Model of the Absolute Value of Cos Z by Richard P. Baker, Baker #446

This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa. Plaster on top, it has wooden sides and base, both painted black. The plaster is color-coded, with a scale on one side explaining the meaning of the spectrum of colors used. A paper tag taped to one side reads: No. 446 The absolute value (/) of Cos Z.

Particularly from the nineteenth century, mathematicians and physicists have found it useful to consider functions of complex numbers, that is to say numbers of the form Z = x + yi, where i represents the square root of negative one and x and y are real numbers. Just as complex numbers have real and imaginary parts, so do functions of them. This model represents the absolute value of the function cos (Z) where, as before, Z = x + yi. This consists of positive values of the equation (cos ²x cosh² y + sin²x sinh²y) ^1/2. This equation can be reduced to (cos²x + sinh²y)^1/2. In the model, values of the angle x run from 0 to 2 pi , while those of y run from roughly -3 to 0 to roughly +3. The colored scale indicates regions corresponding to different values of x. For smaller and for larger values of y, values of sinh²y become much larger and the small fluctuations associated with the first term in the equation would not be visible.

Baker’s 1931 catalog listed seven models which he grouped as relating to the field of complex analysis; examples of four of these survive at the National Museum of American History. These are his #443 (MA.211257.092), #310 (MA.211257.050), #444 (MA.211257.093), and #446 (MA.211257.094 – this model). The models came to the Smithsonian in 1956 after exhibition at MIT.

References:

Baker, R.P., Mathematical Models, Iowa City, 1931, p.17.

Accession file 211257

http://math.pugetsound.edu/~martinj/courses/spring2010/m352/ComplexVariables_Sec33.pdf , accessed August 27, 2020.

https://functions.wolfram.com/ElementaryFunctions/Cos/visualizations/5/, accessed August 27, 2020.

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