Chasing the 6.4 Megapixel Unicorn

Mount Fuji in Lion DP2

The folks over at OS X Daily took note that the latest Developer Preview for Mac OS X Lion includes a new wallpaper for Japan’s Mount Fuji. I usually pay no attention to this minutia — I mean, seriously, wallpaper is news? — but before clicking past, I noticed that they wrote that it is “a rather large 3200 × 2000 pixels.”

3200 x 2000 pixels is large. As a matter of fact, it is greater than the native resolution of the Apple 27” LED Cinema Display and even beyond the specifications of the now-discontinued 30” Apple Cinema Display, the largest display Apple has ever shipped.

It seems odd that Apple would intentionally ship an image with those dimensions when no current display is capable of rendering it. Those well-versed in Apple Kremlinology will tell you that this is a signal that new hardware is on its way, something spec’d with a whopping 6.4 million pixels.

But who knows, really, except Steve Jobs and John Gruber? Besides, raw resolution is much less interesting to me than pixel density, the so-called “DPI” or “dots-per-inch.” The iPhone 4’s Retina Display demonstrates the superiority of this metric to me every day. My iPad has 28% more pixels (1024 x 768 versus 960 x 640) for example, but they are distributed across 4 time the area. The resulting difference in sharpness between the two is nothing less than dramatic.

How long until we see a similar “Retina Display” for other devices? Might this mythical new 3200 x 2000 hardware — let’s call it the “Unicorn Cinema Display” — be capable of showing Mount Fuji pixel-for-pixel in all its glory? And would it represent a new Retina Display for Apple, the first outside of the iPhone? The unsatisfying answer, of course, is “it depends.”

Defining a Retina Display

First, it depends on what a “Retina Display” means. Apple’s page on the iPhone’s Retina Display defines it as “pixel density…so high, your eye is unable to distinguish individual pixels.”

Steve Jobs, at the announcement of the iPhone 4, had a similar description:

It turns out there’s a magic number right around 300 pixels per inch, that when you hold something around to 10 to 12 inches away from your eyes, is the limit of the human retina to differentiate the pixels.

But exactly when individual pixels “disappear” is not a simple calculation. Notice that Jobs doesn’t simply mention the DPI of the display, but also the distance from which you view it. This makes sense, if you think about it. A foot away from your eye, an iPhone 4 is pixel-free. Move it 6 inches closer, however, and now your eye is able to resolve some of the “jaggies” that were hidden before.

The Calculation

Let’s start off, then, with a few baseline measurements for existing displays and compare them to the heavily-anticipated “Unicorn”1:

27” LED Cinema Display 30” Cinema Display iPhone 4 Retina Display New “Unicorn” Cinema Display
Horizontal Resolution 2560 2560 960 3200
Vertical Resolution 1440 1600 640 2000
Size (diagonal) 27” 30” 3.5” ??
Dots Per Inch 109 101 326 ??

But wait. Why the question marks? Well, obviously, we cannot calculate a DPI for the new “Uni” unless we know its size. And since that kind of information is what Donald Rumsfeld would call “a known unknown,” we are going to need to approach this problem from the other side.

Now, wait a cotton, pickin’ arc-minute

We need two things to answer our Retina Display question. We have to calculate the “Unicorn-dog’s” DPI, and we also need to know the distance at which it will be viewed. Let’s focus on this latter one first.

In his exploration into this topic last year on, Phil Plait went into great detail on the biomechanics behind Apple’s iPhone 4 Retina Display claims. It is definitely worth a read, but it comes down to this: if an individual pixel is smaller than the smallest element a person can resolve at a given distance, then that display can be considered a “Retina Display” by Apple’s definition.

But what value should we use for “a given distance”? The United States Department of Labor’s Occupational Safety & Health Administration suggests that the optimal distance to sit from a display is 20 to 40 inches. Forty inches seems quite far to me, and besides, it makes the DPI needed to qualify as a Retina Display much lower. Even 20 inches seems generous, especially for a laptop. There is no one “best” distance, so for the sake of this exercise, I’m going to use three: 18, 20 and 24 inches.

With that information, we can now look at the pixel size demanded of a Retina Display. Plaitt notes that a person with 20/20 vision has a visual acuity of 1 arc-minute, the equivalent to 1/60th of a degree. At 18 inches, that means your eye can resolve pixels larger than 0.00524 inches (5.24 thousandths of an inch). At 20 inches, a pixel needs to be larger than 0.00582 inches to be resolved individually. And at 24 inches, that number grows to 0.00698 inches.2

How big is a unicorn?

We now have everything we need, except the DPI of the “Une.” And since we don’t know its size, let’s consider the various sizes that it might be. In terms of monitors, Apple offers only the 27” LED Cinema Display today, and the “Jimmy-Cracked-Unicorn” could be another, higher-resolution version of that hardware, perhaps even marking the return of a 30” model. But might it also be a component, an LCD panel designed to replace the one inside an existing device? Let’s consider both possibilities.

Apple currently ships products with display sizes that range from 1.5 inches to 27 inches. I doubt we’ll see the Nano sporting 3200 x 2000 pixels anytime soon (ever?), so let’s start at the iPad and its 9.7-inch screen. Working up from there, we have the 11.6-inch one from the MacBook Air, a 13.3-inch component in the larger MacBook Air and the MacBook Pro,3 and so on, up through the 27-inch unit in the LED Cinema Display and largest of the iMacs.

