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How I Discovered a Fractal… And a Bizarre New World

This is the story of a buried treasure brought to light — a precious gemstone unearthed from some foreign, mathematical land, and it begins at the intersection of many different worlds: Switzerland.

During the day, I was a teacher of math and physics at a boarding academy in an alpine village of the French-speaking canton of Vaud. Despite the location, the students were exclusively Japanese, and as an American, I was hired to teach solely in English. To add to the linguistic mix, every evening I drove a half hour through the alps to the German-speaking canton of Bern — undoubtedly the most beautiful commute on earth — where I lived with my Viennese wife and son.

While my formal education was in physics, I had spent several years traveling the world after graduation, living and teaching as far afield as Thailand for some time. I had always been interested in doing academic research but the hoop-jumping horror stories I had come to learn about life in academia had me looking for alternatives.

I decided I didn’t need grants or a university to do research in math or theoretical physics. After all, most of the information I needed was available online anyway.

I just needed time.

I started making a habit of doing calculations during my travels, sometimes in my head if necessary. In fact, the first problem I ever worked out was while driving home from my university one summer. Having a passion for painting, I was interested in the mathematics of perspective projection, a geometric concept related, for example, to how the lines of a highway or railroad track appear to converge at the horizon. During that drive, I worked out the rate at which the spacing between railroad ties would appear to decrease were one to stand on the tracks and paint the scene, extending of into the distance.

Little did I know this calculation would come back to help me years later while living in Switzerland.

It was nearly 3 AM one night at my home in the village of Gsteig b. Gstaad when I first caught glimpse of the geometric wonder.

I had been toying with the idea of algebraically relating the numerical constant 𝜋 with the golden ratio ϕ, through a quite recent discovery of George Phillips Odom Jr in the 1980s.

Odom, an artist and amateur geometer, had discovered that the relationship of an inscribed equilateral triangle to its circumscribed circle generates the golden ratio, such that AB/BC=ϕ, as shown above. I had noticed that one can use the segment of BC as the base of a new equilateral triangle that precisely intersects the circle at its vertex, generating several new triangles with golden proportions to the first.

My hope was to find a recurring pattern such that one could completely tile the circle’s area with ever-smaller golden-proportioned triangles, and thus through mathematical induction, determine an infinite sum relating ϕ to 𝜋 — but my hopes were quickly dashed. No such pattern emerged.

But as with many discoveries, it is through the accidents we find the true gold.

I continued to play with the geometry software I was using — exploring beyond the confines of the circle and generating ever-more intricate patterns of golden-proportioned triangles.

And then I noticed something strange.

The triangles appeared to be extending off into some imaginary distance as though standing, rank and file, upon some abstract field.

Before long, I worked out a pattern that confined all the triangles to a single shape — a grand, outer triangle that enclosed the entire scene.

The result was a fractal — a type of geometric curve featuring self-similar copies of itself.

a new fractal

I was certain such a beautiful figure was already known, especially considering the popularity of the golden ratio among enthusiasts. But despite a long and thorough search, I found nothing in the literature about the shape. Many exhaustive compendiums on fractals, triangles, and the golden ratio all failed to mention this iconic image — a geometric figure at least on par with the famous Sierpinski triangle.

It dazzled me.

I was riveted by its intricate surface — appearing something like the faceted profile of a cut gemstone.

I named it the golden diamond.

It was a profound feeling to consider I may have been the first to ever lay eyes on this marvel. I could sense its secrets were waiting to be mined, as if beckoning to me.

And the more I looked, the more I found.

Much of the information encoded in the golden diamond derives from its use of the golden ratio ϕ.

The ratio associated with the number ϕ, pronounced phi, is often cited as an especially beautiful proportion — a claim that is ultimately a matter of taste — but aesthetic considerations aside, ϕ does represent a particularly impressive mathematical relationship.

At its most humble beginnings, ϕ is simply the solution to the quadratic equation x²-x-1=0.

