RS485 has a bus topology. The big problem is that if you have star topology, every ray of the star has to be terminated. This way, the master transmitter will be loaded with 16x||120Ω = 7.5Ω. Of course such small impedance will overload the transmitter.
There is another solution however. You should use a switch and connect the master only to the ray you want to communicate. I am not sure how exactly this schematic has to be implemented...
One of the solutions is to use a multiplexer and 16xRS485 transceivers - such as SN75176 or similar.
Another way is to use one transceiver and some analog multiplexers with low enough internal resistance.
(Or even electromechanical relays as a some kind of ultra-conservative design :) )
Connect the network in star without termination
It is clear, that if there is no termination resistors, the RS485 network can be safely wired in star topology.
The additional research lead me to this article, that states that for relatively short lines, the termination resistors can be omitted. The term "relatively short" depends on the transmission speed and for 9600bps is about 600m (stated in the article).
I made the same approximated calculations and for 20m line the transmission speed is around 1Mbps.
The condition is that the delay of the signal in the lines for 3 or 4 full trips (enough to dump the reflections) must be shorter than the half time of 1bit transmitted.
Well, in the article there is a presumption that the bit is checked near the middle of the bit width, but some receivers make several sampling of the bit in order to decrease the errors. In this case probably the acceptable speed will be even lower.