About the Knot-Geometry project

Knot geometry

Knots, they are all around us. They tie up our everyday life, they are a structure studied by mathematicians, and they create surprising elegant patterns in space too.

Knots are tangible, yet they often create very complex abstract shapes. In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions. It cannot be untangled to produce a simple loop (​Weisstein, 2019​)

I studied knots before in the 12-30 project, month of June as part of a research on mathematical visualization software. This time, I’ll want to anchor my project in reality and add elements of space and time into it.

According to cartographers, the earth circumference is 24,901 miles, approximately 21,000 nautical knots. Starting at latitude 0, longitude 0, I’ll sail an average of 400 (nautical) knots a week to fully cover its circumference in 52 weeks, 1 (mathematical) knot a week, and visit places and cultures I didn’t know existed. That should give me plenty of background inspiration to interpreting visually the structure of a different mathematical knot each week (Fig.1).


Figure 1. 52 (mathematical) knots a year following the equator line.

To get a livelier sense of what I’m getting into, I’m basing my journey on the 15th century Ortelius map (Fig.2). A good fit, I believe. Like my calculation, the accuracy of the mapping is tentative at best. Furthermore, there is as much to discover in the world today as it was then.

World map

Figure 2. Ortelius world map

Following the equator line in broad strokes will give me a simple framework and provide valuable and consistent resources both climatic and cultural. Depending where I’ll be each week, be it lost in the abstract nature of the ocean or exploring the many rich cultures I’ll be crossing in my journey around the planet, I’ll have plenty of visual references to work from and combine the various parts relating to the mathematical knot in a larger and more holistic visual statement.

Landing on Null on Jan 1st, I’ll set sail westward from latitude 0º longitude 0º at 00.01 am GMT (or UT1 for some) on Sunday January 5th, and in the tradition of the 16th century Spanish explorers. will be moving West toward the Americas. New information will be posted every Sunday thereafter.

To help me track my progress in real time, I’ll be using Cameron Beccario wind map, a feast of abstract strands carrying an unexpected closeness to Knot braid representations. (Fig. 2).


Figure 3. Latitude 00, longitude 00

Additionally, I’ll work with ​GoogleEarth studio​ to extract visuals to complement the narrative and Weather.com​ to add real-time weather information.

My work will have 3 main parts:

  • A wind and climate part that record changes on a weekly basis, from GoogleEarth studio and eath.null school sites,
  • A formal part with a knot description & visual extracted from a modeling program, and last,
  • The final visualization combining the various climatic, cultural and mathematical references collected in the first two parts.

Background notes, sketches and other material will be posted weekly on ​Patreon​ hoping that aside from enjoying my work, it can motivate, inform, and entertain some to come along and follow the evolution of the journey in this fun and unusual exploration of knot theory.

Additionally, and to increase the visibility of the project, I’ll post a brief recap of the week on social media platforms: ​Facebook,​ Instagram, and ​Twitter​.

But first, to celebrate the beginning of this series, I’ll honor the imagination of the US, French, and Brazilian scientists that monitor and maintain the station I’m starting this project from.

Having a healthy sense of humor on top of their technical competence, they instigated the creation of a virtual but very real island on this geographical spot. The lone buoy about 380 miles south of Ghana and 670 miles west of Gabon goes by the name of Null, which also happens to be the word associated with the beginning and ultimate representation of the knot theory.

Since its creation, the “island” of Null has been given its own geography, flag, and history. Appropriately, I’ll use a variation of its color theme for my banner and start my very first knot by celebrating the island flag, a simple un-knotted closed loop on an orange, grey, and blue background.

250 prime knots with ten or fewer crossings, 1,701,936 prime knots (including the unknot) with up to 16 crossings according to ​Jim Hoste, Jeff Weeks, and Morwen Thistlethwaite​, that’s a lot of knots to choose from, not counting composite or hyper knots. 52 weeks seems like a very short time to cover all – or part – of it. Somehow, I’m sure the discoveries I’ll be making will compensate for the limits time is imposing on me.

Many mention that Science’s major impediment is its abstract and complex nature and the challenge its practitioners have to communicate and share their findings in simple terms. Next to words, images do carry a lot of information.

If doing so, I can illustrate that all forms of expression are part of a greater, natural, and all encompassing process, and that by learning from each other and collaborating, we continue to contribute positively to the conversation, I’ll have reached my goal and made the objective of this project well worth my commitment to bring it to completion.

Being a visual artist by training, I’ll have to thank my colleagues in Science for it.

Jean Constant

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