If you were to dust off your grade-school quadratic formula and apply it to solving for x, you would find that

This is of course a pair of solutions: The golden ratio ϕ is associated with the solution having the plus sign, while its conjugate -1/ϕ is the solution with the minus sign, such that

This leads to some curious properties unique to ϕ alone.

From the original equation, one can easily see that ϕ+1=ϕ² — which is pretty weird when you think about it. A related result is found by adding ϕ with its reciprocal 1/ϕ, such that ϕ+1/ϕ=1 — or after rearranging, ϕ -1=1/ϕ — which, again, is quite striking.

Therefore, at its heart, the golden ratio is a number whose square is obtained by adding 1 to it, while its reciprocal is obtained by subtracting 1 from it. It is an irrational number, i.e., it has an infinitely long, non-repeating decimal value, approximately equal to 1.618.

There are hundreds if not thousands of known facts and mathematical identities relating to ϕ — and quite impressively, many of them can be extracted from the golden diamond fractal simply by measuring its features.

The most apparent identity can be observed by labeling the lengths of the various triangles in the fractal’s rows.

This is essentially a proof without words of the well-known identity

Counting the number of triangles in each row, one finds the following sequence of numbers: 1, 2, 4, 7, 12, 20, …

Can you see the pattern?

As you may have guessed, this list of numbers is closely related to the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …

That is, the number of triangles in the nth row of the diamond is

where Fₙ is the nth Fibonacci number.

The Fibonacci numbers are like the integer buddies of the golden ratio and have a tendency to pop up when matters of discrete measurement are concerned. Recall that the Fibonacci sequence is defined as the recurrence relation

That is, the third Fibonacci number is generated by the sum of the first two; the fourth number, the sum of the second and third, and so on.

The defining relationship between the golden ratio and the Fibonacci sequence is that the ratio of any two consecutive Fibonacci numbers approximates the value of ϕ with increasing precision as the index n approaches infinity, that is

Having now identified the Fibonacci numbers in the golden diamond, a slew of new identities readily emerge, such as the following, which I leave to the reader to prove:

(for solutions, see the original arXiv paper)

Being able to recover identities of the golden ratio and Fibonacci numbers was a fun feature of the fractal, but I wanted to go much deeper…

I was aching to understand that mystifying sense of depth that appeared to usher each row of triangles off toward some abstract horizon.

By extending lines through the base vertices of each triangle in a given file, I produced a grid in the plane upon which the triangles stood. The result was mesmerizing.

It suggested all sorts of fascinating variations, such as this one where the triangles have been replaced by octahedra — three-dimensional shapes with eight triangular faces.

There are no observed gaps or overlaps between the octahedra. Whoever or whatever is observing this scene sees but a field of infinitely many octahedra — nothing more and nothing less!

I marveled at the thought while staring out my window at the mountain peaks beyond our valley.

Who or what was observing this scene?!

Of course, we as the readers see the projected image of this abstract arrangement of shapes, but it nevertheless implies an original observer, a sort of photographer, who witnessed this incomprehensible sight — the photo op to end all photo ops.

Naturally, it was absurd to ask who or what the observer was, but I could at least answer the question of where it was.

I thought back to the old question I had posed myself on perspective projection — picturing the painter on the imaginary railroad tracks. It helped me make some progress but soon I hit a problem.

In the fractal, the triangles appeared to be standing in a plane, and the observer was positioned somewhere near the origin — at the corner of an infinitely large square.

How could the projection of a square plane produce an angle of 60° as it did in the golden diamond’s bottom vertex?

I went to the bathroom and stood above a square tile, standing at one of its corners. Protractor in hand, I extended my arm toward the tile to measure the corner’s angle as seen from my vantage point. As I had suspected, no matter my distance or height from the tile, its angle would always appear to be at least 90° — easily more but never less.

I looked again at the fractal.

It was clearly a projected plane — I knew it had to be. It was all so perfect… how could it fail me now?

