The Geometry of Flowers
“There, with irresistible force and tremendous effort, nature formed the flowers and equipped them for works of love.” ~ Goethe
“If we look closely at a flower, and likewise at other natural and man-made creations, we find a unity and an order common to all of them. This order can be seen in certain proportions which appear again and again, and also in the similarly dynamic way all things grow or are made – by a union of complementary opposites.
The discipline inherent in the proportions and patterns of natural phenomena, and manifest in the most ageless and harmonious works of man, are evidence of the relatedness of all things.” ~ Gyorgy Doczi, The Power of Limits
The structure of flowers is some of the most obvious (and beautiful) examples of how geometry is involved in plant life.
Most regular (actinomorphic) flowers and inflorescences have radial symmetry.
Keep in mind that none of this geometry works separate from the other. The geometry of the seeds, roots, stems, leaves, flowers, fruits and even the cellular and molecular structure all work together to form the plant. We have interrelated, interpenetrating and oscillating geometric wave fields that form the flower, just as the wave fields form all matter in physical reality. At the basis of many of these structures lay the golden ratio and Fibonacci Sequence.
Yet also keep in mind that the Lucas sequence numbers (2, 1, 3, 4, 7, 11, 18, 29, 47…) play a role in 1.5% of plants.
“Each flower is ‘in process’ at all times. This means that it usually spirals into being, maintains a full-faced ‘aspect’ for a time, then proceeds to withdraw life from this form as it contracts and often spirals back down into the soil.”1
The Spirals of Flower Growth
The spirals of flower growth are logarithmic and also equiangular, since the angle they describe with the radii remains always the same.
Segments representing consecutive stages of growth can be rotated around the center until they completely overlap, like a folded fan, proving that old and new stages of growth all share the same angles and the same proportions.
Credit: Gyorgy Doczi – The Power of Limits
The golden section in one of the daisy’s spirals. Each stage of growth shares the same proportions (see shaded triangles at right). Credit: Gyorgy Doczi – The Power of Limits
Gyorgy Doczi writes in The Power of Limits: “Patterns generated by spirals moving in opposite directions are frequent in nature. Here they concern us as special instances of a more general pattern-forming process: the union of complementary opposites.”
Follow the different curvatures of these spirals through the squares formed by the radiating and rotating sets of lines in diagram a. One can see that spiral A moves from one circle to the next as well as from one radius to the next within a single row of squares. We will call this a curvature of 1:1, signifying that the rotational and the radiational components of growth are equal. Spiral B moves through two squares while reaching from one radius to the next, crossing two circles: a curvature of 1:2. In a similar manner one can say that spiral C has a curvature of 3:1, while spiral D approximates 5:1.
“In spite of their differences in curvature, Doczi tells us, “all of these spirals share the qualities of being logarithmic and equiangular, through all stages of growth.”
Credit: Gyorgy Doczi – The Power of Limits
The Parts of the Flower
It is helpful here to familiarize yourself with the parts of a flowering plant if you have not already.
These parts include:
- The seed represents the most extreme state of contraction and inner development in the plant.
- The roots are part of the branching structure of the whole plant. The roots reach for darkness as the stems and branches reach for light. These two aspects represent an alternation of function.
- These are the first organs, called seed lobes, nuclei, seed laps or seed leaves. In many plants they are very leaf-like.
- Stem leaves
- Calyx (sepals)
- Goethe writes, “The leaves of the calyx are the same organs that appeared previously as the leaves of the stem; now, however, they are collected around a common center, and often have a very different form. In several flowers [such as the cornflower] we find unaltered stem leaves collected in a kind of calyx right under the flower.”
- He continues, “This, then, is how nature formed the calyx: it collected several leaves (and thus several nodes) around a central point, frequently in a set number and order…Thus, nature does not create a new organ in the calyx; it merely gathers and modifies the organs we are already familiar with, and thereby comes a step closer to its goal.”
- Corolla (petals)
- Goethe writes, “The organs were contracted in the calyx, but now we find that the purer juices, filtered further through the calyx, produce petals that expand in a quite refined form to present us with new, highly differentiated organs.”
