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The Mathematics of Flowers

Gina Margolies

Birthday Blossoms Mathematics of Flowers01

Mother Nature produces tremendous beauty. Every day the natural world presents us with its gifts. Flowers are just one of those gifts and they bring so much joy into our lives. Many factors come together to produce a thing of beauty like a flower, including mathematics. Most people think of numbers when they hear the word mathematics. But math is actually a language, a vocabulary of symbols and words used to express mathematical concepts. Flowers also speak a  type of language. Officially called floriography, the language of flowers refers to the idea that each flower holds a meaning or message which can be used as a way to communicate certain sentiments. What happens when the two collide? You get the mathematics of flowers, of course. What does math have to do with flowers? More than you would think.

The Golden Mean

Birthday Blossoms Mathematics of Flowers04Most math concepts are over my head and Fibonacci numbers are no exception. Fibonacci sounds more like a funny way to say that you told a few too many white lies than something mathematical. I won't attempt to fully explain it, but rather summarize enough to make the connection between math and flowers clear. The word Fibonacci refers to a nickname given to an Italian mathematician. He did a lot of amazing work which is beyond me, but, long story short, there is a concept in math called the Fibonacci numbers. This term refers to a sequence of numbers called, naturally, the Fibonacci sequence, in which every number is the sum of the two numbers before it. This would be a Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144      

One plus one is two, one plus two is three, two plus three is five, three plus five is eight, and so on. The Fibonacci sequence is thought to approximate something called the golden ratio or the golden mean. This is yet another complicated mathematical concept, but to put it as simply as possible, it refers to a relationship or proportion, i.e. ratio between two numbers. The Fibonacci numbers have been used in design fields such as art and architecture because the proportion, i.e. the golden mean, is thought to result in aesthetically pleasing work. Very interestingly, this occurs naturally in many aesthetically pleasing things, including flowers. It turns out that the Fibonacci numbers can really be considered natures numbers.  

The arrangements of leaves on a plant stem and petals in a flower head represent successive numbers in the Fibonacci sequence. For example, seeds on the head of a sunflower are arrayed in two sets of spiral rows in a glorious Fibonacci sequence. Look closely and you will see what is so glorious about it.  

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Plants grow the way they do in order to maximize efficiency. In flowers, the arrangement of the petals is set to maximize the amount of space for each petal or to make sure that each petal has sufficient exposure to sunlight. Many flower heads are densely packed with petals, making these arrangements critical to the success of each individual petal and thus the success and beauty of the flower as a whole. So the arrangement of the petals on a black-eyed-Susan or the scales on a pineapple is not random. To the contrary, it is a precise mathematical arrangement that allows all of the elements to work together to ensure the success of the whole by achieving the golden mean. I might not be able to discover high-level mathematical concepts or produce a perfect sequence of numbers, but I can appreciate the beauty these arrangements communicate. Mother Nature could have shown Signor Fibonacci a thing or two.  

Floras Clock

Another mathematical concept reflected in nature is that of time - time of the day, time of the month, and time of the year. Time seems such a simple idea, just a numerical system or way of measuring the passage of, well, time. Before the invention of mechanical clocks, hourglasses, sundials, and iPhones, people needed some way to mark or note the time in order for society to function. How could anyone arrange a meeting without some way of denoting what time the get-together would be held? Because machines hadnt yet been invented, people used what they had. To tell time they relied on nature.  

The most obvious way nature indicates the passage of time is with the sun. The sun rises and sets in a predictable pattern. However, the sun is a less than perfectly precise way to denote time, so people looked for more exacting ways to do so. There were many different ideas and thoughts on how to do this and, as necessity is always the mother of invention, many brilliant ideas. One that I love is called Floras Clock.

Floras Clock was created by an amazing Swedish chap named Carolus Linnaeus. Doctor Linnaeus is famous as a botanist, amongst other things. He is most known for normalizing the modern system of naming organisms called binomial nomenclature. He also conceived of the idea that became Floras Clock. Flowers were used to mark time, in hours, weeks or months, according to when they bloom. Some flowers open their petals at a particular hour each day. Linnaeus decided to make a list of these. Using this list we can reckon the hours in flower language.

