemetery on Brockway here in town- is a beautiful tribute to him and a stunning piece of art for a man who, in my opinion, is under appreciated. Bob created the design as a 17 year old student and was given a B- as his grade ������ His design, along with appx 1500 others, was given to desk of sitting President Dwight D. Eisenhower, and chosen to represent the Free world. Afterwards, his teacher changed his grade to an A. He passed away in Saginaw, by heart attack and rests here as the first tomb on the left as you pull in.
Happy National flag day!

I have seen several sites that state that the interlaced (4,5)(5,6) pattern was submitted by several, if not many, persons, and was even already "drafted" as a possible 50 state flag by government workers. The selection of Robert Heft as the creator was a public relations effort, and not strictly an award for a singularly original design.

 

[lpetrich]

I've seen discussion of the US annexing Canada, and while I consider that very unlikely, I think it interesting to explore what would happen to the US flag as a result of that.

Canada has 10 provinces and 3 territories -  Provinces and territories of Canada - and each province would presumably become a US state.

That means 60 states with a (6,10) flag, likely a flipped (5,6)(5,6) ones

61 states: (6,6)(5,5)

62 states: ?

63 states: (7,9)

64 states: (8,8) or (4,8)(4,8)

65 states: (5,7)(5,6)

 

[lpetrich]

USFlag.org: A website dedicated to the Flag of the United States of America - Standard Proportions For The United States Flag

Overall:
- Height: 1.0
- Width: 1.9
Blue rectangle (the "union"):
- Height: 7/13 = 0.5385
- Width: 0.76 = 0.4*1.9
Stripes:
- Height: 1/13 = 0.0769
- Width: 1.24 or 1.9 as appropriate
Star offsets:
- Height: 0.054 = 7/130
- Width: 0.063 = 1.9/48
Star diameter:
- 0.0616

The stars have coordinates (ir+1)/(r+1) (ir is 0 to r-1) and (ic+1)/(c+1) (ic is 0 to c-1)
In the current pattern, r = 9 and c = 11, with ir+ic being even.

The aspect ratio of the union is width/height is 1.411 = 1/0.709

For interleaved rectangles, r = r1+r2 and c = c1+c2 for (r1,c1)(r2,c2)

 

[Rhea]

Growing up in Silver Spring, Maryland, right on the north "point" of Washington DC, I have long been fascinated by the geometry of the District. A perfect square, 10 miles by 10 miles, placed on exactly 45 degrees and inset into (well, carved out of) Maryland. So, thinking about adding DC as the 51st state, this is my suggestion of how to add DC, and keep the current pattern.

Also this would allow all existing flags to be retrofitted for the addition of DC . . . . not very capitalistic, but there ya go.

I love it. Especially the easy retrofit.

 

[lpetrich]

The interleaved patterns one can unify as (r,c)(r-1,c-1) = (2r-1,2c-1,I)
Thus, the current 50-star flag is (9,11,I)
For (r,c,I), with a star present on at least one corner, the number of stars is ceiling(r*c/2).
With r and c odd, a star can be present at no corners, giving a number floor(r*c/2)

Let's look at some other patterns.

Hollow rectangle: for r rows and c columns, 2r + 2c - 4 = 2(r-1) + 2(c-1)

Hexagons and hexagrams are easy, since they can be filled in with a plane tiling. Also, their star-point line length is the same as their core-edge length, since the star points are equilateral (regular) triangles.

For size factor n,

• Hexagon - edges: 6*n
• Hexagon - filled: 3*n*(n+1) + 1
• Hexagram - point edges: 18*(n-1) + 12
• Hexagram - inner and point edges: 24*(n-1) + 13
• Hexagram - filled: 6*n*(n+1) + 1

Here's a place that one can find a hexagram "great star": above the eagle in the seal of the US House of Representatives.

Pentagons and pentagrams are more difficult to analyze, but my experience with hexagons and hexagrams should help out.

 

[hyzer]

Growing up in Silver Spring, Maryland, right on the north "point" of Washington DC, I have long been fascinated by the geometry of the District. A perfect square, 10 miles by 10 miles, placed on exactly 45 degrees and inset into (well, carved out of) Maryland. So, thinking about adding DC as the 51st state, this is my suggestion of how to add DC, and keep the current pattern.

Also this would allow all existing flags to be retrofitted for the addition of DC . . . . not very capitalistic, but there ya go.

I love it. Especially the easy retrofit.

Thanks! My son posted to Reddit, where is received mostly derisive comments. But again, there ya go. I really appreciate your comment Rhea.

 

[skepticalbip]

I have long been fascinated by the geometry of the District. A perfect square, 10 miles by 10 miles, placed on exactly 45 degrees.

That was the original start, 100 sq.mi. (10mi. x 10mi) straddling the Potomac River, partly ceded by Maryland and partly ceded by Virginia. The idea being that the capitol of the U.S. not be in any state. It was later decided that the area south-west of the Potomac (that was ceded by Virginia) wasn't needed so was granted back to Virginia. This left DC with 68.34 mi², no longer a square but with a irregular SW border of the Potomac.

 

[lpetrich]

Returning to this subject, I turn to pentagrams.

It has the problem that its point edges are about 1.618 times longer than the central pentagon's edges. That's the golden ratio, (sqrt(5)+1)/2.

