Poincare Doughnuts: An Investigation of Non-Euclidean Orthogonal Circles in Euclidean Space

Finding new and interesting characterizations of familiar mathematical concepts appeals to a wide audience, even those who would never consider themselves to be mathematicians. This investigation takes a specific case of the Poincare hyperbolic disk, and looks at it through the lens of Euclidean geometry. When two circles are orthogonal, they intersect at right angles. Considering a given circle O, and the set of all circles orthogonal to and with the same radii as O, a new set of concentric circles become apparent. By looking at the properties and relationships of these new circles, in the Euclidean plane, unique and interesting relationships can be found. Namely, this set of orthogonal circles illustrates, both visually and mathematically, trigonometric values for the otherwise unassuming angle measure of 22.5 degrees. Whereas finding trigonometric values for this angle measure would usually require the use of the half angle formula, this set of orthogonal circles serves to not only readily produce those values, but also provide simple and elegant visual representations of foundational trigonometric identities