I trained in the Mevlevi dervish "Whirling" beginning in 1979, and began turning regularly in the Mevlevi "Sema" rituals in 1981. First - the dervish does NOT fix their gaze on any object while they turn. This is totally different from dancer, who "spot" their eyes or head on an object while the body turns, and then move the head / eyes once each revolution. The dervish keeps their eyes open, unfixed. The "biology teacher" quoted was completely wrong.
How do the dervishes avoid dizziness? Well, some individuals do have a problem with dizziness at the beginning of the training. (I never had that trouble.) Dizziness, however, is a very small part of the difficulty of the practice, which is very physically strenuous. For example, in the "Sema" ceremony, the arms have to be held up, outstretched, for about 20 minutes; then a short break of less than two minutes, then arms stretched up another 20 minutes - this is repeated 4 times. Try holding your arms up, straight out to the side, while facing a clock. See how long you can last! The steps of the "whirling" are also very precise, and strenuous - and must be kept up continuously for long periods, without variation of speed or rhythm; any alteration of rhythm can cause the "dancers" long skirt to fall and wrap around the legs, tripping them and causing a serious fall. It's hard work!
The dizziness problem is actually minimized if the speed of rotation of the dervish is smooth and continuous - no slowing down or speeding up. A stable rotation speed allows the balance sensors of the inner ear to accommodate to the rotary motion. It is only when velocity or rotation vectors keeps shifting that the instability triggers dizziness. Turns done smoothly do not induce dizziness; martial arts practitioners practice for this, especially in some of the "inner" schools - Aikido, Tai Chi Chuan, Ba Gua. I had studied Tai Chi Chuan for some years before taking up the dervish "turning", and I never felt dizzy while learning the practice.
Vision creates a different sort of difficulty. The eyes are not fixed on anything, so objects in the visual field move across the retina without stopping. It takes some getting used to - it is a different way of seeing, whcih requires growing new neural connections. The process can take some time - weeks or a month or two - but when it is complete, there is no problem with being aware of your surroundings. In the course of the "Sema" ritual, the dervishes have to be able to maintain their spacing while they turn, and in some cases may be moving in complex patterns on the floor. If they approach each other too closely, their whirling skirts may collide and tangle about their legs, unbalancing them or even causing them to fall. Long practice prevents this from happening during the formal ritual - in almost 40 years, I've only seen collisions maybe three times, and only once a fall - and that was someone very inexperienced in the practice.