The Kepler Triangle and Root Phi

The discovery of the “Kepler triangle” (2024) was made by the German mathematician and astronomer Johannes Kepler (1571–1630), who wrote about this shape in a 1597 letter. Two concepts that can be used to analyze this triangle are the Pythagorean theorem and the golden ratio. These were both of interest to Kepler, as he wrote elsewhere:

“Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.”

We think this is one of the more fascinating shapes we’ve ever encountered. It has many interesting properties and there are plenty of great articles about it online. But we wanted to at least cover the highlights and document some of these features for future reference.

1

Construction of The Kepler Triangle

The Kepler Triangle and its relationship to Root Phi is perhaps easiest to understand when you view it in the enlarged model shown in Figure 2. A blue radius line represents the standard procedure for constructing a golden rectangle.

Our center point is established by bisecting the base of the unit square and then drawing an arc, starting from the upper right corner of the square. But the length of the resulting golden rectangle will then be $\varphi$. So we can start from the lower left corner and form another arc of length $\varphi$. This arc is shown in green, and where it meets the top of the golden rectangle will be of length $\sqrt{\varphi }$ from the left edge. But how do we get $\sqrt{\varphi} \; $?

If we let that length be an unknown value and call it $x$ for the moment, we can see from the Pythagorean Theorem that,

\begin{align}
\varphi^2 = x^2 + 1^2 \\\\
x^2 = \varphi^2 \; – \; 1 \\\\
= \left(\varphi + 1\right) \; – \; 1 \\\\
x = \sqrt{\varphi }
\end{align}

You’ll notice above that we substituted $\varphi + 1$ for $\varphi^2$, and we should look at why that’s possible.

\begin{align}
\varphi^2 = \left(\frac{1 + \sqrt{5}}{2}\right)^2 \\\\
= \frac{3 + \sqrt{5}}{2} = \frac{1 + \sqrt{5}}{2} + \frac{2}{2} \\\\
\varphi^2 = \varphi + 1 \\\\
\end{align}

So the Kepler Triangle has side lengths $1$ and $\sqrt{\varphi }$, with a hypotenuse of $\varphi$. These lengths are in geometric progression, and this can be expressed as follows,

\begin{align}
1 : \sqrt{\varphi } \; : \varphi
\end{align}

2

Kepler Triangle Formed By Three Squares

Another nice way to see the Kepler Triangle is by forming a right triangle from three squares with areas in another geometric progression, as shown in Figure 3. The orange square has area $1^2$, the green square has area $\left(\sqrt{\varphi }\right)^2$, and the blue square has area $\varphi^2$.

That new geometric progression can be expressed this way,

\begin{align}
1 : \varphi : \varphi^2 \\
\end{align}

We have three edge lengths for these squares, $1$, $\sqrt {\varphi}$ and $\varphi$ which are the harmonic mean, geometric mean, and arithmetic mean, respectively, of the two numbers $\varphi ± 1$. These three ways of combining two numbers were all studied in ancient Greek mathematics, and are called the Pythagorean means.

Conversely, this can be taken as an alternative definition of the Kepler triangle: it is a right triangle whose edge lengths are the three Pythagorean means of some two numbers. The only triangles for which this is true are the Kepler triangles.

3

Golden Pyramids And The Kepler Triangle

We think a very strong case can be made for a golden pyramid being the foundational math used for the design of Egyptian pyramids, such as the Great Pyramid of Giza. This would be simply a doubled Kepler Triangle used as the cross sections, as shown in Figure 4.

According to the article we have cited below, theories based on physical measurements are difficult to resolve.

“Many other theories of proportion have been proposed for the same pyramid, unrelated to the Kepler triangle. Such theories are difficult to resolve based purely on physical evidence. These different theories are very similar in the numeric values they obtain. This is because of inaccuracies in measurement, in part caused by the destruction of the outer surface of the pyramid. The match in proportions to the Kepler triangle may well be a numerical coincidence. According to scholars who have investigated this relationship, the ancient Egyptians most likely did not know about or use the golden ratio in their mathematics or architecture. Instead, the proportions of the pyramid can be adequately explained using integer ratios, based on a right triangle with sides 11 and 14.”

The Ambiguous History of the Pyramid Builders

If we claim that Egyptians likely didn’t know anything about the Golden Ratio, that’s assuming we know for sure that Egyptians built the pyramids in the first place. And if they did, we have to make even more far reaching assumptions about what they did or did not know in very ancient times. We can probably all agree that it’s a highly mysterious and incredibly impressive structure that we still don’t really know much about to this day.