For a week, I could think of nothing else. I could hardly eat. At work, I spent every minute between my lectures working on a way to solve the problem. My folder of notes and calculations grew thicker and thicker. I toyed with convoluted ideas, such as a two-stage process involving a sphere inscribed within an octahedron, positioned at the origin, whereby an initial, gnomonic projection was performed onto a sphere, followed by a second outward projection onto the triangular face of the octahedron — resulting in the final equilateral image. But it, too, had its problems and seemed, well, inelegant.

I was becoming dejected.

That confounded equilateral shape with its 60°! If only it had been less perfect, it would have been easier to solve.

Then it occurred to me. It was I who had assumed that the triangles were necessarily equilateral, having arrived at the fractal through Odon’s golden ratio theorem of the inscribed equilateral triangle. But the final form of the golden diamond seemed to demand a different interpretation.

In a renewed sense of exhilaration, I opened up some software to stretch the original image of the fractal, until 90° formed at its base.

Naturally, all the fractal’s most defining features — its inner proportions and relationships — remained intact; it was simply a horizontally-scaled version of the original shape.

Immediately after this small change, the solution to the problem unfolded with ease. All this time, I had struggled only due to the blinding insistence of what I had believed to be more beautiful. It was a valuable lesson that has guided me ever since. My new credo became “the truth is always more interesting than what I think!”

Before long, using an important property called the cross ratioa measurable quantity that remains invariant through the process of perspective projection — I cracked the problem. I worked out precisely the height and distance the observer’s eye had to be from the imaginary canvas to project the scene upon it.

I was completely floored by the beauty of it all. Such elegance!

I developed a model in Autodesk Maya, a 3D graphics program, to confirm the results and generate the figure shown above.

Everything seemed to be falling into place and I was eager to start writing and publish my paper. Being the numberphile I am, I searched the calendar to plan for an auspicious publishing date.

My eyes fell on the most perfect date imaginable, and my heart skipped a beat. Unbelievable. In just a few month’s time was the date 1 June 2018… 1.6.18, Phi Day! It was a once in a lifetime coincidence.

Now, the pressure was on! I had to finish by that date.

But a big riddle still remained unsolved. In fact, up until that point I had been working with a less detailed version of the golden diamond than shown above as it was too unwieldy to draw by hand. I had been working on a computer program to automatically generate the design, but it was a struggle to pin down a succinct rule to recreate the pattern.

The real problem, however, was one less to do with convenience and more to do with attaining a complete theoretical understanding of the geometry.

The confusion is most easily observed within the grid lines of the plane; they alternate in a wacky arrangement of two different interval lengths — unlike the uniform integer spacings of the familiar Cartesian plane.

From above, the grid would appear like this:

Here, the values indicate the distance from the origin along each axis, just as one would conventionally list the integers along each x and y axis. One can work out that each large interval spacing has a length of 1, while each small spacing has a length of 1/ϕ.

The problem was that no discernible pattern to the intervals seemed to exist.

Look closely — despite the 1s never occurring more than twice in succession, the pattern appears altogether random.

These standing triangles seemed to live in a bizarre world — a land with a different language where integers had been traded for these two numbers, 1 and 1/ϕ, existing together in a chaotic, back-and-forth dance that continued off to infinity.

I tried my luck with the Online Encyclopedia of Integer Sequences (OEIS), inputting the occurrence of each of the two values: 1,1,2,1,1,1,2,1,2,…, but there was no match. I tried a dozen other ideas, but still, nothing.

I was stumped… again.

I sat staring at the numbers. Outside, the random jangling of cow bells seemed to echo my thoughts… ding, dong, ding, ding, dong, ding, dong, ding, …

Days passed.

Surely there had to be some underlying structure to the pattern. After all, it was an essential feature of the fractal — and although complex — it appeared so orderly. I couldn’t leave this problem unsolved. The diamond had come to me and with such a perfect publishing date, I felt a duty to get it right!

