Gabriel’s Horn Interactive App
We’ve been working on a special project lately, and it’s finally here! This is our new Gabriel’s Horn Interactive App for evaluating ovoid shapes and circles — which result from setting two points for positioning an intersecting plane. So you can perform an interactive analysis of the “Painter’s Paradox” geometry and see the results!
When you mouse over the image in the app window above, you’ll see a small tool bar in the upper right corner. This will allow you pan or move the image, zoom in or out, and rotate the image (by clicking and dragging). The far left camera icon tool allows you to save the current view of your image. So it’s a fully featured program, and it functions the same way as other popular 3D software.
Creating Egg Shapes
This program allows you to download unique ovoid (meaning egg like) shapes that you can create yourself. In addition, the dashboard below the app window will keep track of your metrics data for each unique entry of $x_1$ and $x_2$.
In Figure 1, for example, this egg was created with the values of $x_1 = 1$ and $x_2 = 2$. At the time, we were using a plugin for Sketchup to generate the ovoid outline with an intersecting plane. Then we used Adobe Illustrator to render a 3D view, by rotating half of the outline around a vertical axis.
We then worked out the measurements shown in Figure 1 mathematically. And these results have proven to be identical in our new program as well — as you can see in the metrics dashboard when you enter values of 1 and 2 for $x_1$ and $x_2$. Our new app is far more accurate for rendering the actual ovoid outlines however. So we can experiment with different numbers to see which ones give us the best results (i.e., the best looking eggs).
Other Types of Research
The research doesn’t end with just eggs either, since values of $\; x_1 = 3 \;$ to $\; x_2 = 5 \;$ will yield a shape that looks more like a kernel of grain! The metrics dashboard will give us other measurements as well, like the volume and surface area from a section of the horn between your values of $x_1$ and $x_2$.
A well known researcher named Viktor Schauberger (1885 – 1958) was very interested in the Gabriel’s Horn shape which he used for creating structured water. We think he would have liked the golden fish shape we found with the app, which we describe later in this article. We also found a very strong correlation with the Vesica Piscis, which is known to represent a fish shape as well.
The Painter’s Paradox
What might be the most striking feature of Gabriel’s Horn is that it’s an infinitely long shape, and we’re only looking at a smaller section of it in the interactive app above. Because the $y = 1/x$ graph never really goes to zero, the end of the horn keeps getting smaller and smaller as it goes on forever.
But Gabriel’s Horn has plenty of anomalous behavior right out of the chute, because it can be shown with calculus that it has a finite volume (of exactly $\pi$ units, which is very strange in itself), but it can also be shown that the surface area is infinite. This is called the “Painter’s Paradox,” which means it can be filled with $\pi$ units of paint, but the surface area is infinite so it can’t ever be fully painted.
Even though this can be shown with math, no one really understands why it happens. We’re sure there is a reason though (like a shell or boundary between the finite and the infinite), and the “anomalous remainder” we found later in this article could be pointing at some more of that evidence.
We think it’s also very interesting that an egg shape would be showing up here, since eggs have shells too — which could be existing right on that same boundary!
The Gabriel’s Horn object itself is created by spinning the $\; y = 1/x \;$ graph around the $x$ axis. For more details, you can refer to our related article titled Multiplicative Inverse And Gabriel’s Horn. This article contains additional information on the math used in our analysis. And it also provides a mathematical explanation of the mysterious Painter’s Paradox associated with Gabriel’s Horn!
The Metrics Dashboard Measurements
In the metrics dashboard below the app window, notice that we’re showing two different values of the width analysis for the ovoid. The “Center Width” is the diameter of a vertically oriented circle positioned where the slanted plane intersects the $x$ axis (the red horizontal line shown in Figure 1). The “Max Width” is included as well, since the slanted intersecting plane creates a wider bulge in the ovoid than the Center Width is measuring.
We used half of the Center Width as the radius to construct a Vesica Piscis — fitting almost perfectly over the egg as shown in Figure 1. Note that the Vesica is the intersection of two circles having radii of $r = 3/5$. The height of that two circle configuration would then be $3r = 9/5$, and this is very close to the actual height of the egg — which is
We’re showing the metrics dashboard measurements for this particular egg shape rendered by the app in Figure 2.
The “Anomalous Remainder”
We were curious about why $9/5$ and
|end[align]
Next, we took the inverse of this remainder… just to see what would happen,
|end[align]
But wow, what in the heck is that? The 360 number peaked our interest because it relates to circle geometry. Then the reappearing decimal from the original remainder made it look like a possible harmonic frequency or a geometric progression. There could be a perfectly reasonable explanation for this, and we don’t have all of the answers yet. So we had AI run down the rabbit hole to see what else it could find.