- He continues, “Sometimes nature skips completely over the organ of the calyx, as it were [such as in the tulip] and goes directly to the corolla. We then have the opportunity to observe how stem leaves turn into petals.”
- Sometimes there are secondary corollas which are smaller than primary corollas. “The formation of petals occurs by expansion, but secondary corollas are formed by contraction.”
- Nectaries are a transitional form between the petals and stamens that occur in some plants. They are glandular organs that secrete nectar, a sugar-rich liquid which attracts pollinators.
- Goethe writes, “The so-called nectaries may also appear as independent parts; these sometimes resemble the petals in form, and sometimes the stamens…Other nectaries appear as stamens without anthers…we also find them alternating with the stamens in a whorl.”
- Stamens (Male)
- The leaf ‘organ’ contracts to form the simpler form of a stamen.
- Goethe writes, “Thus a stamen arises when the organs, which earlier expanded as petals, reappear in a highly contracted and refined state.”
- The stamens are pictured below as the 6 narrow green stalks with the dark brown anthers on top. The anthers release the pollen (sperm).
- Pistil (Female)
- What is true of the stamen is also true of the style of the pistil. They are formed by a contraction of the petals.
- The pistil is the single organ in the center of the corolla. It tapers down into the ovary and receptacles. This is where pollen fertilizes the ovary of the flower.
- The greatest expansion of the archetypal organ is the fruit.
- Goethe writes, “With these observations in mind, we will not fail to recognize the leaf form in seed vessels…Thus, for example, the pod may be viewed as a single, folded leaf with its edges grown together; husks, as consisting of leaves grown more over one another; and compound capsules may be understood as several leaves united round a central point, with their inner sides open toward one another and their edges joined…Nature masks the resemblance to the leaf mainly by forming soft, juicy seed vessels, or hard, woody ones.”
Yet Keith Critchlow reminds us, “to describe flowers purely ‘rationally’ is a further example of the trivialization of natural beauty let alone the display of the unfolding which is the celebration of all living things.”
The bud is the next ‘point’ of departure, after the seed.
Maple bud, Lotus bud, Magnolia bud & Rose bud
Recall the flower life cycle: the seed is the first ‘point’. It unfolds into a ‘line’ – the first shoots and roots. This line then becomes a plane with the leaves unfurling. The plane then becomes a solid. The solid is represented by the flower and fruit.
The bud, then is the next ‘point’. Critchlow writes, “Like a seed in shape, the green bud divides to open to the life-giving sunlight.”
He adds, “Within the growing flower bud are the enfolded petals of the emerging flower. These are neatly packed in different ways – most often in a spiral manner of some kind.”
Opening of a Lotus, Plumeria, Rose & Hibiscus – Each flower spirals into being.
As a flower bud unfolds the principle of spiraling is prevalent. Keith Critchlow refers to spiraling as ‘turning into being’.
This spiraling is more marked in some flowers than others. St John’s Wort, Periwinkle and the Rose show a preponderance to unfold in a spiral fashion; the Water Lily and the Passion Flower less so.
On page 206 we see the bud as having a spherical base and opening like a Gothic arch.
Credit: Keith Critchlow
Basic flower petal shape commonly adheres to the Vesica Piscis shape. This is illustrated by Keith Critchlow on page 202 in The Hidden Geometry of Flowers.
As Critchlow states, “Remarkably, a leaf has a very similar, but reversed, profile of geometry: the stalk is at the opposite end. This is not easy to explain but is geometrically precise in both cases. Goethe’s theory of leaf transformation comes to mind.”
This is illustrated on page 203:
Other petals adhere to the equilateral triangle. On pages 204-205 Critchlow shows how the base of an equilateral triangle marks the maximum width of the petal.
Flower “faces” orient in three ways:
“Flower faces ‘shine’ in more than a metaphorical sense just as beauty becomes an inner radiance in any living being.