• Pentagon - edges: 5*n
• Pentagon - filled: (5/2)*n*(n+1) + 1
• Pentagram - edges: 5*(nc + 2*np - 1)
• Pentagram - filled: 5*n*(n+1) + 1

nc = number of core-edge points, np = number of point-edge points

The Bliss Science Museum: 51 Star Flag discusses rectangular star patterns for possible future US flags.

Author Dan Bliss came to the conclusions that I did about rectangular patterns. For a checkerboard pattern, a pattern with alternating full-empty squares that form continuous diagonals, one can distinguish three cases: even (total number of squares is even), odd positive (total number of squares is odd, corners filled), and odd negative (total number of squares is odd, corners empty).

For r rows and c columns, the number of stars is:

• Grid: r*c
• Chkbd even: (r*c)/2
• Chkbd odd +: (r*c+1)/2
• Chkbd odd -: (r*c-1)/2

 

[lpetrich]

Even chkbd: /2, Odd pos chkbd: /2+, Odd neg chkbd: /2-

• Original 13 (CT, DE, GA, MA, MD, NC, NH, NJ, NY, PA, RI, SC, VA) -- 5*5/2+, 3*9/2-
• 15 (KY, VT 1795) 5*6/2
• 20 (IN, LA, MS, OH, TN 1818) 4*5
• 21 (IL 1819) 6*7/2
• 23 (AL, ME 1820) 5*9/2+
• 24 (MO 1822) 4*6
• 25 (AR 1836) 5*5, 5*10/2
• 26 (MI 1837) 4*13/2
• 27 (FL 1845) 6*9/2
• 18 (TX 1846) 4*7
• 29 (IA 1847)
• 30 (WI 1848) 5*6
• 31 (CA 1851) 9*7/2-
• 32 (MN 1858) 8*8/2, 5*13/2-
• 33 (OR 1859) 6*11/2, 5*13/2+
• 34 (KS 1861)
• 35 (WV 1863) 5*7
• 36 (NV 1865) 6*6, 6*12/2
• 37 (NE 1867) 5*15/2-
• 38 (CO 1877) 7*11/2+
• 43 (ID, MT, ND, SD, WA 1890) 5*17/2+
• 44 (WY 1891) 8*11/2
• 45 (UT 1896) 6*15/2
• 46 (OK 1908) 7*13/2+
• 48 (AZ, NM 1912) 6*8
• 49 (AK 1959) 7*14/2
• 50 (HI 1960) 9*11/2+

 

[lpetrich]

Here are some possible flags with more stars, courtesy of myself and Dan Bliss.

• 51 -- 6*17/2
• 52 -- 8*13/2
• 53 -- 7*15/2+
• 54 -- 6*9
• 55 -- (DB) 10*11/2
• 56 -- 7*8, 7*16/2, (DB) 8*14/2
• 57 -- 6*19/2
• 58 -- (DB) 9*13/2-
• 59 -- 9*13/2+
• 60 -- 6*10
• 61 -- 11*11/2+
• 62 -- (DB) 5*25/2- (DB: "too squished!")
• 63 -- 7*9
• 64 -- 8*16/2
• 65 -- 10*13/2
• 66 -- (DB) 6*11
• 67 -- (DB) 9*15/2-
• 68 -- (DB) 8*17/2
• 69 -- (DB) 6*23/2
• 70 -- (DB) 7*10

 

[lpetrich]

The Bliss Science Museum: 51 Star Flag (continued)

Presents some patterns that were much like the actual patterns of some 19th-cy. flags.

Notes Mathematical Model Shows What Future Flags Would Look Like as U.S. Grows

which notes A mathematician figures out the best way to jam an extra star onto the American flag.

Basically discussed the geometries that Dan Bliss and I have discussed.

Flag timeline -- another source on historical US flags. It and Wikipedia both show computer-graphics versions, and despite their mathematical perfection, those versions are not really sources. A good source would be an actual old flag or a photograph of one.

 

[lpetrich]

I'll count up the possibilities for hexagons and pentagons and hexagrams and pentagrams.

• Filled hexagon: 3*n*(n+1) + 1
• Filled pentagon: (5/2)*n*(n+1) + 1
• Hexagram outline: 6*(3*n-1)
• Pentagram outline: 5*(n+2*m-1) -- n for the core, m for the projections
• Filled hexagram: 6*n*(n+1) + 1
• Filled pentagram: 5*n*(n+1) + 1

I have a list of numbers of stars:
(F5,6), (F6,7), (O5r,10), (F5r,11), (O6r,12), (F6r,13), (F5,16), (F6,19), (O5rx,20), (O5r,25), (O6r,30), (F5,31), (F5r,31), (O5rx,35), (F6,37), (F6r,37), (O5r,40), (O6r,48), (F5,51), (O5r,55), (O5rx,60), (F5r,61), (F6,61), (O6r,66), (O5r,70), (F6r,73), (O5rx,75), (F5,76), (O6r,84), (O5r,85), (F6,91), (O5r,100), (O5rx,100)

F5, F6 = filled pentagons/hexagons
O5r, O5rx, O6r = outline pentagrams/hexagrams
F5r, F6r = filled pentagrams/hexagrams

O5r: m = n for projections -- a bit more spaced in projections than in the core boundary
O5rx: m = round(g*n) for projections, g = golden ratio = (sqrt(5)+1)/2 = 1.61803... -- for same spacing in projections and core boundary