This has come to be my favorite fan form, partly because this time around I found and used the teaching videos of Master Zeng Nailiang himself, creator of the form. Read about him here: Kung Fu Magazine-Professor Zeng Nailiang.
To me the most beautiful demonstration of this form is that of Sing May Chen (Irvine World Tai Chi Day 2015). In this grab she is doing 飞凤回首 – Fēi fèng huíshǒu (Flying Phoenix Turns Head):
Here are the links to Master Zeng’s four-part tutorial, totalling about 45 minutes of instruction. I especially appreciate his emphasis on the martial aspects of the form and the use of the fan as a weapon. That is Master Zeng’s daughter demonstrating the moves at the beginning of each segment.
The name of the form is 华武太極扇 (Huáwǔ tàijíshàn). Huawu means “Flowery Martial” Tai Chi Fan, and here is a list of the 42 movements (PDF). For more about Huawu Fan and links to more video demonstrations, see my earlier posts: Huawu Fan (tagged).