page_title: Spiral of Theodorus
section_title: Continuous curve
hierarchical_section_title: Spiral of Theodorus / Continuous curve
context_page_description: In geometry, the spiral of Theodorus is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
context_section_description: The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered in (Davis 2001, pp. 37–38) by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function
  which was further studied by his student Leader and by Iserles (in an appendix to (Davis 2001) ). An axiomatic characterization of this function is given in (Gronau 2004) as the unique function that satisfies the functional equation
  the initial condition  and monotonicity in both argument and modulus; alternative conditions and weakenings are also studied therein. An alternative derivation is given in (Heuvers, Moak & Boursaw 2000).
  An analytic continuation of Davis' continuous form of the Spiral of Theodorus which extends in the opposite direction from the origin is given in (Waldvogel 2009).
  In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral.
  Only nodes  with the integer value of the polar radius  are numbered in the figure.
  The dashed circle in the coordinate origin  is the circle of curvature at .
caption: Davis' analytic continuation of the Spiral of Theodorus, including extension in the opposite direction from the origin (negative nodes numbers).