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 @



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 @


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.”


Endless knot

Endless like this knot, the waves keep weaving their pattern forever.

The Knot geometry project is now sailing toward Macapa – Cidade das Mangueiras (the city of Mango ). No known relationship between the mango tree grain and the 7 – 4 knot, but it was fun to look into it to see if there was any.

In mathematical knot theory, the endless 7 – 4  knot is a 7-crossing knot which can be visually depicted in a highly-symmetric form and is one of the oldest known knot on the planet. It was found on clay tablets over 2500 years (BC) ago and its symbol is featured in many Eastern and Western religions.

Was it at the origin of weaving? It maybe since it is made from a single strand weaved and reweaves symmetrically.

Doing a symmetrical tiling was tempting. But in the context of this project project, I let the waves inspire me and take us into a more complex beadlike composition.

More details on this knot and the knot geometry project @



120-cell knot

The famous 120-cell 4D polytope is a 120 dodecahedral cells. Pick an arbitrary dodecahedron and label it the “north pole”. Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth “south pole” cell. 

Perfect topic for a slow day in the middle of the ocean 18.42º West from the American coast


Isabella’s brooch

Knot geometry project, month of February, 1,650 knots west of Null.

What better place than the middle of the ocean to dream of the 4th dimension. A hologram of Queen Isabella jewelry left by conquistadors on their way west? Actually, the real 4D knot (lower left) is a construct  by mathematician Rob Kusner requiring a 4 dimensional approach to geometry. 50 components, 600 beads, 21,600 polygons went into this model.

More on the Knot geometry project on Patreon



Inverted Möbius

Week 3, 1,200-some (nautical) knots from Null island.

Reflection of an inverted Moebius strip on my compass box. Virtual knot such as this one are made possible using parameters of longitude and latitude on a 3D sphere. The Möbius transformation minimize area distortion and is used to converts 3D brain scans in 2D readable maps.

In addition,  in stereovision,  the 3-rung Möbius ladder helps chemists synthesize molecules. Quite an impressive resume for a surface discovered by German mathematician Möbius in the mid 1800s.

More on the Möbius strip and the Knot geometry project on Patreon @@