This geometric model was constructed by Richard P. Baker, probably in the 1920s while he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
It appears to be part of a series of models Baker made relating to Clebsch’s diagonal surface that are listed on page 10 of his 1931 catalog of models Model 367 is the first model in this section and described as a plaster cast model with the title “Tetrahedral symmetry. 24 finite lines.” While model 367 is not among the models in the Smithsonian collections, there is a second, wire model that Baker dubbed Model 366 and entitled “The lines of 367”
This object appears to be an example of that model. It has twenty-four wire line segments (three others are infinitely far away and hence not shown) and tetrahedral symmetry. See also MA.211257.054.
R. P. Baker, Mathematical Models, Iowa City, Iowa, 1931, p. 10.
David E. Rowe, A Richer Picture of Mathematics: The Göttingen Tradition and Beyond, Springer, 2018, pp.70-74.