Today I thought I’d let my husband get a few words in. He recently posted to his blog in Polish and here is the translation into English. I hope you enjoy it as much as I did.

2007-08-13

To begin with, it’s Monday today and there was a man called Sir William Rowan Hamilton. How is Monday related to him? It will become clear soon.

In the Polish Wikipedia we can find rather little information about Lord Hamilton:

William Rowan Hamilton (August 4, 1805- September 2, 1865) was an Irish mathematician, physicist, and astronomer.

He was a director of an astronomical observatory and a professor of Trinity College, Dublin. His works concerned algebra, theoretical mechanics, optics and differential calculus. He introduced quaternions, complex numbers as ordered pairs of real numbers, and established a certain way of multiplying and adding these pairs. In his youth he had an ambition to learn as many languages as his age. He is supposed to have kept his word until he reached the age of 17.

See also: hamiltonian

In the English Wikipedia we can find more interesting information about Lord Hamilton’s astonishing linguistic abilities:

At the age of seven he had already made very considerable progress in Hebrew, and before he was thirteen he had acquired most classical European languages and also Persian, Arabic, Hindustani, Sanskrit, and a few more. However, mathematics was his passion. In spite of being constantly heaped with praises, Hamilton always remained modest.

We are interested, however, in the mysterious quaternions which he discovered and which have become an integral part of the scientific and conceptual heritage of mankind (quaternions, quaternio: yes, Pythagoras and Jung are smiling from the beyond). It is interesting how they were discovered.

Hamilton for many years was trying to find a method of multiplying not the usual numbers, but three-dimensional points of space, so as to make possible not only their multiplication but also their division. In vain. The children kept asking every morning: Well, Papa, do you know how to multiply triplets yet?

Finally, one memorable Monday, October 16th 1843 A.D., he went, with his un-loved wife Helen, strolling along the Royal Channel in Dublin. Tired of walking, and maybe of each other, they rested under the Broom Bridge.

Hamilton must have had the problem still hidden in his conscious or subconscious mind and instead of thinking about Helen’s qualities, his head was filled with quite different thoughts. And then, abruptly, an illumination came: in order to solve the problem of multiplication in three dimensions, it is necessary to open oneself, get rid of stereotypes and use four dimensions! The formula was there in a split second. He took a pen-knife out of his pocket (yes, yes, Ladies and Gentlemen - back then, people carried pen-knives without fear of being accused of being a terrorist!) and scraped the formula - that today looks exactly the same - on the stone bricks of the bridge:

i2= j2= k2= ijk= -1

In 1958 a memorial plaque, now slightly damaged, was built into this place:

What do we learn from this story, known to many mathematicians, philosophers and historians of science? First, always carry a pen-knife. Second, let’s take our un-loved spouses for walks, otherwise instead of the muse, a deceitful siren may manifest. Third, illumination comes after hard and arduous work, not instead of it.

So, you ask, where is Love in this? Where is the soul?

Let’s begin with the fact, that Sir William Hamilton fell in love at the age of nineteen – it was love at first sight – with Catherine Disney (in the genealogical line with Micky Mouse and Donald Duck). Unfortunately, Catherine, in 1825, under the influence of her parents, married pastor William Barlow, who was fifteen years older than she. From that moment on, the desperate Hamilton had one affair after another, until finally, in 1833 he married Helen Mary Bayly.

Their marriage wasn’t happy, Helen wasn’t psychologically healthy, to put it kindly. Ten years later Hamilton inscribed his famous formula on the bridge. Catherine eventually left her un-loved cleric husband, Helen found letters from Catherine to Hamilton, and several other unfortunate things happened; the cumulative effect being that Hamilton became an alcoholic.

In 1853 Hamilton finished his "Lectures on Quaternions’" and immediately he went to show them to Catherine. After so many years, they clung to each other desperately. Catherine died two weeks later. Hamilton continued to drink heavily and live an unhealthy life. Shortly after he finished his famous monograph, "Elements of Quaternions", he died from overeating and alcohol abuse. "Elements of Quaternions'’ was published posthumously.

It’s all quite sad, isn’t it? Imagine what Hamilton might have done in a happy marriage. Does the world have to work that way? Does the happy love of souls exist at all? And if it does, how can we find it? To answer this question, we have to return to quaternions, four dimensions and then visit Mr. And Mrs. Icosinsky.

So now, on to the quaternions. Let’s calculate, i,j,k represent three dimensions, but where is the fourth? The fourth one hides on the right side of the equation,within the "minus one". Let’s not forget about the inconspicuous "1"! Quaternion q=(w,x,y,z) is noted as

q = w.1 + x.i + y.j+ z.k,

"conjugated" with it is the quaternion q

q = w.1 - x.i -y.j - z.k .

Product of q i q defines thesquare of the length q:

||q||2 = q.q = w2+x2 +y2 +z2 .

Quaternions of the unit length fulfill, therefore, the equation of three-dimensional sphere submerged in four dimensions:

w2 +x2 +y2+z2= 1.

During his life Hamilton was searching for true love and true beauty. He had problems with finding reciprocity in the world of humans, so he sought for it in the world of Platonic ideas. He started to play with his quaternions. He wanted to place the beautiful quaternions on the surface of a three-dimensional sphere, and do it so that the result would represent a possibly beautiful, symmetrical figure. But what do we mean by beauty?

