Pythagoras Tree

Fractal dimension: 2

Visualization

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AIFS program

@
$dim=2
s=$companion([1,0])
g=s+1
A=([0,1]*s*g^-1|g^-1*[-1,2])*A

Overview

The Pythagoras tree is a fractal named after the Pythagorean theorem. Starting from a unit square (representing the trunk), an isosceles right triangle is placed on the top edge, and two smaller squares are built on its two legs. The process is repeated on each new square indefinitely.

In the symmetric variant both child squares are the same size: each has side $\frac{1}{\sqrt{2}}$ and is tilted by $\pm 45°$ relative to its parent.

Definition

Like the Lévy C Curve, the Pythagoras Tree lives in $\mathbb{Z}[i]$ with $companion([1,0]) and $g = s+1 \leftrightarrow 1+i$ .

Map	AIFS	Effect
$f_1$	`[0,1]sg^-1`	rotate $+45°$ , scale $\tfrac{1}{\sqrt{2}}$ , translate $(0,1)$ (left branch)
$f_2$	`g^-1*[-1,2]`	rotate $-45°$ , scale $\tfrac{1}{\sqrt{2}}$ , translate $(\tfrac{1}{2},\tfrac{3}{2})$ (right branch)

For $f_2$ : $g^{-1}\cdot[-1,2] = \tfrac{1}{2}(-1+2,\,1+2) = (\tfrac{1}{2},\tfrac{3}{2})$ — exact, no floating-point approximation.

The only difference from the Lévy C Curve is the translations: Lévy has $f_1$ at the origin and $f_2$ translated by $g^{-1}[0,1]=(\tfrac{1}{2},\tfrac{1}{2})$ ; the Pythagoras Tree shifts both maps upward to build the tree above the base square.

Properties

Fractal dimension: 2 — the attractor eventually fills a bounded planar region with positive area
Tiling: The whole tree fits inside a $6 \times 4$ bounding box for the symmetric case
Self-similarity: Each subtree beyond a branching point is a scaled, rotated copy of the whole tree
Number of transforms: 2, both contraction ratio $\frac{1}{\sqrt{2}}$
Variants: Asymmetric versions (different branch angles) produce visually distinct trees while maintaining the IFS structure

All four of the following IFS share the same algebraic skeleton — two maps contracting by $\frac{1}{\sqrt{2}}$ over $\mathbb{Z}[i]$ with expansion $g = 1+i$ — differing only in the power of $s$ for the second map and the choice of translations:

IFS	Second map rotation	Translation
Lévy C Curve	$-45°$ (`g^-1`)	$(0,0)$ and $(\tfrac{1}{2},\tfrac{1}{2})$
Pythagoras Tree	$-45°$ (`g^-1`)	$(0,1)$ and $(\tfrac{1}{2},\tfrac{3}{2})$
Heighway Dragon	$+135°$ (`s^2*g^-1`)	$(0,0)$ and $(1,0)$
Twin Dragon	$-135°$ (`s^3*g^-1`)	$(0,0)$ and $(1,0)$