Golden Ratio – Why Ken’s Wrong

The Golden Ratio is a quirky number about which mathematicians have noticed all kinds of fun facts. Find out why Ken Wheeler takes these random anomalies to extremes to defend his unfounded, bizarre convictions about metaphysics.

Ever since the days of Pythagorus, philosophers and mathematicians have been fascinated by a number called the Golden Ratio, or phi (φ). It plays a role in the dimensions of the regular pentagon, the so-called Golden Rectangle, and some patterns we see in nature like leaf spirals.

Phi (φ) is a ratio with an irrational number and a Greek letter, somewhat like pi (π), and its value is 1.61803…. Whereas π is the ratio between a circle’s circumference and its diameter, φ is a bit trickier to explain. Imagine a line divided into two unequal segments.

The Golden Ratio, φ, is the number where the relationship between the the longer segment and the shorter segment is the same as the relationship between the overall length of the line and the longer segment. Here’s a diagram to clarify.

Only Ratio that Meets These Conditions is Phi (φ)

Or, in terms of algebra, a+b/a = a/b = φ by definition. The ratio that meets these conditions is φ, or 1.618033…

Ken Wheeler has stumbled across φ’s properties in a 4th century book attributed to the discredited neoplatonist Iamblicus and translated by Robin Waterfield entitled The Theology of Arithmetic. Without crediting Waterfield, Kentucky Ken decrees that “Phi is the ratio and relationship of the Monad to its increasingly phenomenal self-image in emanation.”

Translating that jargon into English, the Angry Photographer claims that φ is the “divine proportionality” connecting his sacred One with our everyday lives. Waterfield translated this notion from ancient followers of Pythagorus, although today we view these beliefs as naive superstitions.

The Ratio φ Does Have Some Cool Properties

Admittedly, φ does have some cool properties. For example, Φ2 = 2.618…, which is exactly φ+1. Another fun fact about φ is that 1/ φ = 0.618…, or exactly φ-1.

The superstitious fascination with φ is easier to grasp if we use exponents. Looked at in that way,

Φ-1 = 1/Φ = φ-1
Φ= 1
Φ= Φ
Φ= Φ+1

Relationship between φ and 1 Seems Meaningful

This anomalous relationship between φ and 1 seems meaningful and even mystical to many people, especially a range of fake experts who are into concepts like monism, the worship of the number one. Pythagorus, Euclid, and presocratic Ancient Greek mathematicians were aware of the Golden Ratio, but their conclusions about it vary widely.

That doesn’t prevent Ken Wheeler from jumping to his own conclusions. He’s fond of saying that “1 is to φ as φ is to 1.” He then asserts without evidence that there’s a kind of φ hierarchy that goes:
Φ-3 Primordial Agnosis
Φ-2 Psyche/Tou/Pantos
Φ-1 Eidos/Matter/Mimesis
Φ Being
Φ3. Totality/Excess/Pan/Pentagram

Although the Angry Photographer claims to have derived this sequence from “the Pythagoreans”, we’ll see below that the historical Pythagorus’ system is quite different and much simpler.

Arithmetic Anomaly Called the Fibonacci Sequence

This fascination with Φ isn’t the Theoria Apophasis producer’s only foray into numerological sophistry. For example, he often expounds on another interesting arithmetic anomaly called the Fibonacci sequence.

The mathematician Fibonacci came up with a numerical sequence where we add two whole numbers together and then the next two and so on. So, for example, starting at the beginning: 0+1=1, 1+1=2, 1+2=3 and 2+3=5…. Deliberately ignoring zero, Ken Wheeler claims the first five numbers of the Fibonacci Sequence are 1,1,2,3,5… and portrays the pair of ones as both ancient and mystical.

According to the Angry Photographer, the first two ones correspond to classical Pythagorean notions of principle and attribute. The 2 and the 3 represent matter and magnitude, while 5 represents ontos or “being.”

No Evidence Pythagorus or Euclid Knew Fibonacci Sequence

This is a logical fallacy since Fibonacci first published his sequence in 1202 CE. Pythagorus, whom Ken Wheeler claims revered the Fibonacci Sequence, died in 495 BC, and Euclid died in 270 BCE.

There’s no evidence that Pythagorus or Euclid had ever heard of the Fibonacci Sequence. There’s certainly no proof that the founders of western geometry thought it was important metaphysically.

Even so, there’s another spooky bit of woo we can discuss. If you take each number in the Fibonacci sequence, multiply it by φ and round it off, you get the next number in the sequence.

Pattern Not Due to Any Metaphysical Properties of φ

For example, 3φ ≈ 5, 5φ ≈ 8, etc… Spookier still, the further along we go in the Fibonacci sequence, the closer the ratios get to φ. For example, when we get up to 610φ ≈ 987, the ratio closes to 1.6180327868852, or 99.9% of φ.

