# Cartesian parametrization of the Greek Meandre

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.

$$\newcommand{\Brak}[1]{\left\langle#1\right\rangle}$$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):

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]


• This looks to be a Mathematica program. Could you confirm ? Dec 9, 2023 at 13:27
• Yes. A TeX equivalent, simplified, $$\left( \begin{array}{cc} \{ & \begin{array}{cc} \{-t,0\} & 8\leq (t \bmod 16)\leq 9\lor 10<(t \bmod 16)\leq 11 \\ \{0,-t\} & 6<(t \bmod 16)<8\lor 11<(t \bmod 16)\leq 13 \\ \{0,0\} & (t \bmod 16)>13 \\ \{0,t\} & 9<(t \bmod 16)\leq 10\lor (t \bmod 16)\leq 3 \\ \{t,0\} & \text{True} \\ \end{array} \\ \end{array} \right)+\frac{1}{4} \text{Floor}[t,16]$$ Dec 9, 2023 at 14:59
• Please have a look at my answer providing a different approach. Dec 11, 2023 at 9:16

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.