Cartesian parametrization of the Greek Meandre

Question

As the title suggests, I've been trying to find a parametric equation to describe the Greek meander pattern that's seen in a lot of historical architecture.

I've created a series of X and Y coordinates that traces it out:

And I have both a set of quasiperiodic data describing the X values:

And a set of periodic data describing the Y values:

How would I find the equation that describes these? At first I thought I had to do a Fourier series but I wouldn't even know where to start with a function like this.

Please help!

Answer 1

$\newcommand{\Brak}[1]{\left\langle#1\right\rangle}$Although Roland's (+1) piecewise function is efficient and accurate, a sketch of setting up a Fourier transform may be useful to posterity.

The two component functions are (quasi-)periodic with period $16$, so we may as well work on the interval $[0, 16]$ with the inner product $$ \Brak{f, g} = \frac{1}{8}\int_{0}^{16} f(t)g(t)\, dt. $$ The functions $$ c_{0}(t) = \frac{1}{\sqrt{2}},\qquad c_{k}(t) = \cos(2\pi kt/16),\quad s_{k}(t) = \sin(2\pi kt/16)\ \text{for $k$ positive} $$ form an orthonormal basis with respect to this inner product.

Let $X$ and $Y$ denote the piecewise-linear component functions, and set $Z(t) = X(t) - t/4$, so that $Z$ and $Y$ are periodic with period $16$. Using numerical integration (perhaps Simpson's rule, dividing $[0, 16]$ into $16n$ subintervals), approximate the Fourier coefficients $$ \widehat{Zc}_{k} = \Brak{Z, c_{k}},\quad \widehat{Zs}_{k} = \Brak{Z, s_{k}},\qquad \widehat{Yc}_{k} = \Brak{Y, c_{k}},\quad \widehat{Ys}_{k} = \Brak{Y, s_{k}} $$ for $0 \leq k \leq 4n$. (Formally, treat $s_{0} = 0$.) For the highest frequency terms, that gives four Simpson subdivision points per period.

Now we can reconstitute the approximate component functions: $$ \widehat{X} = \frac{t}{4} + \sum_{k=0}^{4n} \widehat{Zc}_{k} c_{k} + \widehat{Zs}_{k} s_{k},\qquad \widehat{Y} = \sum_{k=0}^{4n} \widehat{Yc}_{k} c_{k} + \widehat{Ys}_{k} s_{k}, $$ and plot the resulting parametric path.

As proof of concept, here's a plot with $n = 8$ ($128$ subdivisions for integrating, $32$ sine terms and $33$ cosine terms):

Answer 2

You can integrate the directions over line segments. The result is

 X[t_] := 
   With[{s = Mod[t, 16], r = Floor[t, 16]}, 
   {r/4, 0} + 
   Piecewise[{
   {{0, s}, s <= 3}, 
    {{s - 3, 3}, s <= 6},
    {{3, 3 - (s - 6)}, s < 8}, 
    {{3 - (s - 8), 1}, s <= 9}, 
    {{2, 1 + (s - 9)}, s <= 10}, 
    {{2 - (s - 10), 2}, s <= 11}, 
    {{ 1, 2 - (s - 11)}, s <= 13}, 
    {{1 + s - 13, 0},s <= 16}, {0, 0} }]] 

    pic = ParametricPlot[X[t], {t, 0, 64}, 
             PlotStyle -> {Blue,Thickness[0.03]},  Axes -> None]

Answer 3

I would like to add my little stone to the building, even it is not an answer to the question about a parametric plot.

Fig. 1 : Basic pattern $B$ in red. Symmetrical pattern in green.

@Andrew D. Hwang has shown that Fourier analysis could be involved in a natural way in this issue.

I would like here to show that discrete convolution (with a coding of the moves by complex numbers) has also a natural place in this issue.

The generation of the greek meander pattern shown on fig. 1 has been generated by the following Matlab program :

n=5;                           % number of repetitions
B=[0,3i,6,-4i,-2,i];           % (red relative moves) 
C=[B,fliplr(B)];               % (red + green rel. moves)
DC=repmat([1,zeros(1,8)],1,n); % Dirac comb
D=conv(C,DC);                  % (red + green + blue rel. moves)
GM=cumsum(D);                  % (relative -> absolute moves)
plot(GM);

Comments : I have taken into account the fact that the greek meander has a central symmetry ; function "fliplr" (flip left-right) realizes this symmetry. Besides "repmat" (repeat matrix) operates the following repetition :

$$DC=[1,\underbrace{0,0,0,0,0,0,0,0}_{\text{eight times}},1,\underbrace{0,0,0,0,0,0,0,0}_{\text{eight times}},1,0,0,\cdots],$$

The final operation, convolution by a (discrete) Dirac comb, gives the periodization.