Toraja knot

A 7.1 knot honoring the Toraja culture from Northern Indonesia. In their tradition, 7 (1+6) conveys a sense of oneness, togetherness, equality. Their wood carving inspired this week’s iteration of the knot geometry project. The Indonesian colors may have done the rest. 

Thanks to Lande Sudianto and Petrus Simon for their work cataloging Toraja art and putting me on the right track.

Aquatic cinquefoil

A school of cinquefoil knots in the middle of a coral reef in the bay of Weda, Indonesia. 

I should credit Margaret Wertheim’s Ted talk  “The beautiful math of coral” for inspiring that connection.  She stated “The frilly crenulated forms that you see in corals, kelps, sponges and nudibranchs, is a form of geometry known as hyperbolic geometry. And the only way that mathematicians know how to model this structure is with crochet. It’s almost impossible to model this structure any other way, and it’s almost impossible to do it on computers.

Not so – from an artistic perspective. Knots and coral blend quite nicely in this unique and fragile environment.

Background, courtesy of WedaResort 

Mapia knot

You shouldn’t look at this cinquefoil knot as a shape but through the light and shades it projects. Scientists use knots to study the consistency of light beams. Very useful in laser and holographic technology, apparently.

Why a Mapia knot?

Indonesia is comprised of some 17,000 islands, it has hundreds of major aids to navigation, in particular lighthouses. I’m passing by the Mapia atoll lighthouse this week on the knot geometry journey. And light helps define knot contours as Larocque et all submit in their paper Reconstructing the Topology of Optical Polarization Knots

Great segue to explore the visual projection of a cinquefoil knot

Triangular knot

Triangle knot. The people of Papua knew about geometry thousands of years ago. Where did it get lost? How come so few today get a sense of how useful geometry can be – as building a sturdy home in the jungle for example!

This knot figure was inspired by a 2016 paper by ethno-mathematicians Haryanto, Subaji and Nusantara  Thanks for reminding us Mathematics is universal and as ancient as humanity.

More on the triangular knot and the Knot geometry project  @

New Ireland

Week 35 on the knot geometry journey, I’m passing by an island called New Ireland in the Bismarck archipelago. Good segue to explore another Celtic knot!

All figures are a variation of the same 3 braided knot, except for the endless 12/2 grid. 

The beads show the pattern this knot would have if it had no braids. Starry like a night on the Pacific Ocean!


Knot ethnography. Inspired by a conversation with my friend Dan Klarskov on the endless knot, and as continue on the knot geometry journey, a tribute to Lapita cultural tradition.

From Celtic to Vedic, knots have crossed time, culture, and oceans in an endless loop fascinating humans…

 Below the knot, fragments of Lapita pottery, dated 1000 BCE and coming from the Santa Cruz group of islands, south-east of the Solomon Islands, courtesy of the University of Auckland, Department of Anthropology, Anthropology Photographic Archive.

More on the Kno-Geometry journey, knots & windmap updates on Patreon

Tribal knot.

Could it be a mask, a rattan headdress from some unknown native New Guinea tribe floating in the sky? 

Actually  it is an unusual iteration of a Klein bottle, a 4 dimensional knot existing only in the imagination of mathematicians, adventurous artists, and maybe in the mind of some of the island’s inhabitants. Perfect segue for this week’s illustration as the Knot Geometry journey is nearing Asia.

More on the Knot Geometry project on Patreon

Inverting distances

Along with changing West to East, yesterday to tomorrow, I need to reverse my distances too, moving forward past longitude 180º.

Sailing is not a simple task when it comes to reconcile logic and reality! Good thread for working on an inverted 7 crossing knot, though. Its shape fits well with the coral reefs nearby on the ocean floor.

More on the inverted knot and the knot geometry journey @

Knot story

Collision, tension, randomness. Knot theory is an ever-expanding subject. How can art respond to it? Aesthetics? Storytelling? A bit of both using mathematics stable guidelines.

This strange11.4 knot takes on the shape of a Henri Moore sculpture as the sun sets over the vastness of the ocean at longitude 177 º East on the equator line.

More on the Knot-geometry project and the 11.4 knot on Patreon

Musical knot

Collision time for the knot-geometry project, right on the line where yesterday becomes tomorrow. 

Is time dissolving at exactly 180º longitude? Knot-wise this a serious question that could relate to the study of collision. And collision creates sound.

Which brings another question: what is the sound of a knot? Robert G. Scharein who designed Knotplot gives us an interesting option based on collision detection. Slide the ring on the structure of a knot and there you have it, a symphony of modulations of all kinds.

In this image, I use color to achieve the same objective. This is an interesting but well-known oddity: visual art is often described in terms of sound and music — music uses color and images instead!