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Retina Display Math – Why Apple doesn’t need 2X displays on the Mac

With the recent release of the New iPad, there as been a lot of talk about retina displays.

What Apple calls a retina display, its a display where you a can’t see individual pixels at its optimum viewing distance, even if you have perfect vision. It’s not a simple metric as PPI (pixels per inch). For example, the iPhone 4 and 4S’s displays have 326ppi, and on the new iPad there are 264ppi, and they both qualify as retina displays.

After reading this post by Ars Technica, where they found 2X graphics on a developer build of OSX Mountain Lion, i started to think why, or if,  Apple needed 2X graphics. On iOS, the reason is simple.

With screen real estate being very, very limited they want apps build with 1X resolution to work on a 2X display and vice versa. If you have 1X graphics, 1 pixel of your artwork will fit in 4 pixels of a 2X display and you don’t need to do anything for it to work.

But perhaps the most important reason is: Every app is a full screen app. Therefore every app uses the same number of pixels, and there are only 4 different resolutions on iOS devices: 1X and 2X resolutions on iPad, iPhone and iPod touches. But that is not true on OSX!

In fact, OSX will deal with any screen resolution you want, and therefore each app needs a different amount of screen real estate. This is the reason Apple doesn’t need to make every mac screen with 4X the amount of pixels they have now.

And what amount of pixels will be needed, you ask? Well, i’ve done some math.

Here is the size and number of pixels on the iPad and the iPhone. Data from the Apple technical sheet.

First lets calculate the physical size of each pixel:

So an iPhone pixel is 0,0030674 inches wide, and an iPad pixel is 0,003787878 inches wide (all pixels are square, of course).

Now, we need to compute the minimum size of a pixel a Mac Retina display needs to have, in order to qualify for such a name. According to Apple the normal distance for use of an iPhone is 10 inches away from your eyes, and 15 for the iPad. I assumed a distance of 27 inches from your face to your mac display as i’m right now in front of a 2009 13″ MBP. Therefore:

with X being the iPhone pixel size and Y the iPad’s, what is Z? There were a number of ways to compute Z, but i choosed to do it by computing the angle θ.

Now we know that a single pixel can’t be bigger than 0,0055170252 inches wide for a Mac display to qualify as a Retina display.

And that means a display needs to have a pixels density of 182 PPI.

Currently the densest display on Apple’s offering is that of the 11″ MacBook Air with 135ppi, and a resolution of 1366×768. A 2X display would have a resolution of 2732×1536 and a density of 270 ppi. Way more than what is need!

According to my calculations a screen with 1744×982 would be enough! I would like to do the same math for the 27″ displays they offer, but i don’t have one to estimate the viewing distance. If you can help me with that number, please drop me a line on twitter.

Conclusion

On iOS it made sense for Apple to double the screen in both directions, but they dont need to do it for the mac. I think if they can save money on pixels they will, but only if they can get away with calling it a retina display. On the last iPad event they felt the need to reinforce the metric they use to name a Retina Display, and it makes sense reinforcing it if they plan to make Macs with retina displays, but want to save money on pixels. And i don’t think they break any interface design by doing this.

π

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