Star Fort Crystal Geometry & Construction

We can appreciate the natural beauty and geometric precision of the five pointed star, yet there there seems to be very little focus on integrating this ancient math into our modern age. When you explore the designs and architecture from earlier periods in history however, you’ll discover it appearing much more frequently.

As a case in point, we were wondering about how the old star forts were constructed. Some of them appear to be related to the five pointed star geometry, but what was the significance of using this particular configuration? A good place to start with a trial layout would be to use two concentric circles having golden ratio radii of $\varphi$ and $\varphi^2$ respectively, and then inscribing a pentagon into the inner circle. As you can see in Figure 2, what we ended up with appears to be a proportionally correct construction model for a pentagonal star fort.

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Star Fort Construction

We’re highlighting the top triangle in Figure 2 with a red outline to show how it’s related to a pentagonal triangle. In our five pointed star article, we showed how a doubled pentagonal triangle gives us a doubled divine triangle at the same time. So this type of star fort construction provided an aesthetic beauty along with perfect mathematical precision, and these large structures were once integrated right into our landscape!

We also discovered a new three dimensional shape as we were doing this research, and we’re calling it a “Star Fort Crystal.” It’s closely related to our Chestahedron Phi Model since we’re folding a pentagon at the apex in the same way. We weren’t sure if anything significant would come out of this new structure until we folded it and checked the math. But astoundingly yes, this is another equally significant shape! Click the link below to see what an actual star fort looks like, with the geometry button showing the math we used on the next screen.

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Star Fort Crystal Definition

Another Significant New Form!

Taking its position next to the venerable Chestahedron, the “Star Fort Crystal” shown in Figure 4 is another significant new three dimensional form! For that matter, we haven’t been able to find anyone else referencing even the most fundamental element in a pentagonal star fort construction model. So we’re calling this element a “Pentagonal Triangle,” and we had to assign that name to it ourselves because we couldn’t find any other reference to it. Others have worked out the math for its side measurements and angles, but only as the means to an end for finding the center and calculating the edges of a pentagon.

When you study the classic five pointed star (or pentagram), other shapes that make up its construction have been studied and given names. The star points are known as “Golden Triangles,” and the other triangles found in a pentagon (called “Golden Gnomons”) are well known. But what about the other most glaringly obvious triangles you’ll find in a pentagon? No one had even bothered to give them a name until now!

An Alternative Five Pointed Star Construction

The important point to notice about the star fort construction model is that we’re using an alternative method for creating a five pointed star. In Figure 3, this new star is shown on the left side of the illustration. But instead of the star points being golden triangles, the star fort is constructed with pentagonal triangles. This new star variation doesn’t seem to have a name either however, probably because it’s built with the most overlooked triangle in modern history!

We know the ancient cultures who built star forts knew exactly what this variation was called, but in modern times there is almost no reference to it that we can find. So we’ll eventually have to give this star a unique new name also, and we could go ahead and call it either a “Complimentary Pentagram,” or a “Secondary Five Pointed Star.”

Because the radii of our two circles in the star fort construction model from Figure 2 are $\varphi$ and $\varphi^2$ respectively, it’s easy to see how the side lenghts of our pentagonal triangles in Figure 4 are all equal to $\varphi$ and the heights are equal to $\varphi^2/2$. If we let the base lengths of these pentagonal triangles be $S$, with a little math we can show that,

\begin{align}
S = \frac{1}{\sqrt{2}}\sqrt{5 + \sqrt{5}} \; = \; 2 \; \varphi \cos \left(54^{\circ}\right) \\
\end{align}

Pentagonal Tetrahedron at the Apex

As we’re showing in Figure 3, this star can be folded at the apex of a pentagon in exactly the same way as we did on the Chestahedron Phi Model. And all of the math for the Star Fort Crystal follows from the Pentagonal Tetrahedron (shown in Figure 5) in exactly the same way. So there is no need to publish all of the math again since all we’re doing is switching out numbers. Most of the new measurements are already shown in Figure 4, and we’ll be adding more as we need them.