What we can say from our own research is, if the builders of the pyramids didn’t know about the golden ratio, they probably stumbled into it anyway (or came very close). There seems to be more of a wide agreement about the numbers 11 and 14 being involved as the article states. But the Kepler triangle matches up so closely to these numbers that the difference is almost indistinguishable. And if we can’t get accurate measurements because of structural degradation, then how can we make definitive conclusions about 11 and 14?

Calculating Side Lengths of Isosceles Triangles in the Golden Pyramid

Using a little more Pythagorean math, it’s easy to see the side lengths of the golden pyramid in Figure 4. Let $s$ be the side length of any of these isosceles triangles so that,

\begin{align}
s^2 = \varphi^2 \; + 1^2 \\\\
= \left(\varphi + 1\right) \; + \; 1 = \varphi + 2 \\\\
s^2 = \frac{1 + \sqrt{5}}{2} + 2 \; = \; \frac{5 + \sqrt{5}}{2} \\\\
\end{align}

Taking the square root of both sides and rationalizing the denominator, we can solve for the side lengths $s$ of our golden pyramid,

\begin{align}
s = \sqrt{\varphi + 2} \; = \; \frac{\sqrt{5 + \sqrt{5}}}{\sqrt{2}}\frac{\sqrt{2}}{\sqrt{2}} \; = \; \frac{\sqrt{2}\sqrt{5 + \sqrt{5}}}{2} \\\\
\end{align}

Calculating Outer Angles in the Golden Pyramid

Just for kicks, let’s let the base angles of these triangles be $\theta$, and calculate them as follows,

\begin{align}
\cos \left(\theta \right) = \frac{1}{s} = \frac{2}{\sqrt{2}\sqrt{5 + \sqrt{5}}} = 0.5257311121 \\\\
\theta = \cos^{- 1}\left(0.5257311121\right) = 58.28252559^{\circ}
\end{align}

Since these triangles are isoscles, we can call $\alpha$ the apex angle at the top of these triangles which would be,

\begin{align}
\alpha = 180^{\circ} \; – \; 2\left(58.28252559^{\circ} \right) \; = \; 63.43494882^{\circ}
\end{align}

So the facing triangle of a golden pyramid is what you would see by slightly scrunching down an equilateral triangle in the vertical direction.

Pyramid Relationships to Platonic Solids

Platonic solids are said to have the ability to move from one shape to another through a series of transformations. So they can either be in a state of expansion or contraction. We’re speculating wildly here, but do these scrunched equilateral triangles give a pyramid the effect of behaving like a coiled spring, being slightly contracted, with potential energy in the upwards direction? You could maybe also think of them as pulling energy, since it’s already in a slightly contracted state and gravity would be pulling it further inwards. Not sure how we could prove that, but it’s an interesting thought!

Is the Height of a Pyramid Related to Phi or Root 3?

We should notice that an equilateral triangle with a base of width $2$ would have a height of $\sqrt{3}$, and this is also the height of our Vesical Piscus from a previous article. The height of the facing triangles in our golden pyramid, on the other hand, have a height of $\varphi$. So an interesting question is, what is the scaling factor to get from $\varphi$ to $\sqrt{3}$ ?

That scaling calculation would require multiplying by a factor of $\sqrt{3} \div \varphi$. Going the other direction from $\sqrt{3}$ down to $\varphi$, you would multiply by a factor of $\varphi \div \sqrt{3}$. Either way, if the triangles are in a state of moving towards, or being reduced from a Vesica Piscus, that is being done by a factor of $\sqrt{3}$ related to $\varphi$.

Conclusion

A Kepler Triangle is one more piece of the puzzle which is related to a new article we wrote about Squaring the Circle. This is based on an old illustration we found in a paper about Viktor Schauberger, who was also very interested in sacred geometry. The model they used for squaring the circle is based on the same dimensions mentioned in the reference article above, namely 11 and 14. So this ties in with the pyramid numbers also, and it leads to some very interesting research!

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Cite This Page As:

The Math Zone. “Kepler Triangle and Root Phi.” From MathZone.io — A Modern Exploration of Ancient Mathematics. https://mathzone.io/kepler-triangle-and-root-phi/

Work Cited

Wikipedia contributors. (2024, September 21). Kepler triangle. In Wikipedia, The Free Encyclopedia. Retrieved 15:44, September 29, 2024, from https://en.wikipedia.org/w/index.php?title=Kepler_triangle&oldid=1246875160