A helmet structure without any hard outer shell has axially compressible cell units contained in a hemispheric frame by a thin fabric covering stretched over cup shaped cell retainers that have sidewalls of compressible foam. The frame is supported on the wearer's head on plastic foam posts that space the inner ends of compressible bladders from the wearer's head, and ambient air in the bladders compresses at impact, being vented then through openings for gradually absorbing such impact forces. Each bladder is vented into a space between the cup “bottom” and the outer end of a bladder. At least two cell sizes are provided, and some of these are on depending lobes in the frame, for protecting the wearer's ears and neck.