I worked feverishly. Piles of notes were strewn about my desk. Like a ghost, I would suddenly appear by my wife Ava while she lay in bed. “I found something!” I’d say, startling her, and then proceed to rattle off my theories. During these months, she had become a veritable expert in all things about the golden diamond. Although not coming from a technical background, she is a keen listener with a great intuition. She could sense my growing frustration and that the logical threads I had been pursuing were thinning out.

When deep in my thoughts, there is a sensation of being a miner or spelunker, navigating the dark tunnels of the mind. Sometimes when hitting hard stone, a detour is required, and other times, there is little choice but to proceed with brute force as far as one’s will allows. If a tunnel leads to a discovery or breaks through to a new cavern, the sensation is truly transcendental — but when the branching network continues to deepen, further and further into darkness, without reward… the experience can be maddening. One begins to consider abandoning hope and returning to the surface — should one remember how to return at all!

Several weeks had passed. My attempts at decoding the pattern had resulted in some interesting discoveries about logarithms of Fibonacci base values, but I still lacked an elegant explanation for the sequence I was studying.

It occurred to me that the pattern was repeated in each row of the golden diamond as the arrangement of triangles and gaps alternated between each other. As each successive row grew in length, more of the sequence was revealed. Perhaps there was something of value to each of these smaller sequences.

Then, finally, I stumbled upon the solution: Each row was related to a Fibonacci word!

Fibonacci words are sequences composed of two characters, typically 0 and 1 or A and B, but conventionally the choice is of no significance. One starts with two words w₁=A and w₂=AB. Then, similar to determining each Fibonacci number, successive Fibonacci words are formed by concatenating any two, previous words, such that w=ww₁=ABA, and so on. Another way to determine a given Fibonacci word is by taking the previous word and applying to it the substitution rule: A→AB and B→A. The first few words are as follows:

The infinite Fibonacci word with A=1 and B=1/ϕ was clearly the pattern I was seeking!

I was blown away.

Never before had the Fibonacci words been known to emerge naturally in a fractal. Even more profound, the golden diamond demanded a preferred alphabet of Fibonacci words — 0,1 or A,B would no longer suffice — it was required that A/B=ϕ.

It was an incredible discovery. But how was I going to prove it?

Even though the pattern was clear to the eye, I needed a way to prove that the pattern would continue forever. This is mathematics after all!

Then it occurred to me to use the substitution rule, illustrating neatly all of the Fibonacci words within a subset of the golden diamond:

Fibonacci word substitution rule: A→AB and B→A

Applying this rule to the entire golden diamond, one finds rows of palindromic Fibonacci words:

Incredibly, the wonders of the golden diamond didn’t end there. As if I had struck the mother lode, new features of the fractal were uncovered daily.

Remember the interval markings along each of the axes in the plane? Each was associated with a multiple of ϕ plus an integer. Here are the first few:

These values turned out to be something truly fascinating: the integers of the phinary number system.

Let me explain.

Phinary, or base-ϕ, is a positional number system originally discovered by George Bergman in 1957 while still in grade school. Unlike the decimal system with its base of 10, or binary with its base of 2, phinary uses the golden ratio as its base. It is traditionally written in a form that looks very much like binary with 0s and 1s, where each 1 indicates a particular power of ϕ. For example, consider the following:

Unlike binary, however, there are several ways to represent a number in phinary via this method (a special property to which we will return). Therefore, the “standard form” convention is to disallow any representation consisting of consecutive 1s — a rule that happens to ensure a unique form for each phinary value.

Perhaps surprisingly, one can write any real number using phinary, such as the following examples:

If one were to consider all the phinary numbers generated to the left of the radix point (called a “decimal point” in base-10), one would recover all the values along the axes in the projected plane of the golden diamond! (See if you can prove this. Hint: how can you write any sum of golden ratio powers as just a multiple of ϕ plus a constant?)

