In the 19th century, mathematicians became interested in the question how mathematical functions look like. Felix Klein, a German mathematician, had several of such physical models in his lab in Göttingen, and popularized them in America when he brought a boatload to the World's Fair in Chicago in 1893. The photo above shows a model of a Clebsch surface from 1880, designed and presumably built by Adolf Weiler, Klein's grad student. It is kept today at the University of Göttingen. Sources: Photo by Jonathan Chertok. Joshua Batson, This Is What Math Equations Look Like in 3-D, Wired (2014) Göttingen University Clebsch's diagonal surface (thanks to Alba Marina Málaga Sabogal for sending us this reference).

These cams are solid models of bivariate and univariate mathematical functions plotted in cylindric coordinates (left and middle images) and polar coordinates (right image). They were not meant to be visually examined, but were used in mechanical analog computers for aiming battleship guns during WWII. They were also called computing cams. Mechanical fire control aids started to be developed in the 1900s and and were still in use in the late 1980s. During WWII, they could solve multivariate calculus problems in real-time. While cams were used for hard-coded functions, gears were used for arithmetic calculations and other specialized devices were used for multiplication and differential calculus. The design of these analog computers is clearly explained in the video and PDFs linked below. The oldest known mechanical analog computer is from 150BC (see our entry greek orrery). Cams were already used in 300BC to control automata, and were further developed in the 13th century by polymath Ismail al-Jazari. Sources: US Navy (1953) Basic Mechanisms in Fire Control Computers (Video). US Navy (1944) Basic Fire Control Mechanisms (PDF files). Sean Gallagher (2014) Gears of war: When mechanical analog computers ruled the waves. Wikipedia articles on Rangekeepers, Cams, and al-Jazari.

 Since 1999, independent researcher and architect Jonathan Chertok has been digitally reconstructing and 3D-printing historical plaster models from 19th-century mathematical model collections, originally hand-crafted in the 1860s and cataloged in early 20th-century German sources. His work focuses in particular on the Klein-Schilling collection, many of whose originals are housed in European institutions such as the University of Göttingen and the Institut Henri Poincaré in Paris. Chertok’s “1.0 phase” involved a series of digitally modeled and 3D-printed reproductions presented at ACM SIGGRAPH 2008, including classic surfaces such as the Clebsch diagonal cubic, recreated using a combination of mesh-based mathematical modeling and NURBS-based CAD design. The left photo shows 3D-printouts of the Clebsch surface he made in 2004–2005. Chertok’s current “2.0 phase” aims to document and produce over 100 additional models, including forms only fully formulated in the late 20th century. This work is informed by archival research, field visits, historical photography, and computational scripting. The right photo shows some of the models he printed so far. This long-term effort is both a digital preservation project and a contemporary reinterpretation of early mathematical visualization. Chertok’s aim is to revive these artifacts as part of a global cultural patrimony, while exploring their potential use in STEM education, exhibition design, and interactive learning tools. He is currently seeking collaborators—particularly from academic or curatorial backgrounds—to help him organize, document, and publicize the project as he prepares for a second exhibition phase and possible online release. Related: Also see our entry 1880 – Klein’s Mathematical Plaster Models Sources: Most of the information and the two photos come from a personal correspondence with Jonathan Chertok. Jonathan D. Chertok (2008) Artifacts of research: on singularities. ACM SIGGRAPH 2008 new tech demos. Joshua Bateson (2014) This Is What Math Equations Look Like in 3-D. Wired Article for which Jonathan Chertok provided photos. The Phillips Collection (2015) Man Ray–Human Equations. Exhibition about Man Ray featuring some of Chertok’s physical models. Elkadi, Mourrain, and Piene (2006). Algebraic Geometry and Geometric Modeling. Book that mentions Chertok’s work on p.120. Tobies and Pakis. Felix Klein: Visions for mathematics, applications, and education. Briefly mentions Chertok’s work on p.175.