Spooky or not, this pattern isn’t due to any supernatural properties of φ. If we call the shorter line segment 1 unit and the length of the longer segment x, we get the equation (x + 1)/x = x/1

We can simplify that as the quadratic equation x2x – 1 = 0. Solving for x, we get x = (1 + √5)/2, which works out to be the golden ratio. Any quadratic equation with similar variables and coefficients follows a comparable pattern called a limit, with or without the ratio φ.

Later scholars have disproven the numerology about the Golden Ratio as superstition. The Fibonacci sequence and φ are intriguing anomalies, but they don’t represent anything important, and they’re unrelated.

Claims to Have Deduced a Mystical Formula of His Own

Despite all this, Ken Wheeler has mysteriously deduced a mathematical expression of his own that involves the Golden Ratio. His mystical formula is 1/Φ-3

Readers may be forgiven if they haven’t worked with negative exponents since high school and find this expression confusing. That’s what the Angry Photographer is counting on.

The more mathematically inclined will realize that the Angry Photographer’s formula is convoluted and needs to be simplified. 1/Φ-3 is simply Φ3, or 1.618033. x 1.618033 x 1.618033 = 4.23606…another insignificant irrational number.

Wrongly Claims Formula Comes from Plato’s Republic

Ken Wheeler claims to have derived his formula by cherry-picking a passage in Plato’s Republic. In the dialog, Socrates and his friend Glaucon are discussing a line divided into segments.

However, it’s one of Socrates’ obscure analogies, and they’re talking about the difference between opinion and knowledge, not geometry or numerology. The four line segments represent four levels of knowledge from lowest to highest – speculation, belief, thought and understanding.

The Angry Photographer somehow speculates, so to speak, that Socrates is asking him to divide the sections using the Golden Ratio, which Socrates never mentions anywhere. From these segments, he arbitrarily creates the sequence Φ, 1, 1, 1/Φ so that if we add these four numbers together, we get 1.618…+ 1 + 1 + 0.618 = 4.236…, his supposedly mystical Φ3, which he again disguises as 1/Φ-3, apparently to work his magical 1 into the expression somewhere.

Falsely Claims Formula 1/Φ-3 “Not a Mere Number”

The host of Theoria Apophasis goes on to explain “But 1/Φ-3 is not a mere number, rather the expression of the One against itself and manifestation in the most perfect and divine Logos; the proportions of perfection itself as recognized by the immortals. This secret of incommensurability is the deepest arcana of the ancients! Worldly minds cannot penetrate this importance, but wise minds can.”

Don’t try to make sense of the previous paragraph. It’s another example of Ken Wheeler’s infamous word salad.

As a matter of fact, 1/Φ-3 is “a mere number.” When Kentucky Ken says that “worldly minds” can’t grasp the significance of his point, he’s merely trying to set impossible expectations for his critics to disprove his claim.

Then Applies His Bizarre Notions to Geometry

However, that isn’t the end of the Angry Photographer’s bizarre approach to arithmetic. He goes on to apply his notions to geometry as well.

In his Pythagorus, Plato and the Golden Ratio, the Angry Photographer puts his fascination with φ to work in a kind of autodidact trigonometry. Genuine historians think of right-angled triangles as Pythagorean.

Despite this, the Theoria Apophasis creator mistakenly refers to an isosceles triangle with angles of 108˚, 36˚, and 36˚ as a Pythagorean triangle. He claims that these angles have special numerological significance because 3 + 6 = 9 and 1 + 0 + 8 = 9.

Numerology Associates Nine with Completion

The occult notions of numerology associate the number nine with completion because we make ten going back to 1 and 0. Of course, this symbolism has no literal or scientific meaning or relevance.

Kentucky Ken elaborates, “There is only one coherent geometric form which encompasses the four sectors of the Divided line analogy of φ, 1, 1, 1/φ, and that is the Pythagorean triangle below. This is the very same proportional representation for Plato’s cave where the φ Beings below are proportional (logos) to the Nous above and the Monad on high. As seen in the figure, the vertical encompasses the visible realm, and the periphery the noetic.”

If the verbal explanation above seems impenetrable, no doubt the diagram he mentions will clarify everything.

Diagram Depicts Triangle With Sides 1, 1 and φ

Then again, maybe not. What we have here is a triangle with proportions 1, 1 and φ. Ken Wheeler has marked the height as 1/φ or 0.618033… which would be freaky if true because his φ, 1, 1, 1/φ sequence from the divided line repeats itself out of nowhere.

However, applying the genuine Pythagorean Theorem of h2 = a2 + b2 or a ruler, we find the height of a triangle with these dimensions equals 0.588, and not the 0.618 (1/φ) the Angry Photographer needs to complete his mystical progression. Once again, he’s fudging the figures to fit his pet theories.

Also, the Theoria Apophasis creator is wrong when he claims “There is only one coherent geometric form which encompasses the four sectors of the divided line analogy of φ, 1, 1, 1/φ,” Any isosceles triangle has the same characteristics. Here’s one example.

