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Geometry of the The Solar System, part 1
by Frank Hoogerbeets

12 January 2017


Since the discovery of a tenth planet in 2003, our solar system has become a subject of interest once more. Despite the fact that more objects have been found beyond Pluto's orbit in recent years, there seems to be more interest in the latest discovery. This is probably because this planet, Eris (formerly designated 2003-UB313), is larger than Pluto.

Looking at previous discoveries regarding the solar system, there is a phenomenon that seems to have been forgotten. It is about a mathematical law discovered by astronomer Johann Daniel Titius and published by his colleague Johann Elert Bode in 1772. Later this law became known as "the Law of Titius and Bode", or simply "Bode's Law."


I first read about this law in an astronomy book when I was 14. Although I did not fully understand it at the time, I was very intrigued by it and it kept sort of resonating in my mind. Later, after my physics lessons in school, I decided to do a thorough study on the subject. This study led me to the conclusion that the official explanation barely scratches the surface of the very significant underlying geometry that appears to be involved. This four-part series is an attempt to reveal and explain this geometry, and you will see that it not only defines the planetary distances to the Sun (orbital radii) but also velocity, time and even gravity.

If you are not familiar with the law of Titius and Bode, this is how the scientific community describes it:

  1. Consider the following numerical sequence:
    0 – 3 – 6 – 12 – 24 – 48 – 96 – 192 – 384
  2. Add 4 to each number:
    4 – 7 – 10 – 16 – 28 – 52 – 100 – 196 – 388
  3. Divide each number by 10:
    0,4 – 0,7 – 1,0 – 1,6 – 5,2 – 10,0 – 19,6 – 38,8

The numbers of the final sequence appear to closely match the orbital radii of the planets to the Sun in astronomical units (AU). One of the reasons why astronomers these days choose to ignore this mathematical model is the fact that Neptune and Pluto do not seem to fit in. However, this is not entirely true, as we will see later on.

Another problem is that astronomers have created their own system of measurements related to the position of Earth in the Solar System, like the astronomical unit, which is the orbital radius from the Sun to the Earth, approximately 149.6 million kilometers. This unit is being used to express all distances in the Solar System and it is the reason why the second sequence above is divided by 10. But by doing so, the true underlying geometry is being obscured.

The Three Building Blocks of the Universe

So let's have a look at this geometry. We will go back to basics and start with the circle. After all, everything in the Universe is circular, be it planetary, lunar or solar orbits. Now if we consider the mathematical rules, we know that three points are required in order to define a circle, its radius, circumference and surface. It cannot be done conclusively with any other number of points. It requires three, no more, no less. This is a very important detail and very much related to the three possible states of matter: solid, fluid and gas. In an abstract manner these three aspects could be termed positive, neutral and negative.If we do the construction of a circle orderly, we end up drawing three points equally distanced from each other.

figure (1.1)

Consider figure (1.1). We see the three points termed positive, negative and neutral. These three points automatically determine the size of a cricle whereby its circumference goes through each point. We can also draw lines to connect these points. This results in an equilateral triangle whereby each angle is 60°. What you see is a geometric expression of the three building blocks of the Universe, whether you apply this scientifically, psychologically or otherwise, these three aspects always apply. It is very important to understand that everything in the Universe starts with three, in order to exist, move, evolve or regenerate. Without these three, there would be nothing. Think about that for a moment.

Developments of Three

Geometrically, from the Three, different developments are possible and exist in the Universe. One such development is the multiplication of two, very much like cells of the physical body dividing and multiplying themselves in the foetal stage. Geometrically we can do this by taking zero and go three in a positive and negative direction. This gives us sort of an unborn or "gaseous" expression of the Three.

-3 ← 0 → 3+

We can express this division geometrically by taking the triangle and divide it by itself through its own opposite.

figure (1.2)

The original triangle has now been divided into four smaller triangles. We could repeat this step indefinitely thereby creating a micro-universe of an infinite number of triangles within the macro-universe – the original triangle. However, to illustrate what actually happens we only do one more division.

figure (1.3)

The four triangles in figure (1.2) are now divided into four triangles themselves, resulting in a total of sixteen (four times four) triangles. As I said, we could do this division indefinitely, which in numerical form would be:

1 – 4 – 16 – 64 – 256 – 1024 ...

In this form the numbers do not tell us all that much, but if we reduce the multiple digit numbers back to a single digit number, the sequence becomes much more interesting:

1 – 4 – 7 – 1 – 4 – 7 ...

This reduction goes like this. Take number 64, add up its digits (6 + 4 = 10) and repeat until a single digit remains (1 + 0 = 1). If you are not familiar with number theory then this reduction may seem a bit strange, but I can assure you that it is important in geometry in order to reveal underlying patterns.

