In this article we will continue our discussion of the geometry of plants, moving to the structure and function of leaves. In the next two articles we will move into a discussion of flowers then fruits. In each article we will see how regular polygons, the golden ratio and the Fibonacci sequence continue to play a role in the formation and growth of plants and flowers.
Keith Critchlow writes, “It is thanks to leaves and seeds that we are alive and have food.
As each light-transforming leaf is the source of the miraculous continuity of life on our planet, the ‘green’ world should never be underestimated or (one now has to add) abused by mechanical agribusiness and soulless mechanical processing. Financial profit is the least responsible motivation for human behavior.”
Geometry of Plant Leaves
A leaf is “an organ of a vascular plant and is the principal lateral appendage of the stem. The leaves and stem together form the shoot.”1
Look at the cross-section of a leaf and notice the layers of various geometry involved.
The outer layer of cells, the epidermis, is composed of a tessellating structure of epidermal cells.
Below that is a layer of spongy mesophyll cells, the primary location of photosynthesis in the plant.
Some leaves have trichomes (hair-like appendages pictured below). These have a diverse range of structure and function.
As we mentioned in Article 180, the underside of leaves also have stomata. These are the eye-shaped (Vesica Piscis) openings that regulate the exchange of gases and water vapor between the outside air and interior of the leaf. Many of these can be seen in the above image.
The Vesica Piscis is the eye or almond shaped overlapping center area shown in blue above.
Also recall our discussion from Article 179 about Goethe’s six-step progression of plant growth. The leaf acts as the archetypal ‘organ’ that expands or contracts to form all parts of the flowering plant.
As Goethe said, “From start to finish, the plant is nothing but a leaf.”
To review, Goethe’s six part process involved:
1. Expansion from the seed into stem leaves
2. Contraction from stem leaves into the sepals of the calyx
3. Expansion from sepals into petals (corolla)
4. Contraction from petals into pistils and stamens
5. Expansion from reproductive organs into fruit
6. Contraction from fruit into seed
In Article 178 we discussed phyllotaxis of leaves and how it is based on the Fibonacci sequence.
As Keith Critchlow writes, “The disposition of sunlight-feeding leaves arising out of the same stalk or twig is to do so in a neighborly manner so as not to steal another’s light. This is called phyllotaxis and has a series of strategies.”2
This speaks to the important idea in Cosmic Core that nature and life is cooperative and not competitive.
Or as Critchlow says, “Life is always collaborative as well as sacrificial, and not merely competitive or confrontational.”3
We all know that there are evergreen trees that do not lose their leaves in winter and deciduous trees that do lose their leaves in winter.
“As the trees learn to let go, we can release our grip through the yogic practice of aparigraha. Literally meaning ‘non-clinging,’ ‘non-grasping,’ or ‘non-greed’, aparigraha teaches us the importance of letting go of what we don’t truly need. It is unnecessary for trees to hoard the dead leaves, trying to save up for next year. Nature provides the nutrients it needs to create new and abundant foliage in the spring, and in fact it is by releasing the old leaves to the earth that provides the compost for nourishing the soil. Hanging on to what is dead and no longer serves us hinders the ability for the new, stronger buds to open. Yoga philosophy knows what nature long has: it is better to let go of clinging to the past, so that we can become the best of who we are today.”4
“You can never cross the ocean until you have the courage to lose sight of the shore.” ~Unknown
Parts of a Leaf
- Midrib – The central line of each part of a leaf.
- Veins (nervure) – The lines branching from the midrib.
- Blade – The leaf’s surface.
- Petiole – The leaf stalk.
- Cuticle – the outer waxy layer of the leaf that controls evaporation and performs many protective functions.
- Chloroplasts – light-absorbing element in cells which are typically lens-shaped and bounded by a double membrane.
