Model of the Real Part of the Function Cos Z by Richard P. Baker, Baker #444

This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa.

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 = a + bi, where i represents the square root of negative one and a and b are real numbers. 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 – this model), and #446 (MA.211257.094). The models came to the Smithsonian in 1956 after exhibition at MIT.

Just as complex numbers have real and imaginary parts, so do functions of them. This model represents the real part of the function cos (Z). If, as before, Z = a + bi, it represents the values of the product cos a cosh b, where cosh b is the hyperbolic cosine of b.

The model has a wooden base and sides, with a white plaster surface representing values of cos (Z). A tag on the side reads: No. 444 (/) The real part of Cos Z.

Values of the cosine, which run from -1 to +1 and back to -1, form the upper edge of the two higher opposite wooden sides. The upper edge of the other two opposite sides are values of -cosh b, with values of this function running from -1 to 0 to -1 (-coshb is the value of the function cos a coshb whien cosa = -1, as it does in the planes that contains these sides). Values of the product of the two functions within these intervals are represented by the plaster surface.

Baker also designed a model for the imaginary part of the function cos (Z), but it does not survive in the collections.

Reference:

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

Accession file 211257.

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