I've been working in Kleinian groups and have had successes with Apollonian using elliptic quasiconformal transforms: picture attached:

Twin Dragon:

https://www.wolframcloud.com/obj/roger.bagula/Published/IFS_2part_twindragon_52Quasiconformal_1000000.nb

Tame Dragon:

https://www.wolframcloud.com/obj/roger.bagula/Published/IFS_2part_tamedragon_52Quasiconformal_1000000.nb

Akiyama tile: https://www.wolframcloud.com/obj/roger.bagula/Published/IFS_2part_Akiyama_quasiconformal52_1000000.nb
Picture links:
https://www.pinterest.com/pin/293648838213535963/ https://www.pinterest.com/pin/293648838213535977/ https://www.pinterest.com/pin/293648838213536147/

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