Like all multi-band antennas, they use impedance traps to "shorten" the antenna for the lowest frequency (longest wavelength) band. The traps are placed resonant antennas for the six different cellular bands. There is a simple formula that you can use to convert frequency to wavelength and vice versa. The traps themselves will have frequency dependent characteristics to achieve the desired result.
Q: are you familiar with microstripline techniques? They are seldom designed by hand anymore, and the CAD tools used tend to be expensive.
Possibly a reference to hexaband means 6 bands and pentaband is 5 band used in telecom handset used this type of multiband integrated antenna.
The architecture complete with PA, filters, LNA, frequency synthesizer needed to peak 6 bands and have an efficient scatter pattern.
The mobile phone data plan consumers needed higher data rate and performance. This trend to cellular data connectivity influenced regulation for wider telecom band plans back in those days.