The beauty of the face of a flower reaches far deeper than its connection to the insect world. It is the effect on the higher animal kingdoms, ourselves in particular, which goes far beyond a merely material functionality. Flowers can be considered the bearers of brilliant ideas. Just consider how many great poems and paintings have originated from experiencing flowers.”2
Claude Monet’s (1840-1926) Water Lilies Series
Vincent van Gogh’s Sunflowers Series
Vincent van Gogh’s Almond Blossoms
Paul Gauguin’s The Painter of Sunflowers
American artist Georgia O’Keeffe’s (1887-1986) many flower paintings
Japanese artist Makoto Azuma’s (1976 –present) botanical sculptures
Credit: Choo Yut Shing
Dutch painter Rachel Ruysch’s (1664-1750) many flower still-life’s
Salvador Dali’s The Rose Meditative, Female Figure with Head of Flowers, Woman with a Head of Roses
Leonardo da Vinci’s many flower (and Flower of Life) drawings
Japanese artist Utagawa Hiroshige’s (1797- 1858) many flower paintings
Japanese artist Katsushika Hokusai’s (1760-1849) many flower paintings
Symbolism of Flowers
Most people are quite familiar with the concept that different flowers hold different symbolic meanings. These relate to the notion that through flower symbolism we can express what we want to say (and may not be able to speak out loud).
We will cover a few examples now.
Anthurium – open, heart-shaped flowers and tropical disposition; symbolizes hospitality
Birds of Paradise – resembles a colorful bird in flight; symbolizes joyfulness and an upward flight to paradise
Carnations – love, fascination, admiration, deep love (dark red), mother’s undying love (pink)
Chrysanthemums – optimism & joy
Daffodils – rebirth & new beginnings
Gladiolus – strength & moral integrity
Irises – a link between heaven and earth; faith, valor & wisdom
Lilac – the first emotions of love & youthful innocence
Lilies – purity; friendship & devotion; sympathy
Orchids – love, luxury, beauty & strength
Roses – love & passion
Sunflowers – sunshine, warmth, happiness, adoration & longevity
Tulips – passion, elegance, grace
We will now take a visual tour of the geometry of flower petals.
“Whatever else we might conclude about symmetry it is valuable to acknowledge Plato’s first principle of the triad of Sameness, Otherness and Being. In terms of flowers this means that no two petals, let alone leaves, are ever exactly the same in a single bloom yet they are usually clearly identifiable as an Oak, Ash or Willow. In short all scrutiny of the natural world will entail Sameness and Otherness.
In short we can experience a host of Tulips though common experience reveals that each will be slightly different: each is unique yet unquestionably a Tulip. Paradox alone can achieve unity.”3
As Samuel Colman writes in Nature’s Harmonic Unity, “Nature herself is rarely exact in every portion of any one flower, yet if she presents us with an object numbered by five, such as the laurel blossom, for instance, no possible reasoning offered can change the obvious fact that she intended the form to be pentagonal even if it is several degrees removed from a symmetrical measurement, for the law of uniform growth is expressed by the average of geometric correlations and not by single disconnected measurements. However exact her principles may be in the abstract, or however clearly to be proven by scientific methods, it is the appearance that she seems to take most delight.’
“What is evident in the geometry of the face of a flower can remind us of the geometry that underlies all existence. Studying the geometry of flowers is therefore a powerful way to reconnect us with the idea that we are all one.”4
Fibonacci & Petal Number
The Fibonacci sequence leads towards the Golden Ratio. This is covered in great detail in other articles. Its sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…etc.
Keith Critchlow reminds us, “This series of numbers relates not only to geometry and music but also to the flower world.”
- Anthurium (1); Calla lily (Aspidistra – 2)
- Enchanter’s Nightshade (1), Euphorbia (2 & 3)
- the beginning of every fruit and vegetable
- Iris (1), Lilies (2), Trillium (3), Snowdrop (4), Tulip (5), Tulip-tree (Liriodendron – 6), Tradescantia (7)
Many sixfold flowers actually have a double three structure of petals such as the lily and tulip.
“The pentagon and pentagram, like all patterns, are defined by their limits. Incorporated in the harmonious patterns of fruits and flowers, they exemplify an epigram attributed to Pythagoras, that limit gives form to the limitless.” ~Gyorgy Doczi
A pentagon divided into ten equal parts creates 10 3-4-5 Pythagorean triangles.
