As I can attest from experience, this is very hard to do.
Many people dislike bike helmets because they are bulky and uncomfortable. Many are designed for racing rather than cruising around town and are not fastened properly; others are designed for other sports and get hot. Who wants to wrap their head in styrofoam insulation?That's why the Park & Diamond collapsible helmet is so interesting. It is a form-fitting helmet that looks pretty much like a baseball cap; underneath the cap is a proprietary foam that the inventors claim "absorbs and dissipates three times more elastic energy than a traditional bike helmet, which means significantly less energy is being transferred to the head, and making the Park & Diamond Helmet a better bike helmet."
BikeRumor! gives a little more detail:
The folding helmet features a fabric outer skin that gives it the baseball cap look, and an inner skin for a comfortable fit on your head. Both are removable and can be hand washed, or swapped out when new styles are introduced. In between, a proprietary energy dissipating composite shell (not simply EPS like a standard helmet) is made up of many smaller geometric elements that together are able to absorb the energy of an impact. In its expanded position their interlocking shape allows the elements to transmit energy to each other in a crash, while also allowing the entire thing to pack down rather compactly – to about the size of a large water bottle, so it doesn’t take up too much space in your bag.
This ability to fold is important in the world of bike and scooter shares, when you don't want to carry around a bulky helmet.
They also say that it will comply with U.S. Consumer Product Safety Commission (CPSC), Canadian cycling helmet CAN-CSA-D113.2-M, and EU EN-1078 safety certification standards for helmets, which is really tough.
Years ago I tried to develop and patent a helmet that didn't look like a helmet; being Canadian, I built it into a toque, using the elastic fabric to hold the interior pieces together. I could never get it to pass the drop tests. Since I was designing it for snowboarding rather than cycling I didn't worry about that too much, but the lawyers told me that this was not a good idea and I never proceeded. But I can confirm that complying with these standards is very hard.
Another problem I had with my helmet was my understanding at the time that a fabric covered helmet is a problem because it won't slide when it hits the pavement. Most of the testing is done with a vertical apparatus but when you fall, it is often with a slide. If the fabric covered helmet snags on something, it could increase the risk of a broken neck. A slippery helmet is likely going to be safer.
Finally, helmets are not magic and will not protect cyclists in all kinds of accidents, so I wish they wouldn't use scary and not recent statistics (1996-2005!) from a dubious thirteen-year-old report prepared by people who hated cyclists (including the NYPD and Irene Weinshall) back when New York was very different. Including this may not sell helmets as much as discourage cycling.
But if you want to wear a helmet (and I do because I live in a city with crappy bike infrastructure) this looks like a really interesting option. I have ordered one on Indiegogo; after my failures at this, I can't wait to try out a success.