Probably the most important point to clarify however, is that using the same proof strategy we applied to the Chestahedron, we can show how the side lengths of the base equilateral triangle are equal to the same value of the base length $S$ that we calculated above. This proves that the side triangles are also pentagonal, and identical to the triangles that make up the pentagonal tetrahedron at the apex!

So if you include the doubled triangles that make up the three diamond shaped faces, the Star Fort Crystal is made up of nine identical pentagonal triangles, and a single equilateral triangle at the base — with side lengths matching the base length of the pentagonal triangles. In Figure 4, you can see how this is quite a spectacular set of angles and measurements!

It should also be noted that this is a seven sided geometry just like the Chestahedron, and nine out of the twelve edge lengths are identical in the same way as well. So it’s a very closely related model.

The fascinating thing about geometry is that it either works or it doesn’t, and it can’t be forced into giving you incredible outcomes just because you want them. So it’s always thrilling to see results like this!

Finding the Star Fort Crystal Hexagon

If you’ll recall in our Chestahedron Phi Model article, we found the exact horizontal elevation where a perfect hexagon will sit with equal sides. We also found the exact elevation where a hexagon sits in the Star Fort Crystal, and its position within that structure is equally fascinating!

Since everything below the pentagonal tetrahedron at the apex is made up of equal pentagonal triangles, we can see how the horizontal mid elevation line on any of these triangles will be equal, thus giving us an equal sided hexagon running all the way around the shape.

In Figure 6, we’re looking at a parallel projection of our model in X-Ray view, and we can see how the pentagonal triangles can be subdivided further. The key feature to notice is how a single pentagonal triangle can be subdivided into 4 identical smaller triangles. From there, we can see how a pentagonal diamond shape (also a divine triangle diamond at the same time) can be placed to fit perfectly within the vertical height of a pentagonal triangle.

What we need to know is what is the width of this pentagonal diamond, since that will give us the side length $H_s$ of our star fort crystal hexagon. Our illustration in Figure 6 is giving us everything we need to make that calculation which can be expressed as follows,

\begin{align}
H_s = \varphi \sin \left(36^{\circ}\right) = 2 \cos \left(36^{\circ}\right)\sin \left(36^{\circ}\right) = \sin \left(72^{\circ}\right) \\
\end{align}

We can see from the sine double-angle identity formula that $H_s = \sin \left(72^{\circ}\right)$. As we showed in our reference article, $\sin \left(72^{\circ}\right)$ is exactly equal to the height $h$ of a golden triangle divided by $\varphi$. So using the golden triangle math we established in that article, we can substitute exact values for the $H_s$ formula above — since we already have everything we need for those calculations as well.

\begin{align}
\varphi = \frac{1 + \sqrt{5}}{2} \\\\
\sin \left(36^{\circ} \right) = \frac{\sqrt{2}}{4}\sqrt{5 \; – \; \sqrt{5}} \\\\
\cos \left(36^{\circ} \right) = \frac{1 + \sqrt{5}}{4} \\
\end{align}

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Star Fort Crystal Tetrahedron

Features of the Larger Tetrahedron

We can also extend the edges of the pentagonal tetrahedron at the apex until they intersect with the same plane as the base of the Star Fort Crystal. What we end up with is a larger tetrahedron $ \bigtriangleup ACGI$ that fully encloses the original shape as we can see in Figure 7.

As we demonstrated in our Chestahedron article, we can calculate the base side lengths of the larger tetrahedron and prove that our claim in Figure 7 is correct. What this illustration shows is the larger equilateral base triangle $ \bigtriangleup CGI$ is made up of four equilateral triangles which are all identical to the base of the star fort crystal. In addition, all of the three sided pyramids formed on the sides of the star fort crystal, $ \bigtriangleup CEBD$ for example, are identical to the pentagonal tetrahedron at the apex — defined by side triangle $ \bigtriangleup ABH$. With this math established, we can easily make Volume calculations for the Star Fort Crystal with the same process we used for the Chestahedron Phi Model.

