iPads and Retina Displays: Doing the math

Over the weekend, we saw the swirling rumors around the specs for the (presumably inevitable) iPad 2 start to come together. One of the most intriguing suggestions, which Engadget claims to have a reliable source for (and MacRumors some corroborating evidence to boot) is a higher resolution screen to match the iPhone 4's Retina Display -- specifically, doubling in both directions, changing from 1024x768 to 2048x1536.

This has prompted some discussion around exactly what Retina Display means, and whether this would count. The iPhone 4's screen is a mammoth 326 pixels-per-inch (ppi), whereas this rumored new iPad resolution is a somewhat lesser 264 ppi -- quite a bit less. However, I believe it's just as valid for Apple to call this a Retina Display as it was to call the iPhone 4 screen, and after the break I will explain why with some hopefully convincing mathematics.

Firstly though, it's important to stress that these are only rumors and that 2048x1536 is an incredible number of pixels -- 3,145,728 of them, in fact. That's only 17 percent less than the 27" iMac or 27" Cinema Display, and it's 52 percent more pixels than a 50" 1080p television screen! This makes the screen expensive to make; it places greater strain on the graphics chipset to drive the screen, which makes that more expensive, too; it won't do the battery life any favors either. All of this, to my mind, suggests this is one rumor that might come down to wishful thinking. As John Gruber said: "I'll believe it when I see it."

Apple could mitigate these problems by choosing some display size between the current one and the rumored twice-in-each-direction increase, but that brings fresh problems. The iPhone 4's Retina Display increased resolution from 480x320 to 960x640 -- so each single pixel on the old screen became a block of 2x2 pixels on the new screen. This means developers didn't have to do anything to their apps to support this new screen; they would look exactly the same as they did before. Many devs did upgrade their apps to add higher-resolution graphics, of course, but the important thing was that everybody standing in line on launch day could bring every app they'd already bought over to the new phone unchanged. That's the sort of seamless user experience we expect from Apple, after all.

If Apple used some intermediate resolution for the iPad 2, then all existing iPad apps would have to be scaled by some awkward factor by the hardware, and there's no good-looking way to do that. (Update: TiPb have a discussion of this with some nice pictures demonstrating the problem.) If you try to take what used to be two pixels of display, and remap it to fill three pixels instead, then it's a one-way trip to visual artifact city. The day one user experience for everyone upgrading from the mark 1 iPad would be that their existing apps look worse -- which doesn't make for happy users. So if it is going to increase, it seems likely it will indeed increase by a factor of two in both dimensions -- hence the magic value of 2048x1536.

What makes a Retina Display?

The term "Retina Display" seems to be widely interpreted in the Mac blogging world as meaning a display that has more than 300 pixels-per-inch, as that's the limit the human eye can perceive. Hence the iPhone 4's 326 ppi screen is therefore sufficiently pixel-packed that the naked eye can't see the pixels, and a 2048x1536 iPad screen wouldn't be a Retina Display because it only manages 260ppi. This, however, ignores the important factor of viewing distance.

The capability limit of the human eye depends on how close you hold things, and that's not accounted for in the 300 ppi figure. The usual figure quoted in the literature for 20/20 vision is that the eye can tell the difference between two lines that are more then one arc minute apart -- i.e., 1/60 of a degree. Consider two lines a millimeter apart. If you hold it right up to your eye, you'll see two lines; move it 10 feet away, and it'll look like a single line instead. So if the question is "how small does a pixel need to be before I can't see it any more," then the answer has to depend on how far away from the screen you are.

As an aside, it's also worth stressing that 20/20 vision means average; it's not some sort of gold standard for vision. It means that when you're standing 20 feet back from an eye chart, you can read the same amount of text as a normal person can. If you have 40/20 vision, it means you can be 40 feet back from that chart, and still read the same amount as a perfectly average person who is 20 feet back. Similarly, if Hans Moleman has 5/20 vision, he has to walk just five feet away before he can read the chart. All the calculations below are for people with average vision -- eagle eyed users will always see more detail.

So, consider the case of a screen with two adjacent pixels and your eye some distance away, as shown below in a very not-to-scale diagram:

Good old SOH-CAH-TOA trigonometry tells us that tan(a/2) = s/2d. If we assume 20/20 vision, we know a is 1/60th of a degree, and if we consider the iPhone 4's 326ppi display, we know s is 0.0779mm. This gives us a value for d of 26.8 centimeters. In other words, for a person with 20/20 vision, they can not discern the individual pixels on a Retina Display screen at a distance of around 10.5 inches.

I tried out holding my iPhone 4 in a variety of common poses, and found that whether I was sitting, standing or lying down, I generally held it in my left hand with the elbow crooked, so it ended up between 10 and 12 inches away from my eyes. This, then, was the legitimacy of the "Retina Display" term; for users with average vision, holding their phone normally, the pixels are smaller than their eye can make out the details of. The Register has a good writeup of this from last year when the iPhone 4 was released that has a couple of neuroscientists going into a bit more detail.

Now, let's consider an iPad -- a current one, with the 1024x768 screen -- 132ppi. That gives us a value for s of 0.1924mm, and a value for d of 66cm or 26 inches. I'm a fairly short guy, so for me that translates to holding the iPad slightly further away than arm's length.

Now, what about this rumored iPad 2 with 264ppi? That brings the critical distance back down to 13 inches. Sure, this is higher than the iPhone 4's 10.5", but when I'm using my iPad, I'm usually either sitting with it in my lap or lying in bed with it propped on my belly, largely because it's too heavy to hold in one hand for prolonged periods. In all these poses it's between 15 and 20 inches from my eyes.

Conclusion

I think it's fair to say that, for users with average eyesight using their iPad in fairly normal ways (i.e., in their lap, on a desk, or on their chest when lying down), the rumored new display could legitimately be called a Retina Display despite having a lower pixel-per-inch figure than the iPhone 4.

Of course, there is a big footnote here -- Retina Display is a marketing term. It doesn't have a scientific definition, and it's not an ISO standard; it means whatever Apple's marketing folk want it to mean, and they were free to call it a Retina Display anyway -- but if you see any anti-Apple bashing for allegedly misusing the term, you can point 'em here and straighten 'em out.