Back to our "gaseous" form of the Three (-3, 0, 3+), we develop the sequence on the positive or active side by multiplying each number as if we were counting the three aspects of every subsequent division:

-3 – 0 – 3 – 6 – 12 – 24 – 48 – 96 – 192 – 384 – 768 ...
sequence (1.1)

This is just one possible development of number three. There are more developments in the Universe as we will see later on, but this is the one we are interested in right now.

Because each division results in four, we simply add 4 to each number in the sequence:

1 – 4 – 7 – 10 – 16 – 28 – 52 – 100 – 196 – 388 – 772 ...
sequence (1.2)

Now if we would reduce this newly derived sequence, the meaningfulness of number 7 becomes obvious:

1 – 4 – 7 – 1 – 7 – 1 – 7 – 1 – 7 – 1 – 7 ...

Of course, we recognize the first three numbers (1, 4, 7) from the numerical sequence (1.2) that expressed the division of triangles. So by adding 4 to our gaseous form (-3, 0, 3+) we end up with a new expression of the Three (1, 4, 7), which could be termed the "fluid" form. The remaining part of the sequence is a repetition of the numbers 1 and 7.

Now consider the fact that when light is broken through water or a prism, we see seven colours. It is the reason why we see a rainbow of seven colours when we have sunny and rainy weather combined. Also consider the fact that the cells of our physical body regenerate themselves every seven years. It very much seems like number 7 rules the material world.

Geometry of the Planetary Orbital Radius

We have already seen one development of number 3 (sequence 1.1) by multiplying each additional number by two. So 3 becomes 6 becomes 12 becomes 24 etc. Another development would be to simply add 3 each time as an expression of adding one triangle to the next, so 3 becomes 6 becomes 9, becomes 12 becomes 15, becomes 18. This way, after reducing these numbers, we would get an infinite repetition of the numbers 3, 6 and 9, which is basically one times three, two times three and three times three respectively.

Now consider the numbers 6 and 9 for a moment. Symbolically they are each others opposite, very much like the Yin-Yang symbol in eastern philosophy. One is male and the other is female, but we could also term them positive and negative or active and passive. Another way to look at it is that 3 creates, 6 maintains and 9 completes or transforms. What is in existence is being maintained or sustained (6) until it is fulfilled and transformed (9) after which a new cycle of creation (3) starts.

So there appears to be an interesting dynamics of polarity between the numbers 6 and 9 and what they represent in the Universe. If we combine these two numbers we get 15. We could also combine the two numbers in between 6 and 9, which are 7 and 8, which combined also result in 15. Other combinations of single digit numbers won't result in 15. Here things become interesting. If we multiply the numerical sequence by 15 where we left off (sequence 1.2), the result will be this new sequence:

15 – 60 – 105 – 150 – 240 – 420 – 780 – 1500 – 2940 – 5820 – 11580 ...
sequence (1.3)

We multiply because multiplication is the expression of polarity – one opposite the other – which is necessary in order to sustain itself, as expressed by number 6. Indeed, if we reduce the numbers of this sequence to a single digit number the result is always 6. So what we basically see here is the material world being maintained or sustained. But we know that 15 also includes 9 so that at one point completion and transformation occurs. The Universe could not exist without these dynamics.

So what does this have to do with the solar system? Well, if we consider each number of this sequence (1.3), we are literally looking at the orbital radius of each planet. So, basically, what we produced here is the mathematical model of the solar system, based on universal principles expressed through numbers.

planet math. orbit radius astr. orbit radius
(table 1.1)

You may notice that there is one exception: Neptune. It does not directly fit the sequence we derived mathematically, but if we take the numbers for Uranus and Pluto to their average, we come very close to Neptune's observed mean distance:

(2940 + 5820) / 2 = 4380

So mathematically Neptune fits nicely in between. It appears that the Universe always tries to find order and balance in everything and that the mean distance of planets is not random and chaotic.

There is probably a reason why Neptune assumes this orbit instead of Pluto's as we would expect. Our solar system probably has a history we don't know about. For instance, we have dwarf planets, which really are moon-sized planets about the size of our own Moon, originating in what seems to be debris fields, more commonly known as the asteroid and Kuiper belt. Could it be that these fields are the remains of planets and that these dwarf planets were their moons? It certainly looks that way because celestial objects can only build a hydrostatic equilibrium (spherical form) under influence of a strong gravitational force. The erratic oribts of Pluto, Eris and also Sedna suggest a turbulent past, especially if you consider the fact that their orbits nearly cross in the same region of space in the Kuiper belt.

In part 2 we will see how this number 15 as the first orbital radius defines the orbital velocities of the planets.

Copyright © 2006-2017 Frank Hoogerbeets. You have my permission to copy and distribute this article as long as you do not change its content including this copyright notice.

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