Setting Up The Investigation
There are many ways to approach an investigation like this, but the initial research is perhaps easier if we can establish a rule for $x_2$ related to $x_1$, and then focus on $x_1$ as the baseline for numbers to analyze with an algorithm. The rule we chose for $x_2$ was to simply shift the $x_1$ value down the axis by one unit, so that $x_2 = x_1 + 1$.
Then the next part of the algorithm defines what we’re evaluating — which is an equation that describes how we found the original anomaly. This can be seen in the vesica piscis relationship with the egg in Figure 1,
|end[align]
where $H$ is the height of the ovoid, and $r$ is half of the “Center Width” value as outlined above. We’ve included this equation in the metrics dashboard under the width analysis measurements, so it will recalculate the inverse of this interesting little artifact whenever you enter new $x$ values!
The only thing left to do was to have AI look for other interesting numbers, like where does this inverse remainder equal 360, 720, 108, etc.. When we were finished, we had it give us a short summary of findings,
Evaluating The Results
We’ve noticed that AI likes to get creative with its language, so it might have fluffed up these descriptions a little more than we would have liked. But we can live with it, since AI can say whatever it wants as long as the numbers it’s giving us are correct. And who knows, it could even be a reasonable way to describe a bizarre set of results when you don’t yet know what to make of them.
Some of these new relationships are actually quite interesting though, like the “altitude of an equilateral triangle,” since this triangle is directly related to the Vesica Piscis configuration placed over our egg as shown in Figure 1. The Vesica itself (the fish shape formed by the union of those two circles) is defined to be exactly twice the height of an equilateral triangle (which is
Equilateral Triangle Relationship
For any value of $r$, the Equilateral Triangle height would be
In Figure 3 we’re showing the result of an egg created by the app with the following values for $x$,
x_2 = 1 + \frac[2][\sqrt[3]] = 2.154700538 \\
|end[align]
If you try this in the app, you’ll also see that the anomalous remainder output is the same as it is for $x_1 = 1$ and $x_2 = 2$, and we think it’s rendering a better looking egg as well. If we had to pick a single egg for being one of the archetypes, this would be a great candidate!
Also notice that this value of $x_1$ is describing the point where $y$ is equal to the altitude of an equilateral triangle with $r = 1$. So its side lengths are equal to $1$, and the circle configuration forming a vesica piscis would also have radii of $r = 1$. That means this particular egg in Figure 3 could be describing the central boiler plate definition of the “anomalous remainder,” and it would be a great starting point for conducting more research!
The Vesica Piscis
The Vesica Piscis appeared in the first proposition of Euclid’s Elements, where it shows how to use this configuration to construct an equilateral triangle using a compass and straight edge. We can also use this shape to derive other important numbers often seen in sacred geometry, design, and architecture, including the square roots of numbers 1 through 5, in addition to the golden ratio.
Since it appears to tie in so closely with Gabriel’s Horn, we would recommend checking out our reference article on this shape below.
Example Output From Gabriel’s Horn App
In the output image above, we’re showing three different egg variations created with the new Gabriel’s Horn app. We’ve seen each of these different egg types before, so we’re not sure if you could say there is such a thing as a perfect archetype. But the sweet spot seems to be somewhere in the range between $x_1 = 1$ and $x_2 = 2.5$. Our square root of Phi experiment (which relates to the Kepler Triangle) is in that range also since,
\sqrt[\varphi ] + 1 = 2.27201965 \\
|end[align]
Golden Ratio Relationships
Another thing we should point out from the output image above is that on the third version of the egg on the right, with $x_1 = 1.32$ and $x_2 = 2.45$, we were attempting to give the egg more of a classic shape — just by entering experimental numbers to push the $x$ values down the axis a little further. And quite by accident, look what happened to the height value in the output dashboard. We didn’t notice this until later, but it’s very close to the golden ratio!
So another very important line of research is to look at golden ratio numbers in relation to the egg shapes and their anomalous remainders. For example, when we increased the restricted range window of $x_2$ values to $x_2 = x_1 + 2$, we noticed another stunning alignment when we entered $x_1 = \varphi$ and $x_2 = \varphi$ + 2. This is the point where the Center Width (or circular diameter) of the ovoid becomes exactly $2/3$, and that means the triple-radius height $3r$ (of a Vesica Piscis circle configuration) is exactly $1.0$. In this state, the ovoid’s vesica piscis relationship is essentially “normalized” to a unit value.
The Golden Fish Shape
In Figure 4, you can see the ovoid rendered by the app with these new golden proportions, and notice the effect of creating more of a seed or fish-like shape than an egg-like shape — by extending the $x$ value range window.
