Octagonal Complexigram

 Octagonal Complexigram? What kind of animal is that? "Octagonal" means eight-sided. But complexigram? In mathematics, "complex" can refer to something "not simple" or to "complex numbers" with real and imaginary parts. Complex numbers are essential for modern engineering, and without them, we wouldn't have the mesmerizing Julia and Mandelbrot fractals. Complex dynamics? Yes, that makes sense. But why eight?

The Extragalactic Stores offer Octagonal Complexigrams in three versions: Standard, Pro and De Lux. Here is the Pro version:

Octagonal Complexigram Pro

A Cathedral's Inspiration

Here, on our planet, in Strasbourg, stands the Cathedral of Notre Dame—a true masterpiece of Gothic architecture. From 1647 to 1874, it held the title of the tallest building in the world. Today, it ranks sixth among the tallest churches.

Strasbourg - Notre Dame Cathedral

This is what the cathedral looks like from a bird's-eye view.

Notre Dame Cathedral bird's-eye view

A Unique Structure

Here’s the plan of the cathedral:

Strasbourg Notre Dame plan

The cathedral’s tower, reaching 142 meters into the sky, is octagonal. But where are the complex numbers in all of this? Inside the cathedral, its intricate ornaments may hold the answer.

Villarceau Circles - Strasbourg Notre Dame

Spinors and Complex Numbers

Now, let’s compare this with an image from the second volume of Penrose and Rindler’s book Spinors and Space-Time:

Penrose & Rindler - Clifford Parallelism

Do you see the analogy? We meet spinors! And spinors are deeply connected to complex numbers and four-dimensional spaces.

In 2007, the Nobel Prize in Physics was awarded for spintronics! From Wikipedia:

"Albert Fert (French: [albɛʁ fɛʁ]; born 7 March 1938) is a French physicist and one of the discoverers of giant magnetoresistance, which revolutionized gigabyte hard drives."

In 2007, Fert and Peter Grünberg received the Nobel Prize in Physics for this discovery, which contributed to the miniaturization of hard drives.

Toshiba Storage Division

A Journey into Spintronics

From the Internet, we also learn some personal details about Professor Albert Fert. Born in 1938 in Carcassonne, he was surprised by the award, despite knowing he was on the shortlist. Modestly, he acknowledged the many outstanding scientists worldwide. When asked about his interests, Fert mentioned playing rugby for 20 years, windsurfing, enjoying films, photography, and listening to jazz.

So, I invite you on a journey into the fascinating world of spin. In future notes, we’ll dive deeper into the details—where the devil hides. But for today, here’s a historical tidbit.

Spin and History

In 1924, Wolfgang Pauli introduced spin, a new degree of freedom to explain the behavior of spectral lines of atoms in a magnetic field. He believed this new freedom had no classical equivalent—it was purely a mathematical concept.

Yet, in 1926, L.H. Thomas (of Thomas precession fame) wrote in a letter to Samuel Goudsmit:

“I think you and Uhlenbeck were very lucky to get your spinning electron published and talked about before Pauli heard of it. It appears that more than a year ago, Kronig believed in the spinning electron and worked out something; the first person he showed it to was Pauli. Pauli ridiculed the whole idea so much that the first person became the last, and no one else heard anything of it. Which all goes to show that the infallibility of the Deity does not extend to his self-styled vicar on earth.”

P.S. 20-09-24 9:36 The two other edges of the Octagonal Complexigram on the picture are invisible, as they extend into the two extra time dimensions. Time, as it is well known in certain circles, is three-dimensional. Space-time is six-dimensional.

P.S. 21-09-24 11:04 And here is the picture of Igor Bayak's dynamical system with limit cycle on a unit circle (1.1.1):

P.S. 21-09-24 11:54 Eq. (1.1.7) in Igor Bayak's "Mathematical Notes on the Nature of Things" - this time with logarithmic spirals

P.S. 21-09-24 15:42 Tried to plot on the cylinder z=log(rho), but I am not sure if I got it right