1. Apex 2. Midrib (Primary vein) 3. Secondary vein. 4. Lamina. 5. Leaf margin 6. Petiole 7. Bud
The leaf blade is essentially the area of the leaf which includes its flatness and breadth. This ‘area’ facilitates rapid absorption of oxygen and carbon dioxide while also encouraging the maximum amount of sunlight to fall on its surface. The breadth ensures that the necessary gases are transported the shortest possible distance.
“A leaf is living chemistry.”5
It produces food from light and supports the life of all other living beings.
To review from earlier articles, the tree can represent the different layers or aspects of our consciousness:
- The Deepest Roots: The Cosmic Mind (Infinite or All-Mind)
- Mid-level Roots: The Archetypical Mind (Galactic Mind)
- Shallow Roots: Planetary Mind (Akashic)
- Base of Trunk: Racial Mind (Group or Tribe)
- Lower Trunk: Individual Unconscious Mind
- Higher Trunk: Individual Inner Ego
- Lower Branches: Individual Subconscious
- Higher Branches: Individual Conscious Mind
- Leaves: Individual Outer Ego
- Flower/Fruits: Our thoughts, beliefs, actions, intentions, behaviors, creative products…etc. This refers to everything we put out in the world on a mental, emotional, spiritual and physical level.
Now read the following passage by Julie Anne Kilgore in light of the above information.
“If you watch a tree during a storm…if you focus on the swaying branches and leaves, you are sure it will break. If you focus on the trunk however, you will see the power and strength and know it will not fall. The leaves and swaying branches are chaos. The trunk is the polar opposite. And yet they work together. The trunk and its roots cannot be without the nourishment it receives from the leaves. The leaves and branches cannot survive without the nourishment and stability from the trunk and from the water and nutrients the roots pull up. One cannot exist without the other.”
Geometry of Leaf Forms
There are various leaf forms, each one suitably adapted to the species of plant and the climate and environment it grows in.
Notice the geometry of the leaf shapes. There are many that resemble regular polygons and there are many that are compounds, truncations or stellations of these polygons. There are also many leaves with geometry based on the circle and variations of the circle, such as the oval and ellipse.
We will see many of these shapes show up again in the structure of bacteria.
“If the only function of a leaf was to act photosynthetically (transforming light into nourishment) they would all be the same shape, which patently are not!”6
There is no known theory to explain the wide variety in leaf forms, except to say that Nature is an artist as well as a scientist. Beauty, aesthetics, and creativity are as important as function when viewing why the natural world expresses itself in the way it does.
Gyorgy Doczi, in The Power of Limits, teaches us the following about leaf shapes and spirals:
“In figure 18 the outlines of a random selection of leaves were reconstructed with the dinergic method of combined radiating and rotating lines.
If one follows 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, all of these spirals share the qualities of being logarithmic and equiangular, through all stages of growth.
If one looks at these patterns not as still pictures but as traces of the dinergic process which brought them into being, then the outlines of these leaves become their story, recorded in the silent language of patterns.
The outlines of the rhododendron leaf (A) start from the stem at the center with two rising circles, moving from one radius to the next. This pace increases to three circles at midspan, only to slow down again to two, winding up with approximately 1 1/3 at the tip of the leaf. This is the life story of the lanceolate leaf pattern.
The rounded or orbiculate leaf of the bloodleaf plant (B) is produced by a different pattern. The contour starts by crossing five circles while moving from the first radius to the second, and this rhythm gradually decreases to three, then to two circles, finally slowing down to one at maturation.
The cordate leaf of the lilac (G) starts by moving through four circles within the span of the first two radii, then it slows down to three, to two, and to one, only to take a last heroic spurt, rising three circles before the end.
The same dinergic harmonies that delight our eyes in the shape of leaves and flowers also enchant our ears in the chords and melodies of music.”