Linnaeusbasic concept was to plant a garden that, based on the time of day the various plants opened and closed their flowers, would tell time. Linnaeus named his idea the flower clock, often called Floras Clock after the Roman goddess of flowers. Many plants flower on a circadian rhythm. The term refers to a biological process that takes about 24 hours. Some flowers have a particularly strong circadian rhythm and open at the exact same time every day. So Linnaeus created a list of flowers that exhibited this quality and arranged them to mark the passage of 24 hours, just like a clock does for us today.

There are different lists of the flowers that make up the clock. The one I like best is found in the book The Flowers Personified by J.J. Grandaville. According to this book, the flowers that Linnaeus determined to work best for his clock were:

Midnight - large-flowered cactus
one oclock - alpine sow-thistle
two oclock - yellow goats-beard
three oclock - scammony convolvulus
four oclock - smooth crepes
five oclock - day lily
six oclock - hawkweed
seven oclock - small-cape marigold
eight oclock - red pimpernel
nine oclock - field marigold
ten oclock - Egyptian fig marigold
eleven oclock - star of Bethlehem
noon - ice plant
one oclock - profuse-flowering pink
two oclock - mouse-ear hawkweed
three oclock - dandelion
four oclock - madwort
five oclock - marvel of Peru
six oclock - geranium
seven oclock - naked-stalked poppy
eight oclock - upright bindweed
nine oclock - flax-leaved bindweed
ten oclock - cypress vine
eleven oclock - night-flowering catch-fly


Linnaeus actually made a garden plan which indicated where each flower should be planted in order to tellthe time of day. There was one rather obvious problem with Floras Clock. The opening and closing of a flower is generally related to sunlight, weather, and the seasons. So the geographical location of the garden would impact upon the flowering times and hence Linnaeusclock. Apparently this didnt stop some gardeners from trying to follow Linnaeusplan to construct a Floras Clock and some even seem to have had some success at it.  

Floriography and Floras Clock

We obviously dont need to rely on flowers to mark time, but I still like the idea of doing it. I am not the only one. During the Victorian Era, the language of flowers, also called floriography, became very popular in Europe and the United States. This language was based on the belief that each flower has a hidden meaning or message. If one knew all of the hidden meanings, one could use flowers to send secret messages. The Victorians did just this. Books like Grandavilles listed the meanings of the flowers and people used them to decode the bouquets they received. Floras Clock was used to mark the time in secret messages. If your bouquet contained a profuse-flowering pink, you could use Floras Clock to determine that your secret tryst was set for one oclock in the afternoon. A large-flowered cactus meant meet me at midnight.  Microsoft Outlook events seem quite boring in comparison.  

Floras Calendar

Flowers could also help the Victorians mark the time of year. For example, The Flowers Personified also held a Floras Calendar. The idea of it is perfectly simple. Nature herself has arranged this part of the calendar by causing particular plants to bloom at certain times of the year. The Flowers Personified provided this list:

The Calendar of Flora
January - Black Hellebore
February - Mezereon
March - Alpine Soldanella
April - Early dwarf Tulip
May - Common Dropwort
June - Cornpoppy
July - Centaury
August - Scabious
September - European Cyclamen
October - Chinese St. Johnswort
November - Serrated Pallasia
December - Smooth Lopezia


One could indicate that one would return in April by sending an early dwarf tulip or that one wouldnt wait past the end of the year by including a smooth lopezia, a flower that is much lovelier than its awkward name suggests.  

Math + Nature = Beauty

Whether it is the arrangement of the seeds on a purple English daisys gorgeous flower head or the fact that the marvel of Peru is also known as the four oclock because of the time it opens every day, the collision of mathematics and flowers produces not only efficiency in nature, but wonder and amazement in us. Even if you cant remember what you just read about Fibonacci, your eye can tell you that the precise arrangement of the petals on your favorite cabbage rose is a beautiful thing to behold.

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