Love Apple gear? Like math? TUAW’s Doing the Math series examines the numbers and the science behind the hardware and software.

The rumourmill has been busy lately with claims that we might get “Retina display” Macs soon — and of course, a Retina display iPad 3 on March 7, probably, maybe, definitely. For an example of the sort of speculation, consider Bjango developer Marc Edwards, who tweeted: “Retina 27″ Thunderbolt display: 5120×2880 = 14,745,600 pixels. 4K film: 4096×2160 = 8,847,360 pixels. Retina iPad 3: 2048×1536 = 3,145,728 pixels”. This prompted me to dust off my Retina display iPad post from a year ago and revisit the mathematics I applied there to dig a little deeper into what a Retina display Mac might entail. Is Edwards right — would a Retina display Thunderbolt display really need almost 15 megapixels?

## Isn’t this all just marketing?

Before I launch into a long-winded diatribe (“surely not!” — everyone who’s ever read any of my other TUAW posts), I need to address a surpisingly common point of view. Some people say that as “Retina display” was a term Apple made up, it can mean whatever it wants it to mean. If Apple wanted to, the theory goes, it could just declare the current iPad to be a Retina display and be done with it.

I think this argument is asinine. Firstly, although Apple invented the term out of whole cloth, it does offer a definition: “the Retina display’s pixel density is so high, your eye is unable to distinguish individual pixels.” That has meaning, and if Apple were to weaselly dilute the definition for the sake of marketing some future product I think we should absolutely hold its feet to the fire.

Secondly, this isn’t just about Apple. High DPI screens are starting to appear on other devices, like this Android tablet from Asus. The precise phrase “Retina display” might belong to Apple but the advantages of high resolution screens do not. As this is an emerging trend across the whole industry, it behooves us to strip away the marketing pixie dust and take an objective look at what this technology can offer.

## Defining “Retina display”

So what does it mean to say that a screen’s individual pixes are indistinguishable? The launch of the iPhone 4 and the first Retina display was, of course, accompanied by a jump in the screen resolution from 480×320 to 960×640 — from 163 pixels-per-inch (ppi) to 326 ppi. This in turn lead many people to label some arbitrary resolution as “Retina display” — typically 326 ppi itself, or 300 ppi. The latter number is a common rule-of-thumb baseline in the print industry for “photo resolution”.

It’s not that simple, however.

Hold a small-print book at arm’s length. Notice how it’s hard to read the text. Now bring the book up to a few inches from your nose. Notice how much easier it is to read now. Clearly, if Apple is defining a “Retina display” as “one where users can’t see the pixels” then any discussion of whether a given display qualifies or not needs to take into account the distance between the screen and the user — and that differs according to the device. An iMac on a desk, a MacBook in your lap, and a hand-held iPhone all have different viewing distances.

So, how do we determine how small a pixel has to be to be bordering on invisible? To answer this we need to think about subtended angles.

Consider the following scenario:

The viewing angle in this diagram, a, is called the angle subtended by the inter-pixel spacing, s. Whether or not a given detail is too small to be discerned by the eye is down to the size of this angle. This is how the size of an object is related to the viewing distance — as you move an object of a given size closer or further away from the eye, so the size of this angle changes. Conversely, at given distance, a larger object will also subtend a bigger angle. The size of the image on the retina is intrinsically derived from the object size and the viewing distance, linked by this formula:

So what subtended angle is too small to see? The average person has 20/20 vision. This was historically defined as the ability to read letters on a standard eye chart that subtend 5 arcminutes of angle (an arcminute is 1/60th of a degree). What does that mean in pixel terms? Consider that just about the smallest legible fonts, Tinyfont by Ken Perlin and Tiny by Matthew Welch, uses five pixels of height (including descenders for Tiny) for each letter. This suggests the smallest resolvable detail for an average eye is around one arcminute. Indeed, one arcminute is an accepted value amongst academics for the resolution limit of a typical human retina.

## Retina-ness of Apple’s current displays

With the data above in mind, and applying the mathematics from my previous post, we can take some typical viewing distances for different Apple devices, combine it with the screen size and resolution, and calculate how close the screen comes to the definition of a Retina display we have arrived at above.

You can view a Google spreadsheet that shows the details of how this data is calculated.

Just for fun, I threw in a couple of non-Apple devices for comparison — a 50″ TV at a distance of six feet, playing back a BluRay and a DVD; and the announced Asus Transformer Prime Android tablet, which has a 1920×1200 display.

This table shows some things that surprised me.

Firstly, it shows that Apple’s definition of Retina display aligns quite closely with my mathematic derivation. The iPhone 4 screen at a typical distance of 11″ is just barely above the threshold for a Retina display. I believe this justifies my methodology.

