What number is halfway between 1 and 9?
According to the research discussed in this article, when the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear.  Our perception, in other words, is often logarithmic.What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth. My favorite example is that of the twin brothers who invest with annual gains of 12% each: one twin saves $2,000 from age 19 to 26 (a total of $16,000) and then stops saving; the other twin doesn't start saving until age 27 but then puts away $2,000 every year until age 65 (a total of $78,000).  Do the math, and you will see that the savings of the second twin, growing at 12% every year, never catch up with those of the first one.Our perception might often be logarithmic, and we might understand the math of exponentiation, but our brains are not very good at 'seeing' exponential growth!