Wrongly Says “Only One Geometric Form” with φ, 1, 1, 1/φ

It’s a different isosceles triangle ABC with two sides of length 1 and a base of length X. The angles are 72˚, 72˚ and 36˚.

If we bisect angle A, we get another line segment AD of length X. Eerily, the isosceles triangle ACD has exactly the same proportions as triangle ABC. The magic of φ strikes again!

Now, we can get even spookier. The ratio of AB to BD is φ, and so is the ratio of BD to CD. Now, AB = 1, BC = 1, and X = 1/φ, = φ -1. So, this triangle has an equal claim to follow Ken Wheeler’s “mystical sequence” φ, 1, 1, 1/φ.

We Can Derive Angry Photographer’s Triangle from This One

We can go from spookier to spookiest by pointing out that AD creates another triangle ABD, which turns out to be the Angry Photographer’s earlier, so-called “Pythagorean Triangle” with angles 36˚, 36˚ and 108˚ turned on its side. Not only is there more than “only one coherent geometric form” with these properties, but we can derive the Theoria Apophasis host’s triangle from the others. Mind you, none of this has any practical or symbolic significance whatsoever.

Ken Wheeler goes on to claim that his obscure triangle is the basis for something called the Pythagorean Tetractys. It’s not. Although the Pythagorean Tetractys is often depicted as a triangle, its proportions don’t involve the number φ.

The Pythagorean Tetractys is simply the sequence 1+2+3+4 = 10. Scholars sometimes drew it like this.

Pythagorean Tectractys is the Sequence 1+2+3+4=10

Each row of dots represents one number in the sequence (1,2,3, 4) and if you count all the dots, there are ten. This, and not Ken Wheeler’s diagram, is a Pythagorean Tetractys.

Since we’re on the topic, Pythagorus used each level in the Tectractys to represent a realm of being, correponding to Socrates’ four levels of knowledge. One, the monad, represents the unity or the good.

Two, the dyad, is the realm of the gods, while three, the triad, is the level of the eternal ideas, like Plato’s ideals. Four, the tetrad, is our everyday world. None one of this relates to the Angry Photographer’s interpretation of the Tetractys or his hierarchy of the exponential values of φ shown above.

Wrongly Claims Pentagram is “Triangle in Triplicate”

Ken Wheeler’s bizarre notions then leap to yet another level. He claims, wrongly, that a Pythagorean pentagram consists of his “primary triangle composed in triplicate.” It doesn’t.

To draw a pentagram, we begin with an upside down, regular pentagon (one that has five equal sides and five equal angles).

Then we attach five identical triangles, one to each side of the pentagon.

As readers can see, the five triangles are not the same obtuse isoceles shape as the Angry Photographer’s “coherent geometric form.” They’re acute isoceles triangles, like the second triangle (ABC) above.

Five Triangles and They’re Not the Same Shape

If you have the patience to look at a pentagram long enough, you’ll begin to notice that the original forms also combine into larger, obtuse triangles. However, after some study, we can find five obtuse triangles, not the “triplicate” claimed by the Theoria Apophasis host, which is simply because the pentagon has five points.

It’s true that artists incorporate the Golden Ratio into their drawings of the pentagram. Again, it’s hard to describe how they do that with words, so here’s another colour-coded diagram.

The coloured line segments incorporate the Golden Ratio like this:

Red/Green = Green/Blue = Blue/Magenta = φ

Nothing to Do with So-Called “Pythagorean Tetractys”

This is how artists throughout history have applied the Golden Ratio to the pentagram. It has nothing to do with Kentucky Ken’s so-called “Pythagorean Tetractys.”

How do any of the Theoria Apophasis creator’s notions unlock the mysteries of the Universe? Your guess is as good as anyone’s. All of this kind of “sacred geometry” usually falls within the “occult nonsense” Ken Wheeler condemns at his corner book shop.

Although the Angry Photographer is a glaring example, we’re all subject to the same kind of selective perception. If we could objectively examine every one of our beliefs, we’d all find some inconsistencies.

Rejects “Occult Nonsense” – Accepts Numerology

While we denounce some beliefs as foolish superstitions, we may still cling to other, equally naive notions. In the case of the Theoria Apophasis producer, he dismisses such pseudoscience as crystal healing, astrology and the flat earth movement.

Yet, in the same breath, the Angry Photographer will defend his beliefs in such conspiracy theories and “occult nonsense” as ghosts, demons, UFOs, and, in this case, numerology. The double standard by which he reaches his verdicts on supernatural dogma stems from his fanatical devotion to the discredited notion of theurgy, which we cover in more detail under Retroduction – Why Ken’s Wrong.

In a sense, the Angry Photographer is a victim of an all too common human foible. Even so, he’s so arrogant and fanatical that’s it’s all but impossible to care.