Geometry of Batik

Chapter 1 · Batik Kawung

Visualizing history, geometry, and art
Section I · History of Batik Kawung

Origins and Symbolism

Batik Kawung is often cited as one of the oldest batik motifs in Java. Its four-petaled, overlapping-circle geometry appears on stone carvings in temples and on royal textiles, long before it became a modern batik staple. The motif decorates statues of Mahakala in Singosari temple, as well as images of Ganesha, Nandishvara, and Durga Mahishasuramardini.

Interestingly, similar circle-based patterns appear far beyond Indonesia. In Old Babylonian mathematics, a related figure built from circular arcs is known as the apsamikkum, and in Japan a comparable overlapping-circle pattern is called shippo, the “seven treasures” motif often seen in family crests and Buddhist ornament. Across cultures, the same simple geometry is reinterpreted with different meanings—prosperity, harmony, sacred protection.

The popularity of Kawung has easily crossed into the 21st century. Today you can find it on fabric, building façades, graphic design, and even in popular media—if you look closely, echoes of Kawung-like circles appear in the opening credits of Studio Ghibli’s The Tale of the Princess Kaguya (2013). Knowing how widely this motif travels, it becomes even more tempting to explore the geometry that lies underneath.

A motif that travels
World map · Tap / hover the glowing dots or legend items to see local names.
Kawung-like circle geometries across regions
Mesopotamia
Java · Indonesia
Japan
Mesopotamia · apsamikkum
Old Babylonian figure built from circular arcs, used in mathematics problems about areas and diagonals.
Java · Kawung
One of the oldest Javanese batik motifs, found on cloth, temple statues, and modern Indonesian design.
Japan · shippo
Overlapping-circle “seven treasures” pattern, common in family crests, textiles, and Buddhist ornament.
Section II · Geometry of Batik Kawung

Astroid Curve as a Kawung Cell

One way to see Batik Kawung is as a repetition of a smooth four-petaled shape. That shape can be generated by tracing a point on a circle rolling inside a larger circle. Mathematically, the resulting curve is called an astroid.

Interactive demo · Drag/scroll the page normally; the curve loops on its own.
Astroid parametrization:   x = a·cos³(t),   y = a·sin³(t)

In Kawung, this rounded diamond shape is repeated and overlapped in a grid. Each astroid can be thought of as a single “seed” of the motif. By translating and rotating this seed, we can fill the plane with Kawung patterns.

The animation shows a small circle rolling inside a larger circle. The yellow point traces the astroid curve, then the region between the curve and the outer circle is softly filled—hinting at a Kawung “petal”.
Another viewpoint · Kawung as the intersection of four circles with centers at A, B, C, and D.

The same Kawung cell can also be seen as the common intersection of four circles: two along the horizontal axis and two along the vertical axis. Their overlapping region forms the familiar four-lobed Kawung shape.

First the circle centers C, D, A, and B appear, then circles are drawn, and finally the overlapping region is filled in blue—as the Kawung “seed” built purely from circles.
Yet another construction · Kawung from the envelope of Bézier curves inside a square.

Imagine a square rotated by 45° around the center. Using its four vertices as endpoints, we draw families of Bézier curves from the origin to each vertex and back. The envelope of these curves thickens into a smooth, rounded diamond. Repeating this construction four times reveals a full Kawung cell—built entirely from Bézier strokes.

Watch how the envelope grows from thin white lines, then the Kawung stroke is highlighted, and finally the blue fill fades in with the caption:
“Kawung as Bézier Curve with square end points”.
Polar curve view · Kawung as a four-leaf quadrifolium (a rose curve with four petals).
Quadrifolium:   r = a·cos(2θ)

The same four-lobed silhouette also appears as a quadrifolium, a polar curve defined by r = a·cos(2θ). As θ runs from 0 to 2π, the point moves in and out from the center, tracing four symmetric petals. With the right scaling and rotation, this rose curve lines up beautifully with a Kawung cell.

Watch the curve grow petal by petal. Once the full quadrifolium is traced, the Kawung-like region is filled with the same blue used in the previous animations, tying all constructions together.