Paper (纸) is one of China's Four Great Inventions, attributed to Cai Lun (蔡伦) in 105 AD during the Han Dynasty. Before paper, writing was done on bamboo strips, silk, or oracle bones.
Silk-thread radical, simplified form of 糸. Early Chinese paper was made by macerating plant fibres into a pulp resembling silk threads, so 纸 sits in the textile/fibre family with 红 red, 线 line, 经 classic-text, 纳 receive.
Phonetic - 氏 shì shifts to zhǐ in 纸 (a regular Old-Chinese sibilant alternation). Same phonetic surfaces in 低 low, 抵 resist, 底 bottom, 砥 whetstone - all dǐ/zhǐ family. 氏 itself meant 'clan/lineage'; here it is purely a sound element.
