Fibonacci and Bonsai

golden1In my previous blog, which focused on the seven Da Vinci principles, I started to think a bit more about Da Vinci’s use of mathematical proportions in Art and especially Bonsai.

Leonardo Da Vinci has long been associated with the goldleonardo_fibonaccien
ratio. Da Vinci created the illustrations for the book, The Divine Proportion, by Luca Pacioli. It was written in about 1497 and first published in 1509. Pacioli was a contemporary of Da Vinci’s. In the book, Pacioli writes of mathematical and artistic proportion, especially the mathematics of the golden ratio and its application in art and architecture. The golden ratio and the Fibonacci numbers or series, comes from the work of Leonardo Bonacci, also known as Fibonacci, an Italian mathematician. Many writings exist about Fibonacci, Da Vinci and the applications of the mathematical ratios and numbers in art, nature and a few other areas as well. I will focus a bit more on Bonsai design and am hoping to bring some of this “divine proportions” to Bonsai design. First a bit more about the Mathematics involved.

As said, much has been written about the Fibonacci numbers and its appearance in various patterns of spirals in especially leaves and seeds. The Fibonacci series is a series of numbers formed by adding the previous two numbers together to get the next number.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.

In Bonsai, the theory is that when you cut a stem, tfib01he stem puts
out two growing shoots during one growth period. This could be more (pines or where whorls are formed), but the theory is that we keep two stems. If you keep on cutting these shoots as per the illustration, you will increase the number of smaller shoots over time. This is called increasing the ramification of your Bonsai tree. The theory is that if it is done according to the numbers as found in the Fibonacci series, that you will have a visually more pleasing appearance.

Further to this, the ratio of two successive numbers in the Fibonacci series can be calculated by dividing each number by the number before it. The ratio seems to be settling down to a particular value, which is called the golden ratio or the golden number. The value of this ratio is approximately 1.618034. This is often represented by the Greek letter Phi.

The arrangement of stems around the trunk could also be of interest here. When the golden ratio is used in a mathematical equation in relation to circles, it is found that stems / branches could be in a visually ideal position if they are 137.5 degrees or 222.5 degrees (make up 360 degrees in a circle) apart from each other.


For completeness sake, I will also try to explain the Fibonacci rectangles and spirals here. If we start with two small squares of size 1 next to each other and we then place on top of these two squares a square of size 2 (1 + 1) and then carry on like that as illustrated in the diagram, rectangles of various sizes can be drawn. These are the Fibonacci rectangles.


A spiral can also be drawn in the squares, a quarter of a circle in each square. These spirals are seen in the shape of shells and also in the arrangement of seeds in flowering plants. The spiral in the squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. Points in the spiral are 1.618 times as far from the centre after a quarter turn. In a whole turn the points on a radius out from the centre are 1.6184 (6.854) times further out than when the curve last crossed the same radial line.

Am I the only one seeing spirals in these examples of deadwood?


Where does this all fit in? That depends on you and how far you want to take this. For me it is just interesting and something that can be kept in the back of the mind while styling Bonsai trees. I will be looking a bit closer at the shape of curves / spirals in future and I will also be looking at where I place especially the first two branches. Maybe that 137.5 degrees angle is not so silly after all?

Leonardo Da Vinci and Bonsai

Yesda-vinci-profile, you are correct. Da Vinci never designed Bonsai, as a
matter of fact, there is no evidence to indicate that he even saw a Bonsai. What is this article about then, you may ask. In my quest to understand more about the creative process and trying to better myself as a creative person, I explore a lot of different theories and principles and try to open my mind to as many influences as possible to find my own rightful place within the creative space. My chosen field is that of Bonsai creation. We all know that the roots of Bonsai are firmly established in Chinese and Japanese culture. We also know that Bonsai as a creative activity has now spread to all corners of the world and it is clear that other influences are being incorporated into the creative concepts serving as foundations for Bonsai. The recent Artisans Cup in the USA is an example of this, but there are many others spread throughout Europe. A very distinctive style is emerging out of Africa as well. Why not look at the seven Da Vincian principles as a source of inspiration or direction in relation to Bonsai design?


There are seven Da Vincian principles that can be used to give direction to a person’s life in exploring creativity. These are:

