Where 3438 comes from
The rule rests on one number: the smallest gap a normal eye can separate, about one arcminute, or a sixtieth of a degree. That's the gap that defines 20/20 vision. A half circle is 180 degrees, which is pi radians, and each degree holds 60 arcminutes. So one arcminute is about 1 over 3438 of a radian, and a pixel with pitch p fills that angle from a distance of p multiplied by 3438.
Pitch is 1 divided by the PPI, so the distance in inches is 3438 divided by the PPI. A 27-inch (69 cm) monitor at 2560 × 1440 has 109 PPI, which gives a Retina distance of 32 inches (80 cm). Sit farther back and neighboring pixels land inside one arcminute, so they merge. Sit closer and you can pick them out.
Working backward from your seat
Most people don't care what a screen's Retina distance is. They care whether their own seat is far enough. So run the formula the other way: pick the distance, and it gives the PPI you need.
- A phone held 12 inches (30 cm) away needs 291 PPI
- A monitor at 24 inches (60 cm) needs 146 PPI
- A monitor at 28 inches (70 cm) needs 125 PPI
Now hold real monitors up against those lines. A 27-inch 1440p screen has 109 PPI. That's below even the 28-inch (70 cm) line, so single pixels can still show on small text. A 32-inch 4K screen has 138 PPI. It falls just short at 24 inches (60 cm) but clears the bar at 28 inches (70 cm). A 27-inch 4K screen has 163 PPI. It clears the bar even at 24 inches (60 cm).
TVs, phones and the soft edge of the rule
TVs flip the problem. A 65-inch 4K set has just 68 PPI, but its Retina distance is 51 inches (129 cm). Most living room seats are 6.5-10 feet (2-3 m) from the screen, so the pixels are long gone by then. Phones go the other way and need the most density, since you hold one close. Apple's early Retina phones aimed for about 300 PPI at 10-12 inches (25-30 cm), and the formula asks for 291 PPI at 12 inches (30 cm).
The one arcminute line is a rule of thumb, not a wall. Visual acuity references describe 20/20 as the lower limit of normal, and many people see finer detail. Someone who can separate points at half that angle has a Retina distance twice as long, so they can still see pixels where the rule says they've vanished. If you're picky about text, leave some margin on the numbers above.