Consulting the Wikipedia, (the easiest and the quickest way), we find there the following reference under the term "beauty": "See also golden ratio."

So, we go to the golden ratio, where we find the mysterious number"φ":

Its reciprocal, Φ=1/φ, is the same as φ -1=0.618033989, the Golden section, harmonic division, divine proportion – as it is commonly refered to.

So, Hamiltonwas toying with (w,x,y,z) substituting them with different combinations of simple numbers such as 1, 1/2 as well as φ, always in such a way, so that they stay on our 3D sphere. He was calculating their product … And that’s how he came up with … Icosians.

Note: Hamilton reached the icosianswith a slightly different route than the one I presented above, however, for us here, it is not so essential.

Where does the name come from? From icosahedron. Let me remind the reader that the icosahedron is one of the five Platonic solids, a convex regular polyhedron with 20 triangular faces, 12 vertices, 30 edges; 5 faces are meeting at each vertice.

Let us now release the reins of our fantasy and let’s imagine a hypothetical element of atomic number 137 with a chemical symbol of Ic (I am joking here, although not entirely). We will first build Mrs. Icosinsky from these atoms.

Mrs. Icosinsky has a crystal structure and is of a rare beauty. Atoms of the element Ic constituting her crystal are distributed regularly, there are 120 of them. Mrs. Icosinsky lives in a four-dimensional world (but the fourth dimension is not “time”; Mrs.Icosinsky lives in "eternity"). Atoms of the crystal are positioned as follows:

Eight, easy to remember positions: ( ±1,0,0,0),(0, ±1,0,0),( 0,0, ±1,0),(0,0,0, ±1).

Sixteen equally easy:(±1/2,±1/2,±1/2,±1/2)

Ninety-six atoms, whose positions are "divine", namely of the form: (±φ,±1,0,±Φ) as well as their even permutations.

Adding up these numbers we get 8+16+96= 120: this many atoms make up the crystal of Mrs. Icosinsky. Each and every one of these atoms (icosians) can be depicted as a quaternion (four numbers) of unit length. Curiously, multiplying one icosian by another we always get a third one! Icosians form a group with respect to multiplication! A pictureof Mrs. Icosinsky:

Mrs. Icosinsky is a crystal woman, possibly the most noble soul in the entire Galaxy of ours, in all of its Dimensions. However, she did not feel safe. Although her crystal had 120 sharp vertices, it also had 600 unprotected, flat faces – which were attacked by her enemies and psychopaths. So, she was dreaming since her childhood (yes, yes, even those living in "eternity" have their childhoods, for time has not one, but three dimensions: Chronos, Kairos and Aion), that there is somewhere the one and only, the polar opposite, the real love of the soul, and that "the one " will one day learn about her existence and will answer her call.

Mr. Icosinsky indeed existed, only in another Galaxy. His crystal was built from atoms of the same mysterious element Ic (atomic number 137), only the atoms in his crystal were positioned differently. Where Mrs. Icosinsky had a vulnerable and flat surface, Mr. Icosinsky had a sharp vertice, scaring off intruders and aggressive psychopaths. It also so happened that where Mr. Icosinsky was flat, Mrs. Icosinsky was sharp. In short, Mr. Icosinsky’s crystal had 600 vertices placed precisely in centers of the flat faces of Mrs. Icosinsky’s crystal, he also had himself 120 flat faces, which centers were located exactly on the lines defended by vertices of Mrs. Icosinsky.

(At left, Mr.Icosinsky, as he apeared on the screen of Mrs. Icosinsky – at that time she was using the intergalactic version of Windows 95)

So, they were making a "dual couple", one represented the "polar opposite" of the other. Mathematicians call the crystalline structure of Mr. Icosinsky, in which the divine proportion is also embedded, by the name 120cell (sometimes also "120 cell", search in Google). That is to say, the "polar opposite" of the 600cell is the 120cell. They are one "dual" - one to another. They both are "solid bodies" in 4 dimensions. 600cell has the "Schlafli symbol {3,3,5} while 120cell has the symbol {5,5,3}.

Our story is reaching the end. You would surely want to know how it ended? Hopefully, not similarly to that of the story of our sad Lord Hamilton?

I answer that no, it did not end sadly. Mr. Icosinsky heard the voice of Mrs. Icosinsky, although he was in another Galaxy (for there existed the intergalactic Internet). They found each other and ever since, they are always together, having a common purpose and without respite they are sharing their knowledge and experience with others, who still search for nobility, beauty and a true love of the soul. They are deepening their own knowledge - each one serving the other - and together they serve the World and its Creator.

P.S. My daughter recently asked me whether it is certain that there is only one "one and only"? What if there are more of them? (she was saying it certainly recalling her own life experiences, rich in numerous dramas and confusions). Here is what I answered: "as long as you are not a "crystal" of uniform structure, as long as you are a shapeless conglomerate, a cluster of dozens of different "persons" - none of them seeing and knowing about the others - as long as there is no ONE, real complementary person within YOU, then none of the candidates for the complementary person "out there" can be the "real one".

Therefore, there is a need for "individuation", as Jung calls it. Others call it yet differently, but it amounts to the same thing. One has to first "crystallize" his own "I" – so, that the real crystalline soul can shine and be recognized, even from another galaxy, by the polar opposite - the complementary soul. Then the "alchemical process" becomes possible, defined by Jung as "Conjunction". (On the side, I like and respect Jung, but I am not very fond of him).