You might also notice how smaller star fort crystals will fit perfectly within any of these four pentagonal tetrahedrons shown in Figure 7, since they’re all proportionally identical. This is similar to how a smaller five pointed star will fit perfectly within the pentagon of a larger star. And this regenerative effect goes on indefinitely for the star fort crystal as well, from the infinitely small up to the infinitely large!

Star Fort Crystal Inside of a Chestahedron

Because the Star Fort Crystal and the Chestahedron are both defined by a proportionally identical pentagonal tetrahedron (see Figure 5 above), the Star Fort Crystal will fit perfectly inside of the Chestahedron’s apex!

We’re superimposing both of their larger tetrahedron models as shown in Figure 8, and simply scaling up the Chestahedron’s proportions to match the numbers we’ve established for the Star Fort Crystal.

The result of this scaling means we’re using a Chestahedron where the side lengths of the pentagonal triangles (like $ \bigtriangleup ABH$ for example) will be equal to the golden ratio or $\varphi$.

The calculations on this scaled up Chestahedron are simple to perform by using the laws of similar triangles. So the height of the scaled up golden triangle at the bottom end of the new Kite shape will be equal to $Sh_g \;$ — where $h_g$ is the height of the golden triangle in our original model, and $S$ is the value we calculated earlier for the base length $\overline{BH}$ of triangle $ \bigtriangleup ABH$ at the apex.

We already know from Figure 4 that the height $\overline{AI}$ of triangle $ \bigtriangleup ABH$ is equal to $\varphi^2/2$, since line segment $\overline{AK} = \varphi^2$.

And you can verify for yourself that the height of the scaled up Kite shape $\overline{AE}$ is equal to $\varphi^3$ by adding $Sh_g + \overline{AI}$.

We’re showing the values you’ll need for the proof below, and we like using the online Desmos Scientific Calculator for problems like this. So we’re also including a screen image of our calculation below the variables we used. Reducing the algebra down to $\varphi^3 = \sqrt{5} + 2$ would obviously be possible since the calculator shows us the correct value, but it will require using a few extra tricks.

\begin{align}
S = \overline{BH} = \frac{1}{\sqrt{2}}\sqrt{5 + \sqrt{5}} \\\\
h_g = \frac{\sqrt{5 + 2\sqrt{5}}}{2} \\\\
\varphi = \frac{\sqrt{5} + 1}{2} \\\\
\overline{AI} = \frac{\varphi^2}{2} = \frac{\varphi + 1}{2} = \frac{\sqrt{5} + 3}{4} \\\\
\varphi^3 = 2\varphi + 1 = \sqrt{5} + 2 \\\\
\varphi^3 = Sh_g + \overline{AI} = 4.236067977 \\
\end{align}

Frank Chester has discussed the relationship of a cube with the Chestahedron many times, and $\varphi^3$ is an interesting number. Because it represents a cube with side lengths equal to $\varphi$ and a volume equal to $\varphi^3$. This cube would have an internal diagonal length of $\varphi\sqrt{3}$, which would also be the height of what you could call a “golden vesica pisces.”

Additional Measurements and a Visual Proof

We also know that the side length $\overline{AG}$ of the larger tetrahedron in Figure 8 will be equal to $2\varphi^2$, since our Python Number Analytics Library found that this length is equal to 4 times the height of the pentagonal triangle, which is $\varphi^2/2$. This is only one additional outcome out of many more evaluations we could do, but we can also provide a more visual illustration showing why using this new set of metrics for the Chestahedron is a good choice.

Since the Star Fort Crystal and Chestahedron rely solely on the primary and secondary five pointed star geometries, we can learn more about their relationships to the golden ratio $\varphi$ in Figure 9.