  • Curiosity – An insatiably curious approach to life and an unrelenting quest for continuous learning.
    • This principle can directly be implemented in the life of any Bonsai artist. Being curious about life in general and then also the creative process surrounding Bonsai cultivation, is very important in establishing the next steps for growth. Asking questions, exploring concepts, trying new designs on the creative side, but also being curious about the horticultural process underlying Bonsai are all part and parcel of the Bonsai artist’s grounding. I encourage artists to make notes, read often, watch videos, talk to other Bonsai artists, attend conferences and to never stop asking questions. Leonardo da Vinci’s notebooks amount to over 7000 pages.
  • Demonstration / Independent Thinking – A commitment to test knowledge through experience, persistence and willingness to learn from mistakes.
    • As Bonsai artists we are involved in a practical application of centuries of knowledge and techniques. The last few decades saw an explosion of information and it is our job as artists to make sense of all of this and to apply what suits your own philosophy and style to your own designs. Yes, you can copy other people, but I am convinced that real satisfaction and personal growth comes when you use all of your knowledge and skill and create something new. This can be applied in a very practical way in Bonsai as each plant is unique.
  • Sensation – The continual refinement of the senses, especially sight, as the means to enhance experience.
    • According to Da Vinci, experience is delivered through the senses: sight, sound, touch, taste and smell. Bonsai creation is all about using your senses, especially sight, touch and smell. The more sense you can incorporate into your interaction with your trees, the more fulfilling the experience. You learn and experience through your senses. One of the problems that I experience is that modern day living is too noisy and very visual. So much so that it leads to sensory overload. Something that I need to work on more is to refine my sensory experiences by deliberately focussing on those sensory stimuli that matters during the creative process and to block or filter the “noise” out.
  • Smoke (Sfumato) / Embrace uncertainty – Becoming open to the unknown. A willingness to embrace ambiguity, paradox and uncertainty.
    • In her blog, Andrea Balt mentions that we cannot “make use of our full creative potential without the ability to embrace uncertainty. An open, always questioning mind is our creativity’s best ally.” In Bonsai there are many uncertainties, especially when it comes to severe styling processes. Will a tree flourish after severe root pruning, how will it react to wiring, did I cut too much, etc.? There are many different thoughts and ideas around when things should and should not be done when it comes to repotting, lifting yamadori, using chemicals, etc. Embrace the unknown.
  • Art and Science – Whole-brain thinking. The development of the balance between science and art, logic and imagination.
    • Your brain operates as a whole organ. Society has been telling us for a long time that one half is more dominant than the other. As an artist you are using both halves. You are programmed to be creative, it is just a case of exploring and finding your creative roots. The capacity is already there. This principle can also relate to the two parts of Bonsai creation. There is the creative / art side and then there is the science / horticulture side. Both are equally important. Embrace both. It could be that one of these attracted you more to Bonsai than the other. It is your task as a creative Bonsai artist to get yourself up to speed with both aspects. A beautifully created dead Bonsai is as unappealing as an ugly, living Bonsai. Study the art of science and study the science of art. Da Vinci was ambidextrous, something you can try to help with the development of the motor functions of both halves of your brain.
  • The body (mind-body care) – The cultivation of grace, ambidexterity, fitness and poise. Balancing the body and mind.
    • This one is self-explanatory. A healthy body is needed to dig trees, cultivate the trees, carry them around, etc. There is more to it than just being fit enough to do what you have to do around your trees. There is also the healthy body – healthy mind concept that is so important in the creative process. This principle was taken very seriously by Da Vinci in his day-t-day life. Mild exercise and good nutrition with ample rest serve as foundations for what is needed.
  • Connection – A recognition of and appreciation for the interconnectedness of all things and phenomena. Systems thinking.
    • Da Vinci said: “Learn how to see. Realize that everything connects to everything else.” The growing medium that you plant your Bonsai in, the aesthetic effect of pot and tree, the light that all leaves get, the link between water and nutrition, your interaction with the tree, everything is connected. You are connected to the tree and everything the tree is connected to.

For you to be a better Bonsai artist and based on the Da Vinci principles, in summary: Be curious and never stop learning. Test yourself and be independent in a practical way. Use all of your senses when interaction with your creativity. Embrace the unknown and explore. Use your logical and creative powers. Look after yourself and realise that everything in life is connected.


Gelb, Michael. J. 2014. Creativity on Demand. How to ignite and sustain the fire of genius. Colorado: Sounds True

Sphagnum Moss and Bonsai

Sphagnum moss is used in a variety of ways in conjunction with growing Bonsai. The main characteristic of Sphagnum that makes it useful to Bonsai growing is its ability to hold a lot of water. Sphagnum moss plants can hold 16–26 times as much water as their dry weight.sphagnum

Sphagnum also does not decay easily as it contains phenolic compounds in its cell walls. It therefore can be used in applications that can take a long time to complete. It does not break down as easily as other growing mediums. Sphagnum originating from peat bogs is known to aid in preservation of substances due to the phenols, but also due to the fact that it grows in an anaerobic environment. Less oxygen means less decay. It is a well-known fact that bones and the remains of living organisms that end up in peat bogs tend to be quite well-preserved after a long period of time.

These mosses can also acidify its surroundings. This is something to keep in mind when the moss is used as a growing medium or part of a growing medium. It is therefore advised that sphagnum moss is not used with plants that prefer a more alkaline environment.

It is also a natural antiseptic. Sphagnum Moss was used extensively during the World Wars in field dressings to pack out wounds and under bandages to keep wounds clean.

Harvesting these mosses is hard work. It also comes from areas that could be ecologically very sensitive. These facts can contribute to the fact that good quality sphagnum moss can be quite costly. There are environmental concerns about the sustainability of moss harvesting.

In Bonsai cultivation, Sphagnum moss is mostly used in conjunction with roots and root growing. It is a well-known fact that when trees are harvested from nature (Yamadori) that it is a good idea to pack wet sphagnum moss amongst the roots. The reason for this is obvious due to the water holding abilities of the moss. During a recent dig, I not only packed the moss amongst the roots, but also wrapped it in hessian before it went into a plastic bag for the trip home. This not only held the moss in place, but aided in keeping the moisture in. I am also at the moment experimenting with two similar trees, one with Sphagnum in the soil mix and the other with normal soil mix. The question to be answered is whether the one shows more and better root growth than the other one. More on the results in a future blog.

Two Olives

Sphagnum moss is also used for air-layering. Again the moss is tightly packed around the wound created for the purpose of providing a medium for root growth. The water-holding capacity is again the main reason for using the moss. In this case it also helps to allow air flow through the medium which will aid in root growth as well.