Credit: Gyorgy Doczi – The Power of Limits, 1981
- Five petaled flowers are the most common.
- All edible fruits have 5-petaled flowers.
- Pictured below: tomato (1), squash (2), potato (3), okra (4), lime (5)
- Many wildflowers, Delphiniums (1), Larkspurs (2), Columbines (3), Geraniums (4), Pansies (5), Primroses (6), Rhododendrons (7), balloon flower (8), original Dog Rose (9), Herb Robert (10), Periwinkle (11), Borage (12), Plumeria (13), Hibiscus (14), Cowslip (15), Euphorbia Ramipressa (16), Magic Star Flower (17), Oleander-Laurel (18) & Thunbergia (19)
- The largest known flower in the world is a five-petaled: the rafflesia (Rafflesia keithii)
Credit: Col Ford & Natasha de Vere
Keith Critchlow shows up that as the sepals of the Rose unfold, they draw out the five-pointed star in one continuous movement. See page 134 in The Hidden Geometry of Flowers.
- Lesser Celandine (1), Zinnia (2), Bloodroot (3), Cosmos (4 & 5), Clematis(6), Lotus
- Globe flower, Ragwort (1), Mayweed (2), Corn Marigold (3), Chamomile (4)
- Heleniums (1), Asters (2 & 3), Chicory (4), Doronicum (5), some Hawkbits, Black-eyed Susan (6), Osteospermum (African Daisy – 7 & 8) many wildflowers
- Common Daisies (1), Plantain-leaved Leopards bane (2), Gaillardias (3)
55 Petals & 89 Petals
- Michaelmas Daises
“Flowers, as both phenomena and noumena, offer their being as forces of levity through their Beauty, Truth and Goodness. It is good for us to regard their existence with gratitude and awe.”4
Fibonacci Spiraling Flowers
“Patterns generated by spirals moving in opposite directions are frequent in nature.” Gyorgy Doczi tells us, “Here they concern us as special instances of a more general pattern-forming process: the union of complementary opposites.”
- Chrysanthemum (1), Dahlia (2), Unknown (3), Protea (4), Buttercup (5), Camellia (6), Daisy spp? (7), Lotus (8), Dandelion (9), Rose (10), Sunflower (11), Torch Ginger (12)
“Every daisy and sunflower is a window on the infinite, as are apple blossoms and the flowers of other trees and bushes bearing edible fruits. These grow according to the pattern of the pentagon and its extension, the pentagonal star or pentagram in which neighboring lines relate to each other in the dinergic golden relations of neighbors.” ~ Gyorgy Doczi
Credit: Gyorgy Doczi – The Power of Limits, 1981
Other Petal Geometry
Tetrad – 4
- poppy (1), spring cress (2), cutleaf toothwort (3), forsythia (4), mint, western wallflower (5), primrose, fireweed (6), fringed gentain (7), green gentian, Rocky Mountain Bee Plant, Clematis (8), greater celedine (9)
- Dogwoods – It is interesting to note that the dogwood flower petals aren’t ‘true’ petals but modified leaves called bracts that surround a cluster of about 20 tiny flowers. This blends nicely with Goethe’s theory that petals are transformation of leaves.
The balloon flower has both 4-point and 5-point geometry.
The poppy is a unique example of four-petaled geometry. Within the seed pod of the poppy there is further geometry. This geometry is wide-ranging and there is no apparent ‘rational’ or ‘scientific’ reason for such diversity in geometry of the seed pod.
Some of the geometries include sections of: 7, 8, 9, 10, 11, 12, 13, 15, 16, 17 and 19.
Credit: Keith Critchlow, page 418 in The Hidden Geometry of Flowers.
Hexad – 6
- 6-petaled flowers: gladiolus (1), Crocuses (2), Daffodil (3), some Clematis (4), Starflower (5)
Heptad – 7 – very rare
- 7-petaled flowers: wood anemone (1), starflower (Lysimachia borealis – 2), Primula, some tulips
Ennead – 9
- 9-petaled flowers: magnolia grandifora (1)
- certain cacti (2)
Decad – 10
- Passion Flower, Mountain laurel (10-point geometry within 5 petals)
“The number we take from flowers can only be a reflection of a universal truth.”5
“Flowers, the embodiment of beauty, feed the developing intellect by their symmetries, colors, geometries and forms. They hold much in common with musical proportions as the same archetypal number pattern informs both. The soul’s very nature was proportion according to Plato’s Timaeus.”6
Keith Crithlow’s book The Hidden Geometry of Flowers is essential for understanding and contemplating the geometric nature of flowers.
