Saturday, November 12, 2011

51 Star Flag

The 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?

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.

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 square 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:

Number of stars = rows x columns / 2

Also note that we can swap the colors and have a flag that is a mirror image of the original.

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.

That 9 x 11 pattern with the corners white is the standard pattern for the current United States flag.

This gives us two new equations:

Number of stars = ( rows x columns + 1 ) / 2   (corners white) 
Number of stars = ( rows x columns - 1 ) / 2   (corners black)

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:

Number of stars = ( rows x columns )

Putting it All Together:

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:

rows x columns = (Number of stars) x 2       even checkerboard flags
rows x columns = (Number of stars) x 2 - 1   odd checkerboard flags, white corners
rows x columns = (Number of stars) x 2 + 1   odd checkerboards flags, black corners
rows x columns = (Number of stars)           simple grid flags

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.

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:

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.

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:

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.

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.

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.

Friday, October 7, 2011

Deep Space to Scale

Celestial objects are bigger than you think.

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.

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.

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 degrees. For example, the Moon is about 0.5° 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 arcminutes. The Moon is, therefore, about 0.5 * 60 = 30 arcminutes across.

Below I have a collection of to-scale images of celestial objects.  All of these are scaled relative to how big they actually look in the night sky.  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.

The Basics - To Scale
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.

The Great Nebulae - To Scale
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.

The Great Nebulae - To Scale, Continued
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.

The Great Andromeda Galaxy - To Scale
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.

Famous Galaxies - To Scale
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.

The Pleiades - To Scale
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.

The Virgo Cluster - To Scale
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.

Assorted Small Objects - To Scale
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.

Human Visual Acuity - To Scale
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.

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.

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.

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.

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.

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.