Coral knot

Coral reef, reef knot, square knot. Free word association? Maybe, maybe not, as little specks of land are coming at me above and under me, on week 24 of the Knot geometry journey.

Fractal patterns have been observed in the small-scale structure of shallow-water coral colonies. The coral stitch, also called the coral knot stitch or coral knot, is a line stitch made of knots. It provides an impromptu canvas for this square knot shape gliding over it. The 6 -fold symmetry of the knot echoes the 6 fold symmetry of the coral on this iterated square board from  Jaime Rangel-Mondragon



Black pearl

Where do black pearls come from?

Some may say from oysters in some far away Pacific island. But how about a 16 cells, 1040 fragments polytope hexadecachoron?

In 4D geometry knots can be unknotted. Conversely, unknots can start their journey as 3D knots squeezing in between the vertices of this very abstract 16-cell polytope.

Starting with 2D planar geometry we can build a 3D ribbon and project it in 4D.  That’s what I did in KnotPlot to look into the interaction between a knot and a 16 cell polytope.

It may not solve the black pearl mystery, but it creates an elegant string for this unusual occurrence that happens only in one out of 10,000 pearls.



Quipu knot

Quipu” is a Quechua word meaning “knot”. According to  the Quechuan culture, the number 731 is represented by 7 and 3 simple knots and a figure 8 knot. In mathematics, the closest knot relating to this number would be the 7.3 knot.

Visually, it could stand as a proud and a well-balanced accounting statement!

To add to the mood, as the knot-geometry project is within sight of the Marquesas islands, I have borrowed a design from the local complex tattoo  tradition as a background for this week’s illustration. More about the 7.3 knot  and the Knot geometry project @



Unslicing Conway knot

This one in celebration of maths student Lisa Piccirillo  who recently unsliced the famous Conway knot that puzzled mathematicians for decades.

In knot theory, the Conway knot is a particular knot with 11 crossings. It is named after mathematician John Horton Conway

Genius mathematician Conway, who passed away recently, was also a great humanist. He would have loved to thank Lisa in person, I’m sure. 

#knot #knot geometry #Convay #Piccirillo #Kinoshita. More on the Conway – Kinoshita-Terasaka knot mutation @



Satellite knot

I have a great admiration for sailors. Day or night, just the ocean and the sky to look at. 

And then it happens!

Now, a satellite train keeps circling the planet in an endless loop. Is there any link between a satellite knot and a Starlink satellite train? None that we know of, but it’s a great segue for this week’s illustration. The combination of the two creates an intriguing design, mysterious, fearsome, and ethereal at the same time. 

Are we chain linking the planet?  Adrian Jimenez Pascual presented a new family of knots called lassos to construct families of satellite knots. Could they rope the pesky little balls too?


  • Center image: A train of Starlink satellites passing over Pukehina, New Zealand. Image courtesy Astrofarmer.



Braid, symmetry, and quantum physics

A bead curtain? At first, nothing exceptional.

Actually it is a 21 string braid made of 1033 beads. Sailing this week in the middle of the Pacific, with 90% humidity , vapor lines must be raising like hazy braided curtains, a good segue for the Knot-Geometry project!

In mathematics, a braid is an intertwining of some number of strings attached to top and bottom “bars” such that each string never “turns back up.” This arrangement belongs to the Artin braid group and display a symmetry that can expand to infinity.

A little like the background, a simple view of the ocean, the sky and some clouds. I converted a generic view of the ocean into a more abstract figure. The original is included in my Patreon portfolio, wind map #18.

More on Patreon and the knot geometry project @


Of spirals and cyclones

Cyclones. They can be messy. They are characterized by spiral patterns.

Spirals, going back to the beginning of time,  can be found in every culture, worldwide. Was the symbol created out of fear? Out of respect? We’ll never know. 

However, the mathematical spiral knot,  the trefoil knot – cousin of the Triskelion, soars like an elegant signature, or maybe the ghost of a weightless dancer twirling and turning on itself.

Simple, minimal and graceful, the spiral knot can also signal chaos and destruction in the order of things. However, unlike many knots, the trefoil knot get all unknotted in the 4th Dimension. Just like cyclones – they appear, grow, create havoc and die out.

An intriguing connection between spirals, spiral knots, cyclones and cultural symbols.


The knot puzzle

Leaving the Galapagos behind and moving westward, 5,000 (nautical) knots to the next above-water land. What better time to start that leg of the Knot-geometry journey than playing a game called knot puzzle created a few years ago by Akio Karachi & Kendo Kishimoto, two mathematicians from Osaka U.

The puzzle ask me to identify each region and line up the crossing sections in a patch of unified bright color disks.

The knot puzzle I selected is closely related to the 6.2 crossing knot one of 3 prime knots with six crossings. It is alternating, chiral, and invertible.

In the third dimension, this knot is associated with the Borromean rings. Composed of three interconnected rigs, it creates a perplexing figure as no two of the rings are linked with each other, yet all three are linked together.

The open ended version of the 6.2 knot, the stevedore, or docker knot can be found on many shipping docks around the world.

Culturally, it figures in Kolam, an Asian geometrical line drawing tradition drawing rice powder pattern on the floor of the house to invites birds and other small creatures to eat it, thus welcoming other beings into one’s home and everyday life: a daily tribute to harmonious co-existence.

From the House of Borromeo to Indian floors and shipping docs,  this knot demonstrates quite unusual but very versatile qualities.


The turtle knot

Week 15 of the knot geometry project. An 8-15 knot to celebrate the 15 subspecies of the Galapagos giant turtle according to Academy herpetologist John Van Denburgh.

Standing on a 8.15 braid-shaped iron work balcony this invertible knot rests on what is an actual giant turtle shell. 900 lb or more, the Galápagos giant tortoises used to populate the entire planet but today can only be found in two small islands, the Galapagos being one of them.

More on the Knot-geometry journey, wind maps and mathematical knots on Patreon


The Perko pair dilemma

For a hundred years the best mathematical minds thought knot 1 and knot 2 in the background were two different knots until K Perko in 1973 found they were one and the same. Who would have guessed? Knot classification is not a simple task.

From my vantage point, knot 10.161 could also qualify as a memorable roller coaster ride!

More on the Perko pair and the Knot geometry project on Patreon