tooth, cog
齒 centers on tooth-like projections: a tooth in the mouth, or a cog on a wheel. The cog sense is a natural extension of the tooth shape.
齒 is a pictograph of a mouth showing teeth. The modern form retains the top 止 (originally representing the upper teeth/lip area) and the lower enclosure with inner rows suggesting teeth.
The top 止 looks like the upper gum line, and the lower box with inner strokes resembles rows of teeth. Together, 齒 is a tooth.
For シ, imagine a toothy grin showing a shiny tooth: she -> シ, and her shiny tooth is a perfect cog in her smile.