Amazing Numbers for the Fruit of Life

By Giuseppe Gori

_© This article can be copied, with attribution.

The “Fruit of Life” is the name given to a geometrical figure of 13 circles assembled along the three main directions at 60 degrees angles, thus creating a composite structure analogous to a hexagonal shape.

This Fruit of Life figure is a subset of a figure (called by the same sources the “Flower of Life”) composed by 19 non-intersecting circles, disposed in a hexagonal formation.

The complete Flower of Life figure can be found in records of different ancient civilizations, usually decorated with same-size, internally intersecting circles along the three axes.

For centuries, mathematicians have included the Normal series of integer numbers (1,2,3, …) in each cell of squares, and other regular geometric shapes, discovering “Magic Squares” and other magic figures.

Incidentally, we notice that the 19-circle Flower of Life is equivalent to 19 hexagons, similarly positioned. The figure below shows the smallest possible Magic Hexagon, which is probably the most unique of all “Magical figures”, with a magic constant of 38.

I refer to the Magic Hexagon in my article about the amazing Magic Rhombicube at: https://bit.ly/2wAJpg4

The Smallest Possible Magic Hexagon - equivalent to a Magic Flower of Life

Now, let’s turn our attention to the Fruit of Life. This figure has become known for its geometrical properties, but to my knowledge, it has not yet been associated with the Normal series of numbers 1, 2, 3, 4 . . . (In this case, the series from 1 to 13). 

Filling the circles with a Normal series

Examining one unique distribution of the Normal series, shown in the diagram below, I have discovered a number of interesting properties that seem to spotlight the number 7 and several of its multiples: (7x2=14), (7x3=21), (7x4=28), (7x5=35), (7x6=42), (7x7=49), (7x9=63), (7x12=84), and (7x13=91) !

By choosing regular geometrical figures, which point to, or highlight certain circles, and by adding the numbers in those circles, we always obtain multiples of 7 !

Our unique distribution

Geometrically, our figure is related to equilateral triangles (3 sides), forming hexagons (6 triangles, 6 sides) in patterns that exhibit 3 axes, and filled with a series of 13 numbers.

But when filled with our numbers, this pattern seems to change nature. As you read this article, keep in mind the following questions: Why would the number 7 acquire such prominence? How does the number 7, a prime number, relate to 3, 6, and 13? Why the multiples of 7 come into play almost exclusively in our patterns?

In the following, we may not provide all the answers, but we will discover these amazing patterns and many additional properties.

Referring to the diagram below, let’s start by observing that:

     1. The number 7 appears at the center of our figure, a prominent and unique position.
     2. The numbers 5, 6 and 3, at the vertices of the large equilateral triangle add up to 14.
     3. The numbers 5, 4, 6, 2, 3 and 1, traversed by the large equilateral triangle add up to 21.
     4. The numbers 9, 8 and 11, at the vertices of the small equilateral triangle add up to 28.
     5. The numbers 7, 9, 8 and 11, indicated by the center Y, add up to 35.
     6. The numbers 5, 10, 6, 12, 3, and 13, corresponding to the large hexagon, a very visible set of cells, add up to 49 (or 7 x 7).

The first five observations

In addition, we notice, as highlighted in the figure below:

     7. The numbers 7, 4, 1, and 2, indicated by the center inverted Y add up to 14.

     8. The numbers 9, 4, 8, 2, 11, and 1, corresponding to the small hexagon add up to 35.

The center inverted Y and the small hexagon

Let’s look at equilateral triangles, pointing upwards, as shown in the figure below:

     9. The numbers 1, 2 and 4, at the vertices of the small equilateral triangle, add up to 7.
   10. The numbers 10, 12 and 13, at the vertices of the large equilateral triangle, add up to 35.
   11. The numbers 5, 9, 7, 8, 6, 11 and 3, corresponding to the big red Y add up to 49 (7 x 7).
   12. The numbers 10, 8, 12, 11, 13 and 9, traversed by the large equilateral triangle, add up to 63.

Upwards-pointing equilateral triangles and the big red Y

Furthermore:

   13. The numbers 9, 4, 8, and 7, corresponding to the red rhombus at the center add up to 28. 
   14. The numbers 7, 8, 2, and 11, corresponding to the pink rhombus add up to 28.
   15. The numbers 9, 1, 11, and 7, corresponding to the orange rhombus add up to 28.
   16. The numbers 13, 1, 7, 2, 12, 4 and 10, corresponding to the inverted Y add up to 49 (7 x 7).

The three center rhombi and the inverted green Y

Now, if we rotate the rhombi by 60 degrees clockwise, we can see that:

   17. The numbers 8, 2, 7, and 4, corresponding to the red rhombus add up to 21.
   18. The numbers 2, 11, 1, and 7, corresponding to the pink rhombus add up to 21.
   19. The numbers 9, 1, 7, and 4, corresponding to the orange rhombus add up to 21.

The three rotated rhombi

What happens if we expand the rhombi?