He offers a series of drawings from pages 208-287 that illustrate these concepts very precisely and beautifully. This book is highly recommended.
What he shows is that the geometry of flowers is far more than mere petal number. It is shown that the flower itself is built upon a matrix of geometry. Important features fall at crossings and lines of the geometry. This backs up the scientific model presented in Cosmic Core. All matter is composed of a symphony of interrelated waves. These waves oscillate back and forth from time/space to space/time creating interference patterns. The interference patterns create (by the very nature of reality) geometry upon which vortices of photons, subatomic particles, atoms, molecules and cells coalesce upon. These ‘particles’ of matter concentrate upon the nodes (crossings), edges and corners of the geometry. This is seen in subatomic particle interactions, atoms, molecules, minerals, snowflakes, crystals, plants, flowers (as we see here), insects, animals, humans, features of planets, planetary orbits, galaxies and galactic clusters. We have the same principles and processes at work on all scales of nature. Nature is consistent in her laws.
Without undertaking geometric analysis of these life forms, the underlying matrix of geometry upon which life is built is difficult to see in all its precision and complexity.
Critchlow states, “Flowers perpetually remind us that at the top of our list will inevitably be the mystery of life itself. We are immensely clever and erudite in telling what life does and how it constructs but completely inadequate in saying what life is. We irresistibly manipulate living organisms predominantly for our own narrow benefit. We are losing the ancient wisdom of the wise farmer who works with the natural world. We are thankfully extremely good at ‘mending’ living things when they are in ill-health. But this still does not bring us to a definition of life itself. Primarily we have to live it.”
Here are nine examples of Critchlow’s geometric analysis of flowers:
Samuel Colman presents us with several other geometric analyses of flowers in Nature’s Harmonic Unity. These can be found on pages 101-112 and include: milkweed, henbane, parnassia, jonquil, tiger lily, Easter lily, zygadenus elegans, loosestrife, hypericum, sarracenia purpurea, and Rudbeckia.
“The Lotus flower has been held sacred by the greatest number of people over a long history, from Egypt to China and Japan…That the Lotus has been called ‘enlightenment itself’ is not surprising as the Lord Buddha sits on the seedpod in the center of the unfolding petals quite without disturbing the flower! An example of profound symbolic meaning. As the flower has arisen above the mud and water of the material world so the ‘Buddha Nature’ rises above it. From light to enlightenment.”7
In ancient Egypt the number 1000 (10 x 10 x 10) is represented by the lotus flower:
As we learned in Article 181, the lotus needs both wind and water for seed dispersal.
“The center of the large, waterlily-like flower is occupied by the enlarged floral axis with the shape and appearance of a shower head. Sunken into this structure are the individual carpels, each of which develops into a single-seeded nutlet. The nutlets remain embedded in their compartments in the floral axis until they are ripe, at which point the compartments open wide enough for them to pass through. When the long fruit stalks are shaken by the wind, the nutlets are flung out and thrown into the water where they immediately sink to the bottom. If the stalk of the fruit breaks off, the nutlets may be dispersed over a longer distance. The ‘shower head’ is wider at the top than the bottom and so lands face down on the water. Its spongy air-filled tissue ensure that the fruit floats on the surface, releasing some of the fruitlets instantly, and others later as the remaining air escapes from the chambers.”8
Interestingly, the oldest living seed whose exact age could be established belongs to the sacred lotus, at 1,288 (+/- 250) years.
