This spiraling structure is the work of Gustavus Hinrichs. It was one of the many contenders for the Periodic Table of Elements. It was actually better than most, but there's a very good reason why it lost out to the current design.
Around the mid-1800s, people started noticing that the elements, when arranged from least to greatest, seemed to form patterns of behavior. The periodic table was decades in the making, but many people came up with early periodic laws, and periodic charts. One of these people was Gustavus Hinrichs. Born in Germany but destined to be the head of the Iowa Weather Service, he studied, and contributed to the knowledge of sciences as diverse as magnetism, meteorology, and chemistry.
This periodic spiral was his creation, and one of the many popular periodic models of his time. Hinrichs' spiral successfully incorporated many of the known elements, and made it easy to see which elements were grouped together. (Some of the tables were incredibly complicated. One had to be mounted on a cylinder that needed to be spun to be understood.)
Unfortunately, Hinrichs was working at a time when chemists knew only a few elements, ordered their tables by weight instead of atomic number, and didn't know about isotopes. Although his spiral incorporates a few patterns, it's a far cry from the information-rich and accurate table that we now have. It's a pretty spiral, though.