Chinese Lamp

Fractal dimension: ≈ 2.334

Visualization

Open in IFStile ↗

AIFS program

@rotate3d
$a=h
q=[0,0,1]
ang=0
k1=1
k2=1
arad=ang*asin(1)/90
sn=k2*sin(arad)
cs=k2*cos(arad)
qn=(q[0]*q[0]+q[1]*q[1]+q[2]*q[2])^(1/2)
x=q[0]/qn
y=q[1]/qn
z=q[2]/qn
a=[(x*x),(x*y),(x*z),(y*x),(y*y),(y*z),(z*x),(z*y),(z*z)]
b=[0,z,-y,-z,0,x,y,-x,0]
ret=k1*a-(b*sn)-((a-1)*cs)

@
$dim=3
$camera=[[-0.1,0.55,-3],[0.5,-0.5,0.62],[0.17,0.74,0.15],30]
$root=ChineseLamp
ChineseLamp=(1|[1,-1,0]*rotate3d([0,0,1],180)|[0,0,1]*rotate3d([1,0,0],90)|[0,-1,0]*rotate3d([0,0,1],90)|[1,0,0]*rotate3d([0,0,1],-90))*SnowBall
SnowBall=([0,0,0]|[1/3,0,0]|[2/3,0,0]|[0,0,1/3]|[2/3,0,1/3]|[0,0,2/3]|[1/3,0,2/3]|[2/3,0,2/3]|[1/3,1/3,1/3]|[1/3,0,1/3]*rotate3d([0,0,1],90)|[2/3,1/3,1/3]*rotate3d([0,0,1],-90)|[1/3,0,1/3]*rotate3d([1,0,0],-90)|[1/3,1/3,2/3]*rotate3d([1,0,0],90))*(1/3)*SnowBall

Overview

The Chinese Lamp is a 3D fractal object built by assembling five translated and rotated copies of the SnowBall — a self-similar fractal surface. The construction originates from IFS Builder 3D and is related to the quasisymmetric surface theory of D. Meyer (2002).

The SnowBall

The base building block is the SnowBall: a self-similar fractal surface defined by 13 affine maps each scaling by $1/3$ . The 13 contraction centers are distributed over the surface of a unit cube — 8 on the $y=0$ face plus 4 rotated edge copies plus 1 interior-facing map:

Position	Isometry
$(0,0,0)$ , $(1/3,0,0)$ , $(2/3,0,0)$	identity
$(0,0,1/3)$ , $(2/3,0,1/3)$	identity
$(0,0,2/3)$ , $(1/3,0,2/3)$ , $(2/3,0,2/3)$	identity
$(1/3,1/3,1/3)$	identity
$(1/3,0,1/3)$	rotate 90° around $z$
$(2/3,1/3,1/3)$	rotate −90° around $z$
$(1/3,0,1/3)$	rotate −90° around $x$
$(1/3,1/3,2/3)$	rotate 90° around $x$

The SnowBall attractor $S$ satisfies:

S = \bigcup_{k=0}^{12} \tfrac{1}{3} R_k(S) + t_k

where each $R_k$ is a rotation and $t_k$ the corresponding translation. Its Hausdorff dimension is:

d = \frac{\log 13}{\log 3} \approx 2.334

The ChineseLamp Assembly

The ChineseLamp is formed by placing 5 copies of SnowBall under different rigid motions:

Copy	Isometry	Translation
0	identity	$[0,0,0]$
1	180° around $z$	$[1,-1,0]$
2	90° around $x$	$[0,0,1]$
3	90° around $z$	$[0,-1,0]$
4	−90° around $z$	$[1,0,0]$

The five copies interlock to form the lamp silhouette.

References

Meyer, D. (2002). Quasisymmetric embedding of self-similar surfaces and origami with rational maps. Ann. Acad. Sci. Fenn. Math., 27, 461–484.
IFS Builder 3D (original source)

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