If the displays on each of these devices were replaced with one of equal physical size but with the “Unicorn-on-the-Cob’s” 3200 x 2000 resolution, you would end up with the specifications shown in the table below:

Diagonal Size 9.7” 11.6” 13.3” 15.4” 17” 21” 27” 30”
DPI 389 343 290 252 222 180 140 126
Pixel Size (in.) 0.00257 0.00292 0.00345 0.00397 0.00450 0.00556 0.00714 0.00794

The question now is which one of these possible options represents a Retina Display? Using the visual resolution for 20/20-sighted people we calculated above, we simply need to note whether the pixels on the display at a given size and distance from the screen are smaller than the resolving power of the human eye. If so, then Apple could reasonably be expected to market it as a Retina Display:

Diagonal Size 9.7” 11.6” 13.3” 15.4” 17” 21” 27” 30”
Retina Display @ 18″? Yes Yes Yes Yes Yes No No No
Retina Display @ 20”? Yes Yes Yes Yes Yes Yes No No
Retina Display @ 24”? Yes Yes Yes Yes Yes Yes No No

Notice that even for the closest viewing distance of 18 inches, a 3200 x 2000 resolution represents a Retina Display for display sizes up to and including 17 inches. And of course, that covers the entire range of Apple laptops on the market today.

For smaller display sizes, a “Full Unicorn” would obviously not be needed to achieve the same Retina Display label. A 13.3” MacBook Air, for example, would need a screen providing approximately 2200 x 1375 pixels (191 dpi) to be considered a Retina Display at an 18-inch viewing distance.

So, what does this tell us, other than the fact that I had a lot of spare time on a Sunday evening? Well, just as I said at the start, “it depends.”

There are a lot of “ifs” in this exercise. If the 6.4 megapixel Mount Fuji wallpaper is not some sort of fluke, and if Apple does indeed have a “Unicorn” in its labs that can display Fuji at its full resolution, and if they ship it in their laptop or iPad line, then Retina Displays might be coming to nearly all Apple products.

Or, it could just be a very large photo of Mount Fuji.

  1. All DPI calculations were performed using the DPI calculator available here

  2. All acuity calculations done using the same tool Plaitt used, available here

  3. Same physical size, but different resolutions, though this is technically irrelevant to this discussion. 

Posted 04 April 2011 under Technology

Comments (17)

  1. It’s fun to speculate, isn’t it? Thanks for doing all of those calculations. I think that it is natural for a new generation of displays to come on line soon. We’ve been using the region of 1024-768 to 12xx – 800 and not much larger for many, many years now. I also expect that Apple will lead the curve with adopting these new high res displays. If I could also make a bet, the next gen Macbook Pro with be a redesigned model and will carry this new display. If we were so lucky, all of their laptops would use them, in addition to the next iPad.

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  3. Nice use of maths, there – I did something similar for TUAW some time back:

    Question though. If this new wallpaper was designed for new displays, why is it square? It’d be 16:9 or 16:10 surely?

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  6. No need for calculations and sheets. Replace the pixel per inch (ppi) to Pixel Per Degree (ppd) (and recalculate the values), then we get the real measure unit of displays’ resolution quality.

    For example now I look on a HP 17″ monitor, resolution is 1280×1024 pixel. And I don’t see its pixels because it is cca. 80 cm far from my eyes. My simple monitor acts as a RetinaDisplay because its PPD value from that 80 cm is similar then an iPhone 4′s PPD value from that 30 cm. :)

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  9. That resolution would lend itself to suggest a larger pro screen, perhaps 34″ or 35″. Here are a few reasons I think this is the case:

    #1 The resolution is simply wrong for “retina” display. When the iPhone increased its resolution, it didn’t do it incrementally. It doubled the resolution. 3200×2000 is not doubling any resolution of any screen Apple is currently making. Some of the tech advancements for high-resolution displays support this doubling idea. The iPhone’s upgrade increased the resolution by 100% where this would only be an increase in the 27″ display by about 20%. If Apple were really to sport a new set of retina displays, we’d see much MUCH higher resolutions, which would be a boon to photographers because we’d be able to look at dSLR photo at pixel-for-pixel resolution filling the screen. It would also allow for direct viewing of 4k and possibly 5k video streams for movie makers.

    #2 This is not likely to replace either of the cinema displays unless the put the entire line back to a 8:5 aspect ratio (the 21″ and 27″ are currently running at a 16:9 aspect ratio). Also, if you look at the previous lineup, Apple made a 20″, 24″, and 30″ display. Since size does matter, I doubt Apple would expect us to accept the 27″ as a true replacement of the 30″ despite having the same horizontal resolution. The vertical resolution is still shorter, as is the physical size. If you conclude that the new 21″ replaced the 20″, and the 27″ replaced the 24″, then it’s also logical to conclude that Apple’s flagship display would also be replaced with something larger, and a 35″ display at 3200×2000 fits the lineup almost perfectly aside from the aspect ratio. Then again, this could be specifically geared toward the Mac Pro, not laptops like the other displays are.

  10. Well the old 22″ IBM T220/221 LCD monitors were 3840×2400 which would the correct aspect ratio for 16×10 (e.g. 30″ Cinema display). However since Apple seems to be moving away from 16×10 towards 16×9, 3840×2160 would be the target. Those old IBMs were 221ppi. I remember really enjoying the one we had at work, and have been thinking about since the iPhone 4 came out.

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  14. It should be noted that at that resolution, even the 27″ iMac could be marketed as “Retina Display”… starting from a 24.5″ viewing distance, that is.

    And, if i did my math correctly, event the current resolution si technically “Retina Display” when viewed from about 31.5″ and beyond.

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