Freddy Silva writes in Secrets in the Fields, “In Hindu philosophy, the lotus is the ‘flower of light’ a symbol of matter and spirit, cause and effect. Its leaves, flowers, and fruit are said to form the figure of a circle, so it is considered a symbol of perfection. Its petals represent spiritual unfolding, and the seedpod the fecundity of creation – the ‘superhuman’ rising out of and above the mud of the physical world…In essence [it represents] an individual stepping out of ignorance, and through knowledge one’s true nature opens like petals to reveal the inner, enlightened being. On another level, the myth serves to illustrate the constant unfolding process of manifestation. As with the sphinx and the horse, the lotus has its Western counterpart in the rose, the symbol of truth.”
The Geometry of the Lotus
The sacred Lotus seedpod head is “a cipher for awakening the objective consciousness in humans,” as Keith Critchlow says.
There are many different number patterns that the lotus offers.
Some of these include:
- Nine around three
- Twelve around six around one
- Six around three
- Ten around five
- 17 around 13 around 6 around 1
The Flower of Life is represented by the lotus:
The Flowers of our Soul – The Chakras
The seven main human chakras, or energy centers, are often depicted as flowers – specifically lotuses. These represent the ‘flowering of our consciousness’.
The seven chakra system can be overlaid on the Flower of Life symbol. The crossing points of the vertical axis determine the placement of the chakras. This illustration is not only beautiful but it represents the ‘flowering of consciousness’ among an individual that is part of a greater ‘flowering of consciousness’ of the universe.
The Flower of Life
The Tree of Life comes from the Flower of Life. Overlaid upon the human figure shows the seven major chakra points.
Keith Critchlow reminds us, “The Lotus is a mythological symbol of the flowering of the human soul.”
He adds, “It must not be missed that this crowning glory [flower garlands or wreaths] was also symbolic of the crown chakra flowering in Vedic symbology, a symbol of the highest spiritual achievement while in human bodily form.”
“The renowned Vedic centers of the human body called the Lotuses or chakras relate to our gross anatomy as recognized by contemporary Western science but in themselves are not considered as of a gross nature. This system offers a series of flowers (Lotuses) each petal bearing a letter of the Sanskrit (Devangari) alphabet. Probably the oldest system of human anatomy, it is still very much taught and used today.”9
Chakras are energy centers, or “wheels of light” that are key elements of the spiritual system of the body. They are connected to the endocrine glands and major nerve centers in the body.
They are like lenses through which everything in existence is interpreted.
Chakras are representative of our state of consciousness and they are directly related to the evolution of our consciousness.
This evolution of consciousness related to our chakras is sometimes referred to as the “Rainbow Bridge”.
The journey to fully activate and balance each chakra is a journey through varying states of consciousness.
It represents the seven stage transformation of our inner self.
See Articles 200-203.
“Whatever the Divine quality of the soul, its inner being strives with all its capacity to express that Divine quality on the physical plane. The nature of the soul is self-expression, and therefore there is no being who, given the opportunity to exist in freedom, would not seek to express its deeper nature and what was central to its heart. The creation of conditions on the earth that would maximize this freedom-to-be are what spiritual evolution is meant to achieve, so that the embodied self can in all ways conceivable come to express all that it experiences of itself as its own deeper nature. This is the true meaning of freedom. It is the freedom to be.
In the end, the flowering of the soul on earth for individuals and for a collective humanity will herald a new age of peace in the world, for the soul’s presence is a guarantor of peace and the soul’s expression is a guarantor of alignment with the higher purposes of light and truth. When souls are capable of manifesting their inner life within the realm of the physical, then life will no longer be separate from the sacred, and what is the highest and best within each human expression will finally be given a vehicle for doing so and the chance to flower within the realm of time and space.”10
Keith Critchlow writes, “Flowers nourish us at all levels if we let them and are awake to them.”
“By uniting the fragments in their own life and soul to a living body, a person unites the fragments of the world around them and transforms them into the ‘heavenly city.’ Engaged in that one pointed end, they will use more and more of their self-consciousness and become ever more loving and serviceable to others; not like a machine turning out a manufactured article, but like a flower opening up its heart to the sun.” ~ G.H. Mees
- Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
- Kesseler, Rob and Stuppy, Wolfgang, Seeds: Time Capsules of Life, Papadakis, 2014
- The Flowering of the Soul, http://www.lightomega.org/Ind/Flowering-of-the-Soul.html