Why I won't care about 4K resolution until I have a bigger TV

So along with the OLED wave coming, 4K TV seems to be coming.  4K TV is roughly 4 times the pixels of 1080p: 2x horizontal and 2x vertical.  Now, I think a bajillion pixels per inch would be cool, but would it really matter for the average viewer?  To be more specific, would anyone sitting 6 feet from a 47" TV display even notice a difference between 1080p and 4K resolution TVs?

Let's get some numbers and do some quick geometry.

Fact 1: According to Wikipedia, a healthy (20/20) human eye can discriminate two points that are 1 arc-minute, or 1/60 of 1˚ apart.  How this translates to width depends on the distance between the viewer and the points, as will be discussed.

Fact 2: 1080p means "1080 pixels tall by 1920 pixels wide",  4K means "2160 pixels tall by 4096 pixels wide".

Fact 3: A 47" TV is 23" tall and 41" wide. (This assumes a 16:9 aspect ratio.  If you are really good at mental math, you'll have noticed that 4096:2160 is a 256:135 aspect ratio.  For the rest of this, I'll just pretend that everything is 16:9).

Okay, now the geometry.  We know the minimum angle your eye can resolve, and we're assuming you sit 6 feet from your TV, so we can calculate the minimum distance between points (x below) your eye can resolve.

So, at a viewing distance of 6 feet we know that your eye can distinguish points as long as they are 0.021 inches apart.  What we want to know now is whether the pixels on some TV are far enough apart that your eye can tell the difference.  Instead of pixels per inch, we really want to know the inches per pixel, which we can calculate like this:

Now, let's look at the pixels on the 1080p standard on a 47" TV.  If we just take the horizontal axis, 1080p means 1920 pixels horizontally, and 47" TVs are 41" wide, so we can calculate the distance between horizontal pixels as follows:

So how would it look if we had a 47" 4K display?

The Verdict

The goal for a good TV is to get as many pixels as possible close together, but once you pass the threshold where your eye can't tell the difference any more, it does you no good!  So the test is this: is the distance between pixels on a TV smaller than the distance between points your eyes can distinguish?  

For a 47" TV from 6 feet, pixels are about 0.022 inches apart, and your eye can only distinguish points 0.021 inches apart.  So, if you, like me, are watching a 47" TV from 6 feet, then congratulations!  Your TV has about as many pixels as your eye could ever care about!  So smile and relax, knowing that if someone swapped your set for a 4K display, you wouldn't even notice!  Unless you are sitting closer than 6 feet, you have it made!

If, however, you catch your shows on a 65 incher, or you really like to sit close when you play Halo, be excited for the 4K future of TV!  Of course, at 4x the file sizes, can you imagine how much buffering you'll sit through watching 4K videos on YouTube?!

To appease your curiosity,  I'm including some lines from my spreadsheet looking at other TV sizes.  Remember, the 0.021 inches "resolution" of your eye only applies if you sit 6 feet away, and these show the numbers for size and pixels in the horizontal axis (though vertical should be the same ratio).