What PPI does the human eye have?

Ever since the iPhone 4 was announced there has been a lot of talk about "how many PPI the human eye can see" and if the iPhone 4 really had a retina display. Today I thought I'd clear up some confusion.
Is the 326 PPI the iPhone 4 and above enough to be classified as a "retina" display?



Short answer: Yes

Long answer: For the average person at a normal distance it is. For a lot of people, and in a lot of situations it is not enough though. At the bottom of this page I have made a graph showing exactly what PPI you need to not see individual pixels depending on your eye sight and distance.

Before I show you the graph I am going to explain why it looks the way it does, what everything means and how I calculated it.
The first thing we have to understand is how to measure eye sight. To measure this we use something called an "arc minute".
The reason we don't use for example millimeter is because that does not take distance into consideration. 60 arc minutes is the same as 1 degree. If you can see 60 arc minutes, then it means that you can separate two dots that are 1 degree from each other (imagine a triangle coming out from your eye, and then measure the degree of difference between two dots to figure out how many degrees they differ, pic related):

(image source: wolfcrow)

OK so now that we know what an arc minute is, let's calculate how good our human eyes are.

A human with 20/20 vision can see down to 1 arc minute of difference. So if something is 1/60 of a degree away from each other, you can see them as two separate dots. Please bear in mind that 20/20 is considered normal vision. 20/20 means that you can see what a normal person see at 20 feet away. The first number is the distance the person needs to see something, and the second number is what a normal person needs. So if someone has 20/12 vision, they can see something at 20 feet away a normal person needs 12 feet to see. The reason I mentioned 20/12 is because that is actually not too rare to have. Some of the best recorded eye sights have been 20/8 which is just insane (a hawk has 20/2 for those of you who are interested).

So here is what some different eye sights can see:

20/20 - 1 arc minute

20/12 - 0.6 arc minute

20/8 - 0.4 arc minute

Most people with good eyes have between 20/12 and 20/20.

Your average adult can focus at as close as 4 inches away (100 millimeter). Younger people can focus at things closer than that but let's do our calculations on your average adult to keep things fair and simple.

Here is the formula for calculating the perceived size of an object: p = 2 * d * tan (α/2) where p is the size, d is the distance in millimeters and α is the angle in degrees. (here is an online calculator, you want to use "linear size from angular size and distance)

Here are the pixel sizes assuming you look 100 millimeters away.
1 arc minute = 0.029 millimeters
0.6 arc minute = 0.0175 microns
0.4 arc minute = 0.0116 microns

All that is left now is to calculate the PPI if each pixel is a specific size. (I couldn't find a pixel pitch to PPI calculator so I had to use this site and change the resolution until I found the correct dot pitch and check what the PPI corresponded to)
1 arc minute - 874 PPI
0.6 arc minute - 1447 PPI
0.4 arc minute - 2181 PPI

Please note that I have done some rounding up/down on certain numbers so that I didn't have to use like 20 decimal points.

TLDR: Your average adult with normal vision can see up to 874 PPI. Some people with really really good eye sight (not super human, just really really REALLY good) can see over 2000 PPI. Don't underestimate the human eye. That's from 4 inches away though, and that's not really a normal viewing distance.

Here is the graph. Please note that I have done quite a bit of rounding up and down so your results may vary.

You should be able to use this as a rough estimation though. As we can see, someone with 20/20 vision can only see 291 PPI at 30 centimeters away, which is a typical viewing distance. That means that the iPhone's screen really is better than our retina, if you got average vision and hold it ~30 centimeter away from your eyes. If you hold it closer and/or have better than average vision then you should be able to see the individual pixels in the best possible conditions.

(click to enlarge)