A sigmoid tax is the idea of basing income taxes on a sigmoid curve, where the X axis is net income (after allowable expenses), the Y axis is the amount of income over the average which the recipient should be allowed to retain, and 0 is the mean income for that tax year.
Those earning the mean would be neither taxed nor supported. Those earning below the mean would have their incomes enhanced in an amount corresponding to how far they are below it, while those earning above would have their incomes curtailed similarly.
Looking at it mathematically: if we use sinh() for the sigmoid, and use 'a' for the average income and 'p' for pre-tax net income, then the net income after taxes would be sinh(p-a)+a ...or, to put it another way, the tax amount would be a+p-sinh(p-a). I think.
originally proposed in a comment here