So now we not only have eggs to deal with, but fish? Good grief! Where is Viktor Schauberger when you need him?
Gabriel’s Horn Research Data
To analyze results like this even further, we realized how convenient it would be to do all of the calculations we need with a Python script, and then have it output spreadsheets and charts into a nice readable html format. If we had a tool like that, we could present some of those findings on a “Research Hub” with links to our various output results. So we just recently launched the output from a Python app like this on our development server.
We have short explanations of what we’re researching in the sidebar, and you can see all of the different kinds of things we can look into with output data. When we have clean readable charts and graphs in an online format, it can be a lot easier to make sense of the many odd things we’ll be encountering in the world of math!
Fractal Geometry Research
And speaking of odd things, we also set up a new Mandelbrot Fractal Generator on our development server so we can start doing more research into fractal geometry. If you haven’t seen the famous Mandelbrot Set before, get ready for something that could even be more anomalous than Gabriel’s Horn! We’re not sure what the anomalous remainder is yet, but we thought it might be an interesting value to play around with in a fractal generator. So an interactive Mandelbrot app was a good place to start.
Mandelbrot Fractal Zoom Sequence
If you’re new to the Mandelbrot Set, this video zoom sequence shows one of the things it’s famous for. Rendering a high-resolution Mandelbrot zoom can take these guys anywhere from a few hours to several months, depending on their hardware, the depth of the zoom, and whether the software uses GPU-acceleration.
Reciprocal Mandelbrot Set Fractal Generator
We had no sooner than finished the Mandelbrot app when we found out that there is such a thing as a “Reciprocal Mandlelbrot Set” which is related to Gabriel’s Horn. By simply modifying our original Mandelbrot script with the updated inverse equation, we can display this new app on a separate page… where we might notice some familiar shapes showing up as you can see below in Figure 5!
If the basic Mandelbrot equation looks like this,
|end[align]
The Reciprocal Mandelbrot equation would be this,
|end[align]
So all we had to do was swap out the logic with this new equation in our original Mandelbrot Viewer program.
Generating Reciprocal Mandelbrot Images
We’re showing an image below in Figure 5 which has coordinates that you can enter into the app yourself. Enter the values -0.32692400000000, -0.53859100000000, 8017.00, 339 separated by commas in the Jump field, and it will bring up the image we’re showing below. These coordinates refer to the values for RE, IM, Zoom, Detail, and you can copy them out of the app’s header.
But we also included a handy new feature where you can press the “c” key on your keyboard, and it will copy the coordinates automatically. Then you can paste these settings into a text file if you want to save them for later, and pasting them into the Jump field in the header will bring up your exact image view. If you find something interesting, you’re welcome to post your coordinates in the comments section below so we can all have a look!
We’ve also included a dropdown menu in the app with some additional Presets, where we highlighted the coordinates for a variety of other interesting structures when we found them.
Figure 5: Reciprocal Mandelbrot Set Fractal Generator Image
Cubic Inverse Mandelbrot Set Generator
So in Figure 5 we can see what looks like the ovoid structures we’ve been studying, and we’re not making this up! If we hadn’t just gotten done researching the heck out of egg shapes in Gabriel’s Horn, we wouldn’t have understood what we were looking at. It’s very easy to see why the eggs are there in our intersecting plane app, but why this is happening in a reciprocal mandelbrot set is harder to get our arms around. It’s unbelievable, but we can see it for ourselves.
The Detail Slider
If you try any of these mandelbrot generators, we’ve improved the precision of the “Detail” slider because we’ve noticed how important this feature is. What it’s doing programmatically is imposing “iteration limits,” and these act exactly like a focal lens or a mathematical filter.
So you can use the detail slider as an active exploration tool to isolate structural boundaries:
The 3 Core Detail Slider Lens Adjustments:
- Low Settings (20–80 Iterations): This isolates the massive asymptotic structural walls. It removes the noisy fringe filaments entirely, allowing you to trace the pure geometric sweep of the curves as they mimic the smooth walls of Gabriel’s Horn.
- Medium Settings (100–300 Iterations): This is the sweet spot for general exploration on your device. It brings out the primary branching structures and localized spirals without overloading your processor.
- High Settings (500+ Iterations): This acts like a deep-space telescope. The broad structures will appear to shrink or sharpen, and the extra math will resolve the hyper-dense Mini-Mandelbrots and microscopic thread-filaments tucked deep within the edge boundaries.