If we show both of these stars together in the same image, we can construct the inner red circle with a radius of $\varphi$, the next blue circle with a radius of $\varphi^2$, and the final green outer circle with a radius of $\varphi^3$. Circles with these dimensions will fit precisely over the two stars in the diagram as shown, and it gives us another way to visually prove our statements about the illustration in Figure 8.

So this shows how we can easily establish a powerful new set of metrics for our original Chestahedron Phi Model by simply scaling up the original triangles by a factor of $S$ to match the Star Fort Crystal proportions. In fact, the only thing we’re missing now are the side lengths of the four equilateral triangles, and these would each be equal to $S\varphi$.

Star Fort Crystal 360 Animation

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Geometry of the heart corollary

We’ve already done enough math to show how the Star Fort Crystal shape is legitimate, but the big question now is what does it represent in the real world? We know it’s closely related to the Chestahedron which is said to be “the geometry of the heart.”

If that’s the case, could the star fort crystal be representing a closely related organ to the heart? We’ll go out on a limb here and say it looks like it might be matching the structural geometry of a helmet. So that’s one possible hint for what the closely related organ might be!

In Figure 10, we’re showing an ancient Greek helmet which appears to contain some related symbolism. Also interesting are some of the ancient Spartan helmets which even show a flat folded plane on the upper front side, similar to the folded planes of a star fort crystal. You can see an example of this in the link we’re posting below.

> Ancient Spartan Helmet Example

At this point we need to make it clear that we’re only speculating about what the star fort crystal might be related to. The Chestahedron has a longer history of research behind it which can back up the claim that it correlates with the geometry of the heart.

The Star Fort Crystal on the other hand is a new discovery with none of this research behind it yet. We’re still in the early stages of looking for possible clues that can be correlated with what it might be representing in the real world. So we invite you to share your thoughts in the Comments section below if you have other ideas about what it could be related to…

The Pentagonal Triangle Rhombus

There is one last thing we wanted to stress further about the new Star Fort Crystal, since this is turning out to be a very important feature. The three “Kite” shape faces we encountered in the Chestahedron have now become the new diamond shape we’re showing in Figure 11.

In our Five Pointed Star Reference Shapes article, we can see many fascinating relationships between the angles of the pentagonal triangles, the golden triangles, the divine triangles, and obviously with the pentagon. But what’s really interesting is what happens when we stack and invert two triangles to form what we’re calling a “Pentagonal Triangle Rhombus” or a “Pentagonal Diamond.”

We’re showing two pentagonal triangles attached at the base horizontally, but oriented in opposite directions vertically. But look at what happens with the angles. In that same image without changing anything, you also have two Golden Gnomons (Divine Triangles) attached at the base vertically, but oriented in opposite directions horizontally! So a Pentagonal Triangle Rhombus is also a Golden Gnomon Rhombus at the same time!

This is an important feature to recognize, since it demonstrates one of the most fundamental oscillating properties found in a classic five pointed star. Namely, we can observe how the pentagonal triangle and the golden gnomon are actually two different faces of the same coin! And likewise, it’s interesting to see how all of these same properties will also apply — even more directly in the “Secondary Five Pointed Star” and our Star Fort Crystal Geometry as well!

Conclusion

Putting it all together, what the Star Fort Crystal really appears to be is the Chestahedron’s long lost relative — finally revealing itself and coming home after being previously unknown throughout all of history! The angles and measurements are equally miraculous, and it’s always thrilling to see results like this in such a strict discipline like geometry. As we know all too well, geometry either works or it doesn’t, and it can’t be forced into giving you incredible outcomes. It’s always been there with no one ever having invented it, so anything you might discover with geometry was already there to begin with. You just didn’t see it yet, until you were able to uncover it and reveal the magic…

Star Fort Crystal Video

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Cite This Page As:

The Math Zone. “Star Fort Crystal Geometry.” From MathZone.io — A Modern Exploration of Ancient Mathematics. https://mathzone.io/star-fort-crystal-geometry/

New Chestahedron Phi Model PDF Press Release
> A New Paradigm In Euclidean Geometry (PDF)