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 <i>unconsciously</i> perceive the increase as linear.  Our perception, in other words, is often logarithmic.<p>What I find most fascinating about this is that our <i>conscious</i> 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, <i>never</i> catch up with those of the first one.<p>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!