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If you run a retina display at exactly half resolution (i.e. 1280x800 on a 13" Retina MacBook Pro [native 2560x1600]), that it will not look blurry at all and will look the same as if the display was native resolution.

Is this true?

I've tried using these half-resolutions but they look extremely jagged. I wouldn't describe it as blurry, but it definitely looks pretty awful.

However, if I use the same resolution (1280x800) on a native panel (i.e. MacBook Air 1366x768), it looks MUCH better.

What's going on here?

Why doesn't half-resolution on Retina look the same as a panel with the same native resolution?

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It depends of what you mean by "will look the same".

It looks the same only in the way that the elements on your screen are displayed in the same size as if it was a 1280x800 screen.
The aim of the retina displays is to display more details, to do that, what it does is that it displays things at the same size a half resolution screen would have done but uses all the available pixels to show you more details, nothing is better than an example to explain you what I mean :

enter image description here

In fact, the pixels of a retina display are designed to be small enough so that you retina won't be able to distinguish them at a standard use distance, that's why it looks as sharp as a printed page, and that's why a retina display will never look the same as a display that would be natively 1280x800 pixels, actually it should looks much better.

Now let's answer to your question ! You didn't explained if it always looks blurry or only on some applications/websites.
I will assume that it does not always looks blurry because it shouldn't.
As I explained, although they are displayed in the same size than on any other display, the elements are "constituted" of 4 times more pixels which means that an element that would have been 10x10 pixels (100 pixels) needs to be 20x20 pixels (400 pixels) to take advantage of all the available pixels ; it looks blurry if the element hasn't got a resolution that's high enough "to fill all the available pixels", in fact, this is exactly like when a picture is pixelated but here it looks more blurry because the pixels of your screen are extremely small.

There should be more and more retina ready websites/applications in the future (in think that in something like 4 years most of the websites will be).

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The real 1280x800 on native 2560x1600 will generally looks similar to native 1280x800, unless there is subpixel font rendering.

Subpixel font rendering use colored pixels that fit the display subpixel (RGB) cells to smooth font edge. When naively zooming 1 pixel to 4, the colored pixels does not fit the subpixel cell anymore, and then it will 'look extremely jagged'.

subpixel rendering of letter e

PS:

  • Subpixel font rendering is turned off by default in Mojave
  • 'Looks like 1280x800' (aka 1280x800 HiDPI) is not real 1280x800. It is 2X rendering to 2560x1600.

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