Updated Equation and Output Results
For the Cubic Inverse Mandelbrot Set, our updated equation will look like this in the new app,
|end[align]
We’re showing one of our output images from the app in Figure 6 below, and the new cubic inverse equation apparently had the effect of making the ovoids look more like grapes!
Figure 6: Cubic Inverse Mandelbrot Set Fractal Generator Image
Three Dimensional Mandelbrot Sets
Once we had the Reciprocal Mandelbrot Explorer finished and we were able to explore the shape on a flat two dimensional plane, we were beginning to suspect more than just two dimensions based on the appearance of the structures. For one thing, the ovoids we’re seeing look like three dimensional objects. This could just be a trick of the algorithm because of how it applies shading based on the more compressed areas. But then why wouldn’t these just be a single solid color if they were just flat featureless shapes?
In addition to that, we kept noticing what looked like tiny little pin holes in all of the ovoids. And this would suggest a relationship with something more like a horn shape. We know the Reciprocal Mandelbrot Set is related to Gabriel’s Horn which has a tiny pin hole going to infinity at one end. So is this what we’re seeing in the ovoids?
Evidence of a Third Dimension
Beyond just the dimensional appearance of the ovoids however, we’re clearly seeing the overlapping of shapes in our Mandelbrot images. Some shapes are appearing in front of or behind others, or smaller shapes are being cut off by larger ones and so forth.
So we were becoming increasingly more convinced that there must be another dimension. The most intriguing question to us was about the ovoids. Are they three dimensional eggs, are they some kind of strange horn shaped inversions, or are they just plain old flat cutouts? If we could add an extra dimension to the Mandelbrot’s two dimensional equation, we might be able to write a 3d program to tilt that base plane back a little and see what’s going on when we look at it from a side view!
Ovoid Mystery Solved with Code
The Reciprocal Mandelbrot algorithm generates the fractal by testing points on a complex number plane. For each coordinate, it runs a simple repeating equation:
Points that stay mathematically bounded belong to the set (colored black), while escaping points are colored by how fast they diverge. It turns out that we can add that extra dimension by including a $z$ axis and remapping the coordinates into a new spatial orientation. Then we can explore the structure in 3d space by using the same algorithmic strategy.
There are lots of different methods for modeling 3d surfaces, but we thought the best bet would be to approach it like a particle physics experiment. That way we can start out with larger particles to see what the algorithm is able to pick out in 3d space. Once we could verify that it was working, we could steadily reduce the particle size while increasing the number of particles until we could visualize what was going on with the shapes.
In Figure 7 below, we took a screenshot of an upper side view from the final program. And it finally answered our nagging question about why it looked like we were seeing three dimensional ovoids with tiny pin holes!
Conclusion
Eggs can’t be constructed with a compass and straight edge like many other shapes in geometry, since the curves of an egg are continually graduated all the way around the perimeter. We’ve seen other attempts to describe them mathematically, but their shapes never look as convincing as the ones you can get from Gabriel’s Horn. An equation or geometric strategy might only be describing a single egg shape, whereas our method can describe an infinite number of different variations (which is exactly what you see with eggs in the real world).
What’s been fascinating to us is how all of this work keeps folding back in on itself and everything is tying together. The golden triangle in the Chestahedron has the $y = 1/x$ equation contained right in its definition, and that’s Gabriel’s Horn. Then we discovered that Gabriel’s Horn is directly related to the Reciprocal Mandelbrot Set. So what does that mean? It means the Chestahedron is also directly related to the Reciprocal Mandelbrot Set!
Notes on the Computations We Used
The numbers in our dashboard output images from the intersecting plane app needed to be updated a couple of times during the development process, because we were testing and improving the precision on the computations that involve calculus. These numbers are matching the latest version of the app’s output now, and we have some additional notes about the calculations below.
We also have a new Documentation Portal to explain the math and logic we used in the Mandlelbrot Generators.
* Note that this app was developed and tested with modern browsers, so rendering results may vary with older browsers that are no longer being supported or updated.
** Also note that some of the values we’re including in the output dashboard — like the ovoid perimeter and area, and the horn section’s volume and surface area, are computed with calculus. This means the precision depends on the number of steps used for the integration, and larger numbers put more strain on the server. So we have to balance out how high we can go with accuracy and precision, while also paying attention to page performance. With that in mind, some of the calculations where calculus is involved should be considered as close approximations rather than exact values. Other output metrics can be considered to be more exact however, like the height, tilt angle, and center width, since these were calculated with more fundamental math.
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Cite This Page As:
The Math Zone. “Gabriel’s Horn Intersecting Plane App.” From MathZone.io — A Modern Exploration of Ancient Mathematics. https://mathzone.io/gabriels-horn-intersecting-plane-app/
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