   20. The numbers 5, 10, 6 and 7, at the vertices of the large red rhombus add up to 28.
   21. The numbers 6, 12, 3 and 7, at the vertices of the large pink rhombus add up to 28.
   22. The numbers 5, 13, 3 and 7, at the vertices of the large orange rhombus add up to 28.

The three larger rhombi

Now, if we rotate the large rhombi by 60 degrees clockwise we can see that:

   23. The numbers 7, 10, 6, and 12, at the vertices of the large red rhombus add up to 35.
   24. The numbers 7, 12, 3, and 13, at the vertices of the large pink rhombus add up to 35.
   25. The numbers 5, 10, 7, and 13, at the vertices of the large orange rhombus add up to 35.

The three larger rhombi rotated 60 degrees clockwise

We did not consider yet the easiest pattern to notice (or possibly the easiest to miss):

   26. The total of each five-circle major diagonal, such as 10, 4, 7, 11, and 3, is 35.

Major diagonals

Let’s take another view. Our figure can be thought as a central circle (the circle with number 7) with six spikes coming out of it.

For example, the two-circle spike at the top of our figure contains numbers 4 and 10.

Perhaps you have already noticed that:

   27. All the two-circle spikes contain numbers adding up to 14 !

Up to here we have identified the prominent position of the number 7, at the center of our figure, and twenty-five six in which we obtain a sum that is a multiple of 7, by adding the numbers in special positions, symmetrical, or corresponding to regular geometrical shapes.

In 7 of these cases that sum is 28.

But we did not finish discovering the properties of our particular sequence and disposition of numbers in the Fruit of Life figure.

Since the numbers in each spike add up to 14, this obviously means that if we select any couple of these spikes, they add up to 28.

Just for visual gratification, the following diagrams show selections of contiguous spikes and alternate spikes, that are reminiscent of the Flower of Life with intersecting circles:

Spike couples: 5+9+4+10=28, 5+9+1+13=28, etc.

And, of course, opposite spikes in every direction (E.g., 10+4+11+3) add up to 28.

We can also take three spikes. For example, looking at the diagram above to the right, we can add up 5, 9, 8, 6, 11, and 3, to obtain 42 (14x3). In the same way, adding up 10, 4, 2, 12, 13, and 1, we obtain 42.

We can observe some other interesting properties (see diagram below):

1. Three of the cell next to the central cell contain numbers smaller than 7. If this number is added to the 7, we find the result in the opposite cell: 1+7=8, 2+7=9, 4+7=11.
2. This is also true when we consider the spike’s tips: 5+7=12, 6+7=13, 3+7=10 !
3. All two-cell spikes contain two numbers, one less than 7 and the other greater than 7.
4. The differences between the two numbers in each spike happen to be all even numbers: 2, 4, 6, 8, 10 and 12. These are all the even numbers between 1 and 13.

More interesting symmetries

We can also observe (see diagram below) that three of the spikes, contiguous to each other (top and right), have circles containing only even numbers, while the other three have circles with only odd numbers.

If we divide the figure according to this oblique axis, excluding the central cell, we have all the even numbers on one side and all the odd numbers on the other side! 

Amazing odd / even balance!

The sum of all the even numbers one one side is equivalent to three spikes. Since each spike adds up to 14, that gives us a sum of 42 on each side of the axis. No matter which axis we draw, each side adds up to 42 (7x6) and both sides combined add up to 84 (7x12).

Now, if we only consider the tips of the spikes, on the odd side their numbers (5+13+3) add up to 21, while on the even side (10+6+12) they add app to 28.

Conversely, if we consider only the three circles touching the center circle, the sum on the odd side (9+1+11) is 21, while the sum on the even side (2+8+4) is 14.

Amazingly, both of these properties remain valid no matter which axis we use to split our figure. Lots of 14s, 21s, and 28s !

Amazing symmetry!

After the above observations, it’s easy to understand that most properties are a consequence of the symmetry of the spikes in every direction.

For example, let’s take cells that face each other on each side of the red axis.

Mirror cell pairs

If we add up the two couples of numbers at the tips of the spikes: 3+12, and 5+10 they add up to 15, while the more internal couples, 11+2 and 9+4, add up to 13. So, necessarily, the average between the two sums (green and blue couples) is 14 (as the total of internal plus external couples is two spikes adds up to 28). This is true for each of the three axes of symmetry.

In conclusion, the numbers in our series seem to ignore the geometric foundations of the Fruit of Life shape (i.e., triangles, hexagons, 60 degree rotations, and the 3 axes of symmetry) and the shapes we use to identify our patterns (hexagons, triangles, rhombi and other figures with the main axes of symmetry). Instead they use that geometric foundation to create an alternative numeric reality dominated by the number 7 and its multiples.

One last thing…

If we add up all the numbers, 1 to 13, in all of the circles, what do we get? The sum of the series (13 x 14 / 2), which is 91. That of course is 7 x 13: The number in the middle of our figure, our ubiquitous multiplier and prominent number 7, multiplied by 13, the number of circles in our figure !! - Don’t ask me why.