Watching a goldfish swimming in a fishbowl

Abstract The problem of determining the position and magnification of a fish swimming in a fishbowl can be solved by means of the small-angles approximation. However, an exact solution, free from such simplifications, yields a more realistic and quantitatively accurate description of what is observed. In this analysis, the curve of the transverse magnification <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> is determined as a function of the goldfish’s position within the bowl. It is found that the magnification increases as the goldfish moves away from an external observer. This qualitative behaviour can be readily confirmed through simple experiments. Quantitative differences between the exact analysis and the small-angle approximation become significant when the size of the goldfish is not negligible compared to the radius of the bowl.