by George Buehler

Do you know what a Fibonacci (pronounced fee buh 'nahch ee) series is? When you look at a bonsai, do the proportions of one tree look better than others? Do you remember that when you choose a pot, the length of the pot should be about 2/3 the tree height? Then perhaps you do know what a Fibonacci series is but just don't know that it is a Fibonacci series.

This is not a mathematics course, nor will there be any tests nor will you need a calculator to finish this article. However, we need to set up the background of a Fibonacci series before we show how it relates to bonsai.

Leonardo Pisano, better known as Leonardo Fibonacci was a mathematician who was born in the 12th century. In 1202, he posed the following problem in his treatise Liber Abaci (pub. 1202): How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? Fortunately for us, Fibonacci was also interested in botany, and he proposed a problem similar to the rabbit question: Suppose that when a plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing point, how many branches will there be in 7 months?

So let's look at how to solve both problems by starting with the rabbits [if rabbits actually grew like this we wouldn't have any chances of growing a garden]:

1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
5. And so on

The series starts like this - 0 + 1 = 1

Then the series continues - 1 + 1 = 2

Then we have 2 + 1 = 3

Then 3 + 2 = 5

And the series continues as 5 + 3 = 8, 8 + 5 = 13 and so on. Do you see the repetitive nature of the problem? As the numbers increase, you add the sum to the previous sum to get the result.

So in the 12th month there would be 144 pairs of rabbits.

After the above problem was studied, mathematicians of the era assigned the name Fibonacci numbers to this sequence. [The answer to the number of shoots that would be present in the 7th month would be 13 from the above table.]

The next thing we must cover before we relate all this to bonsai is the Golden Number. Without going into a lot of mathematical details, if you take the ratio of two successive numbers in Fibonacci's series, 1.61538 is obtained (1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538. If the sequence was carried out for many more ratios, the number turns out to be 1.618033). This number, 1.618033, is called the Golden Number or Golden Ratio (Mathematicians assigned the Greek letter Phi to this Golden Number).

If instead of dividing the sum of the two numbers by the previous sum, the reverse is done, the number approach 0.61803 (mathematicians assign the Greek letter phi to this approach). One other Fibonacci fact is that if you look at this 'golden division' an interesting fact is determined as indicated in the table below. When the series is divided by the sum of two of the series (3 + 5 or 8 + 13, etc.) there is a definite repetitive result.

You see that every other division results in a number of 0.3 while the alternate is 0.6 or approximately 1/3 and 2/3. This is an important fact when proportions of bonsai are examined (as shown below).

It seems in nature that many plants have a Fibonacci number for the number of petals in its flowers: 3 - lily, iris; 5 - buttercup, wild rose, columbine; 8 - delphiniums; 13 - ragwort, corn marigold, and some daises. There are only a few flowers with the number of petals that are not a Fibonacci number. I went out to my garden and checked several plants and this holds up very well. I must say that there were a few malformed flowers on the same plant that didn't have the Fibonacci number, but in general, most follow the Fibonacci series.

In architecture, the Parthenon's facade is considered a perfect design. It should be noted that it has a 2:3 height to width ratio. Now what does all this have to do with bonsai?

We could go on with other facts that follow the Fibonacci series or the golden ratio, but we need to relate all this to bonsai. As best I can tell, John Naka in his "Bonsai Techniques I"¹ was the first to state that a Fibonacci series could be used in bonsai although I also found in Simon & Schuster's Guide to Bonsai² a drawing derived from the Fibonacci series but it had no mention of the series itself. Naka lists a number of things to look at as a general approach to bonsai, including symmetry, balance and proportion. He states "Proportion is a rate or ratio, which is one of the important elements needed to create a bonsai. A method call the "Golden section of division" has been used to obtain the proper proportions." Perhaps Naka obtained this information from other sources but there is nothing mentioned. Perhaps because of Naka's 'eye' for bonsai, he might have been using the golden section but like most of us, didn't know it.

Looking at figure 1, the line AB is 21 inches. Point C is the middle of line AB (10 1/2 inches). An arc is drawn using BC as the radius and point B as center. The line BD is drawn at a 90 degree angle to line AB. A line is drawn between point A and D forming the triangle ABD. Another arc is drawn using BD as the radius and point D as the center (arc BD and BC are equal). Finally, another arc is drawn using point A as the center and AE as the radius. The point of intersection of this arc on line AB (point F) marks the golden section of division. Notice that the section BF equals 8 inches and the section AF equals 13. Now remember that in the Fibonacci series, one of the number sequences is 8, 13, 21.

Now if we look at a tree that is 21 inches tall, to obtain the "ideal" proportions, the first branch should be at the golden section or 8 inches. Naka also pointed out that the height of the pot should equal the diameter of the trunk where the roots go into the soil. De Groot³ states that "if the proportions of the (Fibonacci) sequence are used to design a bonsai, it will have an unfailingly pleasing effect.

Now all bonsai are not 21 inches tall and even if they were (which would be very boring) the first branch probably wouldn't be at 8". Therefore, this design is just a curiosity and not a practical model. However, most bonsai artists try to follow the golden division or rule of thirds. In order to get the most pleasing look in bonsai, the proper proportions of any size tree is that the first branch should be approximately 1/3 the height of the tree (Figure 2), and the tree should be placed in the pot, 1/3 the way across the pot (for a rectangular pot). I have also read that 2/3 or the foliage should be in the top 1/3 of the tree.

If we look at branches, the distance from one node to the next ideally would be 2/3 the distance of the previous one to give the most pleasant look.

If pictures of specimen bonsai are examined, the ones found most pleasing have the first branch about 1/3 the height of the tree. I checked several pictures and for me this holds up very well. Due to the spread of the nebari it is difficult to place the center of the trunk and the 2/3:1/3 placement in a pot is more difficult to show.

Now we all know that Mother Nature doesn't place branches at the 'proper' place, therefore the bonsai artist must use his skills to manipulate the tree to give the most pleasant look. If the first branch is not at the 1/3 height, a limb may be able to be bent to the ideal height, the tree height may be adjusted, or the golden section rule may be ignored. As with a lot of other guides in bonsai, we must just deal with what we have. Also when we deal with the various styles used in bonsai, use of the golden section would destroy the style, as in the literati style.

We must also remember as Tom McCurry says, the bonsai 'look' must please the artist. If the tree doesn't follow the guidelines, design the tree to what pleases you. Although there are plenty of bonsai rules, at the end of the day, it's your tree.

1. Naka, J.Y. Bonsai Techniques I, Bonsai Institute of California, Whittier, CA, 1999, pg. 15-17
2. Victoria Jahn Ed. Simon & Schuster's Guide to Bonsai, Simon & Shuster Inc. , New York, 1990, pg 52
3. De Groot, D., Basic Bonsai Design, American Bonsai Society, Toledo, OH, 1995, pg. 16