The rich do pay taxes, just a smaller portion of their income as compared to your average taxpayer. But that smaller portion is leaps and bounds larger than the bigger portion paid by the average taxpayer.

I seem to recall some graph indicating that even though they pay 4-5x (or higher) lower percentage of their money in taxes, the higher amounts mean they shoulder most of the tax burdon.

I don't think the percentage difference is ultimately beneficial to society, but it's not correct to say they pay nothing.