This means that the grid lines in the plane that looked like wacky integer lines actually are integer lines — but in phinary!

When I realized this, I was speechless… The golden diamond truly was a window into another world.

To my surprise, not much had been written about the phinary integers in the past, but having now seen them on full display in the land of the golden diamond, I was compelled to investigate them further.

I’m glad I did, because what I found was the most incredible discovery thus far.

During this time, my family and I decided we would move back to Austria to be closer to relatives in Vienna, as I had been offered a teaching position at a prestigious private school in the area. It was a bittersweet change. I had submitted my letter of resignation to my English-speaking French-Swiss-Japanese school, but having grown so close to the community there, I was in need of a bit of closure. Having never been to Japan before, I felt that a trip to the far east during spring break would be a good way to close that chapter of my life — but with only a couple months remaining before Phi Day, I was sure to continue my research while away.

So many new leads had come to light, and just like Alice down the rabbit hole, I was doing my best to make sense of the wonderland I was in.

Throughout my studies, I had been reading countless papers on topics related to my findings and had recently come across a fascinating — and in my opinion, totally underrated — subject in number theory called the hyperbinary sequence, a list of numbers that generate a special pair of branching structures known as the Stern-Brocot and Calkin-Wilf trees. Elsewhere, I had stumbled upon another, somewhat obscure, yet related sequence of numbers called the Fibonacci diatomic sequence, which seemed to imply the existence of number trees as in the former case, but no such trees had been known. I was certain that the phinary numbers together with the tree-like structure of the golden diamond were the key to unlocking this secret. My intuition was about to be proved correct, but I was not prepared at all for what I would find.

The Stern-Brocot and Calkin-Wilf trees are astonishing mathematical objects. Each tree systematically lists every positive rational number in reduced form, with the Calkin-Wilf tree doing so in order of magnitude, as read from left to right!

The Stern-Brocot tree
The Calkin-Wilf tree

One can easily generate each tree by following its branching rule as indicated in the upper-left of each figure (Notice, however, that the Stern-Brocot tree requires the additional fraction 0/1 and pseudo-fraction 1/0 to achieve this branching rule, as indicated in gray above the tree).

Alternatively, one can utilize the aforementioned hyperbinary sequence to generate the numerators and denominators of each fraction in each tree.

Do you see the pattern?

The Stern-Brocot tree uses the even-indexed entries of the hyperbinary sequence — with the first entry having index 0 — to generate the numerators along each row of the tree as produced from left to right; the denominators are generated symmetrically in the same manner from right to left. The Calkin-Wilf tree, on the other hand, iterates through the entries of the hyperbinary sequence, using as many values as possible for a given row to generate its numerators and denominators — again, in a symmetrical fashion — before moving on to the next row. The set of values used for each row are indicated by the alternating gray/bold typeface in the displayed sequence.

So, what is the hyperbinary sequence anyway, and what does this all have to do with phinary and the golden diamond?

The algorithm that generates the nth entry of the hyperbinary sequence H(n) asks the following question: How many ways can n be written as a sum of powers of two, with each power occurring at most twice?

The first few entries result as follows:

If we had instead permitted the powers of two to only occur at most once, then we would have simply recovered the binary representation of n — for which there is only one. This should clarify the meaning of the word “hyperbinary.”

As you will recall, the phinary integers are defined by powers of ϕ, and a similar question posed by the hyperbinary algorithm can be applied to them.

Let p be a phinary integer. My question was this: How many ways can p be written as a sum of powers of ϕ, with each power occurring at most once?

Recall that there are potentially several representations of a number in base-ϕ notation if one disregards the standard form convention for phinary. Therefore, the answer to the above question results in enumerating what I dubbed the hyperphinary representations of p.

I called the resulting list of numbers the hyperphinary sequence, as shown in the second column of the above figure.