Secondly, it repeats my previous conclusion that a pixel-doubled iPad running at 2048×1536 is easily enough definition to count as a Retina display — even at a 16″ viewing distance, which is on the close side from my experimentation with an iPad and a tape measure. Similarly, that Asus tablet is a Retina display too.

It also shows that many current Mac displays are a lot closer to Retina display levels than you might have thought. The 27″ iMac at a distance of 28″, a 17″ MacBook Pro at 26″, an 11″ MacBook Air at 22″ — these screens all have pixels small enough to border on invisible.

Furthermore, the 480×320 iPhone screen is notably worse than everything else Apple makes today, at 53% of a Retina display. Even the second-worst 1024×768 iPad screen has finer detail at 61%. The worst Mac display is the 24″ iMac at a distance of 28″, at which distance its pixels are one-third too large to be individually indistinguishable.

Finally, this also shows why BluRay looks so good. On a largish TV at a shortish distance (50″ at 6′), a 1080p image is at 92% of Retina level, whereas a DVD is a downright poor 36%.

There are two very important points here.

The first is that in order to achieve, or even handily exceed, the threshold for a Retina display, Apple does not need to double resolutions on most of its displays. Far from it. It would suffice to boost a 27″ Thunderbolt Display from 2560×1440 to something around 2912×1638.

The second point is that people shouldn’t get their hopes up for how much better a Retina display Mac would be compared to the current offerings. The iPhone 4 was a huge step forward from the iPhone 3GS mostly because the 3GS’s screen was comparatively poor. Existing Macs have much better screens to start with, so any improvement will be much more modest.

## Looking beyond one arcminute

From the above, you might think that there is hardly any reason to Apple to change anything, because the benefits of higher resolution screens are so modest. But clearly HiDPI mode exists, and specialist medical imaging screens are between 508 and 750 ppi. What’s the benefit to these high pixel densities?

The answer is that our definition of the limits of human vision — details that subtend an angle of one arcminute — is rather simplistic. There’s a lot more to think about when considering how real human vision interacts with computer display technology, including atypical viewing distances, different sorts of patterns, and so forth. Reading words, for example, is possible at smaller sizes than reading random letters, because your brain has more context to guess at the characters. Your brain is a sophisticated pattern matching tool and it will use information from the surroundings to try and interpret details your eyes can’t quite make out clearly.

Here’s a number of test patterns for you to try this out on your own display. If you want to try this on an iOS device, you need to get the appropriate file for your device — iPhone or iPad — and save it to the Camera Roll. This is because iOS will helpfully try and zoom and pan images but we want to ensure that one pixel in the test image takes up one pixel on your display. Once you have them in the Camera Roll, view them full screen through the Photos app with your device in the portrait position. If you compare your Mac, iPad, and iPhone, you should see quite a difference in how well each screen performs.

## The pixel doubling argument

Rene Ritchie for iMore makes a solid argument for why an iPad retina display must be pixel-doubled — i.e. 2048×1536 — and not some intermediate resolution (just as was the case for the iPhone 4 before it). Anything else means every single existing app either has to re-scale art assets — resulting in a fuzzy display — or let them appear at a different size on-screen — resulting in usability problems as the tap targets are resized. This is because every single existing iPad app is hard-coded to run full screen in 1024×768.

The situation is fuzzier on desktop, however. Apple’s current displays already vary between 92 and 133 pixels-per-inch. Users are more tolerant of UI element resizing, within reason.

Consider the 109 ppi 2560×1440 27″ Thunderbolt display, and let’s suppose Apple wanted to Retina it up. It could up the resolution to 4192×2358 — which works out to 178 ppi — and achieve a display with finer details than the iPhone 4. This is one-third less pixels than the native pixel-doubled resolution (which would be 5120×2880). UI elements would look proportionally larger — but no more than they do on the 24″ iMac display today, so it wouldn’t look clumsy or odd.

## Wrapping up

Hopefully, I have convinced you of several things in this post.

• “Retina display” carries more meaning than pure marketing.
• The definition of what, and isn’t, a Retina display must consider viewing distance.
• The improvement you’d see from a Retina dispaly Mac is significant, but less than the improvement the iPhone 4 offered over the 3GS.
• A 2048×1536 iPad would be a Retina display and would look quite a bit better than the current model (but, again, be less of an improvement than the iPhone 4).

Still not convinced? Sound off in our comments!

I’d like to thank fellow TUAWers Brett Terpstra and Erica Sadun for helping me with the Retina Tester graphic.

Retina display Macs, iPads, and HiDPI: Doing the Math originally appeared on TUAW – The Unofficial Apple Weblog on Thu, 01 Mar 2012 22:00:00 EST. Please see our terms for use of feeds.