Tooth soap is very simple. You can easily make it yourself at home, and recipes abound on the Internet. All include glycerin, baking soda and water; some also suggest hydrogen peroxide, salt, and flavoring such as orange zest or cinnamon oil -- since soap, after all, doesn't taste that great. One popular store-bought brand of tooth soap is made with saponified extra virgin olive oil, coconut oil, water and essential oils. (Saponification is the chemical process by which an oil or fat becomes soap.)
Soap cleans because one part of its structure is hydrophilic (attracted to water), while the other is lipophilic (oil-loving). The lipophilic parts stick to anything greasy or oily, breaking it up into smaller globs until the whole thing can eventually be lifted off the surface and washed away. Your teeth get washed the same way dirty dishes do.
Dental powders are another, less fashionable alternative to the tube on your bathroom counter. Another DIY project, you can make tooth powder several different ways. Chalk plus soap dust and flavoring is one recipe, but others call for clay, salt, baking soda and flavoring such as peppermint oil.
You're in good company with dental powder. The ancient Egyptians had a powder of their own, made with pumice, eggshells and the ashes of hooves. In the centuries after, oyster shells, charcoal and bark also made it into dentifrices. The first real paste entered the market in the 1850s, but ingredients like SLS didn't show up in toothpaste until after 1945 -- when they replaced soap.
These methods do clean your teeth, and they're very cheap when compared to toothpaste; however, they don't contain fluoride, which the American Dental Association (ADA) recommends for oral health. And no matter what you use to brush your teeth on a daily basis, you should also floss and go to the dentist every 6 months for a thorough cleaning and exam. Periodontal disease is common, but luckily, it's also reversible if it's caught early.