As it turned out, the hyperphinary sequence is precisely the same list of numbers as something called the Fibonacci diatomic sequence, which asks the question: How many ways can the integer n be represented as a sum of Fibonacci numbers, with each Fibonacci number occurring at most once?

The equivalence of the hyperphinary sequence to the Fibonacci diatomic sequence is worthy of some pause. It exposes the relationship of the Fibonacci numbers to the integers in a particularly revealing light. Moreover, it demonstrates that the relationship of the powers of ϕ to the phinary integers is directly analogous to that of the Fibonacci numbers to the normal integers — generating an exact, one-to-one correspondence between the two number systems.

The fact that the phinary integers had a hyperphinary sequence wasn’t quite enough to convince me that they ought to have their own number trees, as well. The big clue came from noticing that the binary trees, which make up the Stern-Brocot and Calkin-Wilf branches, could also be used to generate a “diamond” fractal — and it, too, was the result of a perspective projection.

I called it the natural diamond.

Just like the golden diamond, the natural diamond represents the perspective projection of triangles standing in the plane; however, this time, the grid lines are evenly spaced apart — they indicate the natural numbers!

For another comparison of the golden and natural diamonds, consider the “shadows” of each as viewed from above. That is, imagine that the photographer who captured each image had done so with their flash ON, sending long shadows behind each of the triangles. If you were a bird flying above the scene, this is what you’d see (Each dot indicates one of the base vertices of a standing triangle).

Shadows of the golden diamond
Shadows of the natural diamond

Notice the striking similarities, and how each row of triangles is spaced by ever-increasing powers of ϕ or 2 as they intersect their respective axes.

Having noticed all of these analogous features between the golden and natural diamonds, I had a strong hunch that a “phinary tree” was waiting to be discovered, but try as I may, I couldn’t fit the numbers of the hyperphinary sequence into the nodes of the golden diamond.

It drove me nuts.

I can’t remember how many variations I tried. The poor trees I sacrificed to my endless scribbles!

At this stage in my research, I came across a recent paper by Sam Northshield who had discovered a branching structure related to the golden diamond. He, too, was considering number trees related to the hyperbinary sequence but did not provide a solution to the problem. My curiosity was now thoroughly piqued.

For me, part of the issue was understanding how to generate an analogue to the Stern-Brocot tree, which relied on the even elements of the hyperbinary sequence. Somehow, the notion of “even” and “odd” felt terribly out of place in the world of the phinary integers. The irregularity of the embedded Fibonacci word pattern in the distribution of phinary integers seemed to demand something new, but what exactly?

I was stuck on a layover at the Beijing Daxing International Airport on my red-eye flight to Japan. Unable to sleep, I spent the night hunched over my notes beneath the vaulted glass ceilings of the futuristic building.

I had been struggling to foolishly guess the solutions to the phinary trees, so I resolved to find a new approach. I looked to another known property of the hyperbinary sequence — its recurrence relation.

Every odd-indexed element of the hyperbinary sequence is in fact a reoccurrence of an earlier value in the sequence. And every even-indexed element is generated from the sum of two consecutive elements occurring earlier in the sequence. The exact recurrence relation of the hyperbinary sequence H(n) can be succinctly formulated as follows:

An illustration of this recurrence relation can be depicted as follows:

The rounded arrows indicate the recurrence of the odd-indexed values, while the angled arrows indicate the pairs of values that sum to later even-indexed elements.

Unfortunately, there was no succinct recurrence relation known for the Fibonacci diatomic sequence that I could adopt for the hyperphinary sequence, and if I were going to find one, I would need some new mathematical tools — new operations that could handle the aperiodicity of the phinary integers.

The problem with the phinary integers is that they are not closed under the conventional operations of addition, subtraction, multiplication, etc. — for example, adding 1 to an arbitrary phinary integer will not necessarily yield another phinary integer.

I needed a way to stay inside the phinary world while doing calculations.

