I finally found a use for this set of concentric WS2812B rings: http://www.aliexpress.com/item/RGB-LED-Ring-WS2812-5050-RGB-LED-with-Integrated-Drivers/32372470495.html
I assembled them into new guts for my lamp. The old ones were a little too 2 dimensional (and borrowed from another project). Each of the rings is rotated 90 degrees from the last. They’re driven with a Photon.