Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 TheodorusT

Source Notebook

Evaluate the Theodorus function

Contributed by: Jan Mangaldan

ResourceFunction["TheodorusT"][z]

gives the Theodorus function T(z).

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The Theodorus function interpolates the spiral of Theodorus, and is given by the expression .
For certain special arguments, ResourceFunction["TheodorusT"] automatically evaluates to exact values.
ResourceFunction["TheodorusT"] can be numerically evaluated to arbitrary numerical precision.
ResourceFunction["TheodorusT"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate numerically:

In[1]:=
ResourceFunction["TheodorusT"][1.5]
Out[1]=

Plot the spiral of Theodorus in the complex plane:

In[2]:=
ParametricPlot[ ReIm[ResourceFunction["TheodorusT"][\[Alpha]]], {\[Alpha], -5, 5}]
Out[2]=

Scope (4) 

Evaluate for integer arguments:

In[3]:=
ResourceFunction["TheodorusT"][3]
Out[3]=

Evaluate for complex arguments:

In[4]:=
ResourceFunction["TheodorusT"][0.6 + 0.7 I]
Out[4]=

Evaluate to high precision:

In[5]:=
N[ResourceFunction["TheodorusT"][7/2], 50]
Out[5]=

The precision of the output tracks the precision of the input:

In[6]:=
ResourceFunction[ "TheodorusT"][3.5000000000000000000000000000000000000]
Out[6]=

TheodorusT threads elementwise over lists:

In[7]:=
ResourceFunction["TheodorusT"][{2.5, 3.5, 4.5}]
Out[7]=

Applications (1) 

Plot of TheodorusT for complex variables (this computation may take a while):

In[8]:=
ComplexPlot[ResourceFunction["TheodorusT"][z], {z, 9}]
Out[8]=

Neat Examples (1) 

Show the discrete and continuous Theodorus spiral together:

In[9]:=
ParametricPlot[ ReIm[ResourceFunction["TheodorusT"][\[Alpha]]], {\[Alpha], 0, 16}, Axes -> None, PlotStyle -> Thick, Prolog -> {Directive[FaceForm[], EdgeForm[Opacity[0.4, Black]]], Polygon[PadRight[#, {3, 2}] & /@ Partition[ AnglePath[{{1, 0}, \[Pi]/2}, Prepend[ArcCot[Sqrt[Range[15]]], 0]], 2, 1]]}]
Out[9]=

Version History

  • 1.0.0 – 23 March 2021

Source Metadata

  • Citation:
    • Davis P.J., Spirals: From Theodorus to Chaos. Wellesley, MA.: A.K. Peters, 1993.
    • Gautschi W., "The spiral of Theodorus, numerical analysis, and special functions." Journal of Computational and Applied Mathematics, vol. 235, no. 4, 1042–1052, 2010. DOI: 10.1016/j.cam.2009.11.054
    • Gronau D., "The Spiral of Theodorus." American Mathematical Monthly, vol. 111, no. 3, 230–237, 2004. DOI: 10.1080/00029890.2004.11920068
    • Waldvogel J., "Analytic Continuation of the Theodorus Spiral." Unpublished manuscript, 2004. http://www.math.ethz.ch/~waldvoge/Papers/theopaper.pdf

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