I devised some new operations ⇁ (hook) and † (dagger), among others, in a rigorous formalism that could handle the task. The hook and dagger operations were the phinary analogues of subtraction and addition, respectively. For example, the successor of any phinary integer p was defined as p † 1 — and the value after that was p † ϕ.

With the new operations, I was well-equipped to tackle the hyperphinary recurrence relation.

It was around 5 A.M., Beijing time, when a sudden screeching echoed throughout the terminal. A dozen window-washers repelled down the massive windows from hidden rafters above the wall. The eastern world was waking up, and after a night of tireless calculations, I had arrived at the solution to the hyperphinary recurrence relation.

Surprisingly, I managed to find a solution comprised almost entirely of ordinary operations, and needed only the extra, phinary successor operation († 1) to finish the job.

I was struck by its similarity to the hyperbinary recurrence relation but was fascinated by how it differed…

Recall that the hyperbinary recurrence relation was defined by its initial value H(0)=1 and two rules: one for its odd-indexed elements and one for its even-indexed elements.

But the hyperphinary relation had three rules…

The last of the hyperphinary rules was directly analogous to the even-indexed rule for the hyperbinary sequence; not only was it the sum of two, consecutive elements in the sequence but it was associated with — in addition to other values — all p values that were powers of ϕ — similar to the association of the even-indexed rule of the hyperbinary recurrence relation with all n values that were powers of 2.

Likewise, the first of the hyperphinary rules resembled the odd-indexed rule of the hyperbinary sequence in that it was a direct “copy” of earlier elements in the sequence and that it was always associated with a p value that followed what justifiably appeared to be an “even” element of the hyperphinary sequence. Consequently, the number of “even” and “odd” elements of the hyperphinary sequence were equal in quantity.

So far, so good — but what in the world was the second of the three rules defining? In many ways, it was similar to the “odd” rule as it represented direct copies of previous elements in the sequence, but it differed in that its occurrence was seldom — appearing, on average, only 1/ϕ as often as either the even or odd elements in the sequence. It was like some strange “third-wheel” of the party and implied that the phinary integers had a third type of parity in addition to the familiar even and odd variety!

It was odder than “odd,” so I dubbed it curious.

An illustration of the hyperphinary recurrence relation can be depicted as follows:

Again, the rounded arrows indicate the direct copies of elements in the sequence, resulting in the odd and curious indexed elements in an alternating fashion — and the angled arrows depict the sum of consecutive elements that generate the even-indexed elements. Notice how the spacing between each type of element follows the Fibonacci word pattern and how the curious-indexed elements occur more infrequently.

If we let p=0 be even, then the even, odd, and curious phinary integers are defined as the following sets of numbers:

Quite strange! Even if one were to clump the odd and curious numbers under one name, they would still outnumber the quantity of even values by a factor of ϕ:1, which is of course a departure from the symmetrical nature of parity under the integers. It is my hope that one day this property may benefit further research in number theory and perhaps even offer clues to understanding the distribution of prime numbers.

Finally, with the recurrence relation in hand, I was able to identify the even-indexed elements and pin down the phinary analogues to the Stern-Brocot and Calkin-Wilf trees:

The phinary analogue of the Stern-Brocot tree
The phinary analogue of the Calkin-Wilf tree

Compare these trees with the ones shown earlier and note the similarities and peculiar differences.

To my surprise, the golden diamond appeared as just a subtree within these structures, indicated by the gray lines. Another surprise was to find that each tree contained infinitely-many copies of its Stern-Brocot/Calkin-Wilf counterpart — with each copy featuring fractions in a particular unreduced form — thereby accounting for all the possible, unreduced versions of these trees.

By the time I had solved these problems, I was back from my vacation abroad — both in body and in mind.

Altogether, from the first discovery of the golden diamond to the final word of my written article, six months had passed.

It was a journey I will never forget.

To top it all off, I managed to publish the 61-page article exactly on Phi Day, 1.6.18.

What a gem!

To read the original article, visit

American expat artist, musician, and mathematician, based out of Vienna.

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