Retina Math

Yesterday, Apple released its latest version of the iPad. One big selling point is the new 2048-by-1536-pixel screen - touted as a retina display. People have asked whether the new screen is actually retina, especially since its pixel density is 19% less than that of the iPhone 4/4S.

I will quickly show you the math that goes into the term and discuss whether or not the new iPad qualifies.


Phil Plait offers a great explanation of what resolution means in terms of the human eye. In essence, resolution is the ability to discern two objects that are close to one another.

Resolution is measured as an angle, and as a result, can be translated into a separation. Basically, at some distance, two objects appear as one because the angle between them grows very small. That distance represents the eye’s resolution limit.


Perfect eyesight is roughly 0.6 arcminutes. Of course, not many people have perfect eyesight because of imperfections in our lens. A more likely eyesight is 20/20, which has a resolution of one arc minute.

An arcminute is 1/60 of a degree, or π/10,800 radians. That means we need to use a scaling factor in order to determine our resolution limit.

10,800/π = 3,438

That means the iPad must be held a distance equal to 3,438 times the size of an individual pixel in order for two pixels to become indistinguishable.


The new iPad sports a display with a 264-pixel-per-inch density. That means an individual pixel is of size:

1/264 = .0038 inches

Now, me must calculate the distance at which the eye reaches its resolution limit.

d = 3,438 * .0038 inches = 13 inches.

Thus, for the new iPad to qualify as Apple’s definition of retina, the device must be held at least 13 inches from one’s face. Beyond that distance, the eye cannot resolve individual pixels.


The new iPad screen does qualify as a retina display - with a few caveats. The user must hold the iPad 13 inches or farther from their eyes. This assumes an eyesight of 20/20. The greater one’s eyesight, the farther it must be held from the face.

In fact, Apple was somewhat conservative in its definition for the iPad. They stated the screen was retina if held at a distance of 15 inches from the eye. This corresponds to an eye resolution of 0.87 arcminutes. In either case, I can personally attest to those as being normal use-cases for the iPad.

In summary, Apple was correct in calling the new display retina based on their consistent definition. No goal posts were shifted or anyone deceived. Just keep in mind that the term is largely for marketing purposes (Apple is a for-profit corporation after all). In the end, the important thing is how awesome the screen will look rather than what it is called.