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


04-05. Coffee conundrum

What is the relationship between a coffee bean and Alexander horned sphere?


Pursuing my knot geometry journey around the equator line, I’m now above Pasto, coffee capital of Colombia.

It is said that mathematicians can’t make the difference between a coffee cup and a donut. Princeton mathematician Alexander described the problem back in the early 1900s. More recently, S. M. Blinder composed a persuasive demonstration for this problem on the Mathematica site.

Leona de Los Andes, as Pasto is also known, gave me the perfect coffee-knot segue to the Alexander horned sphere. Using Michael Rogers’ script for the sphere, I composed this unusual 3 knot-dance to celebrate both the old mathematical joke and the place I have reached this week.

I selected the Awa flag color scheme for this design. The Awá, also known as the Kwaiker or Awa-Kwaiker, are an ancient indigenous people of Ecuador and Colombia living in the Narino-Pasto area.

More info on Alexander horned sphere and the knot-geometry project @http://bit.ly/knotgeometry


Gueta knot

5,000 miles from Null, from the land of the Muisca, a set Gueta gold earring knots. Gueta means 20 in the local language. A 10.10 knot was good fit. Starting with a 10.10 braid pattern scripted by Tom Verhoeff that I positioned in the background of a pre-Columbian – like structure, I explored a 10.10 knot geometry and discovered this intriguing shape matching perfectly the location I’m flying over this week, 2 (nautical) knots per hour.

The Muisca had so much gold, they were thought to be the real inhabitants of the mythical Eldorado.

Maybe not a coincidence, the Muisca territory is in Colombia, Putumayo national park. Coming from the Quechua language, putumayo means spring-forth or burst out. A little like what some of us are feeling like this week.

More details on the Knot-Geomerty project and the Gueta knot on Patreon


The legend of Baku

I’ve had a hard time focusing on my Knot-geometry project this week, for reasons we all know & share.

However, I am sailing over the Japura river by now. This area of the Amazon at the edge of the  Brazil – Colombia border has a very large concentration of tapirs. Tapir have for unique feature four toes on the front feet and three on the hind feet. Great for an all-terrain run stability. That’s where I found my bridge to a 4 and 3 Conway knot visualization.

The background is a repeat tiling of Conway notation 4 and  the two knots (4&3) were done in stereography,  to give an idea of what an abstract representation of a tapir emerging from  its frame could look like.

Even better, as Do- Mana explains, tapirs are mythical creatures in many Asian lores known for devouring bad dreams and nightmares. Very timely indeed.



A very old knot

Sailing over the Sao Marcos reservation this week, I had a glimpse of the top and the famous Caverna da Pedra Pintada (Painted Rock Cave ) where petroglyphs were found, dating back 11, 000 years ago.

It gave me the idea to explore the shape of the oldest recorded knot in history, the Ryssänsolmu featured on the Antrea net found in Finland’s Karelian basin and dating back to the mesolithic Stone Age.

Still used in many places today ( also called the truss knot, pack or butcher’s knot), in knot theory it converts into a 6.2 knot, one of three prime knots with crossing number six. 

It will be my small contribution to our common ancestors and petroglyph art through the ages.

More details on the 6.2 knot and the Knot geometry project  @http://bit.ly/knotgeometry


The unbreakable knot

Sailing over the Tumucumaque National Park this week inspired me to carve this 7.7 knot in one of the planet most hardest and most precious wood – lignum vitae (tree of life).

This is a nod to our fragile ecosystem as well as its often forgotten dwellers. For good measure, I borrowed for my background a local symbol I found in a wonderful booklet on the heritage of the WAYANA E APARAI culture published by the Brazilian Indian Museum in Rio. 

More details on the Knot-Geometry project, on Patreon


The 10.136 knot

The 10.136 knot. Fitting knot if anything to celebrate my actual position, as I am approaching the Amazon river estuary  in this 52 week Knot Geometry journey along the equator line.

10.136. If one count all the many  tributaries that contribute to this mythical river, we may come close to that number. I used an Amazon-like color scheme for this illustration: from black coffee to cappuccino white and all the variations in between.

Coincidentally, 10.136 is also the number of a poem found in the  ancient Sanskrit,  Rig Veda. As I am not indifferent to the news around us, I’ll dedicate to you all this uplifting quote: “Following the wind’s swift course, go where the gods have gone before and behold our natural bodies.”