Some others of these many artistic forms include:
Simple leaf – undivided blade
Acicular – needle-shaped – Pinus
Acuminate – tapering to a long point – Osage-orange (Maclura pomifera)
Auriculate – Arabis Caucasica
Aristate – spine-like tip – Agave
Bi–lobed – two distinct lobes – Bauhinia variegata
Cordate – heart-shaped (golden spirals) – Ivy; Ficus religiosa
Cuneate – wedge-shaped – Clinopodium georgianum
Deltoid – triangular – Populus deltoids (Eastern cottonwood)
Digitate – finger-like lobes – Tapioca; Cannabis
Elliptic – oval-shaped – Ficus pumila, Rubus ellipticus
Falcate – hooked, or sickle-shaped – Encephalartos
Flabellate – fan-shaped – gingko biloba
Hastate – triangular w/ basal lobes – Convolvulus arvensis
Lanceolate – pointed at both ends – Codiaeum variegatum
Linear – parallel margins – Iris tectorum
Lobed – deeply indented margins – Sassafras
Obcordate – heart-shaped, stem at point – Cercis Canadensis (eastern redbud)
Oblanceolate – Ilex glabra
Obovate – egg-shaped, narrow at base – pawpaw
Obtuse – bluntly tipped – Cat greenbrier (Smilacaceae Smilax glauca)
Oblong – Ficus elastica
Orbicular – circular – pennywort; nasturtium
Ovate – egg-shaped, wide at base – Hibiscus rosa–sinensis
Palmately compound – leaflets radiating from the end of the petiole, like fingers of the palm of a hand (5-pointed star-like) – Maples
Pedate – palmate, divided lateral lobes – chestnut
Peltate – stem attached centrally – pennywort
Perfoliate – stem seems to pierce leaf – Uvularia perfoliata
Pinnately compound – leaflets arranged along the main or mid-vein – White Oaks (rounded lobes, left) & Red Oaks (pointed lobes, right)
Odd pinnate – with a terminal leaflet – Wisteria frutescens; Robinia
Even pinnate – no terminal leaflet – golden shower tree; Ceratonia
Bipinnately compound – Gleditsia triacanthos
Pinnatisect – deep, opposite lobing – Solanum aviculare
Reniform – kidney-shaped – Asarum canadense
Rhomboid – diamond-shaped – Sapium sebiferum
Scale-like – Junipers
Spatulate – spoon-shaped – Myrica pensylvanica
Spear–shaped – pointed, barbed base – bamboo spp.
Subulate – tapering point, awl-shaped – Juniperus communis
Trifloiate – pinnate leaf with just three leaflets – clover
Tripinnate – leaflets also bipinnate – Nandina domestica
Truncate – squared-off apex – Reynoutria japonica
Unifoliate – having a single leaf
Leaf margins also widely vary.
Michael Schneider, in his essential book A Beginner’s Guide to Constructing the Universe, shows that many leaf forms are based upon the pentagon, either stretched, compressed or rounded-off. See image below.
Samuel Colman provides a nice geometric analysis of the pentagonal nature of the maple leaf on page 108 of Nature’s Harmonic Unity.
On page 57 he shows how leaves fit within specifically proportioned rectangles:
These are rectangles formed from the:
- Pentagon: 36°, 54° (oak, yellow water lily) and 72° (milkweed)
- Equilateral triangle: 60° diagonal
- Square: 45° (poplar) and 63° diagonal (cherry, arrow-head)
- Egyptian triangle: 51°30’ and 38°30’ and 58° diagonal
- Ideal Angle: 42° (maple, bloodroot), 48° (convolvulus), 66° diagonal (raspberry) and 61° diagonal (elm)
He writes, “No clearer illustration of the persistence with which Natural forms fall into these rectangles could be found than in various families of leaves. Nature’s use of these harmonic rectangles is not confined, however, to leaves, but they will be found to escribe all living things in a more or less exact way; for the nuts, as well as leaves and petals in the various plants and flowers are governed in a like manner, and the same principle holds good in the case of insects, animals and even man.”
Some leaf shapes are based upon the triangle, rhomboid, Vesica Piscis and circle as well.
Leaf shapes resembling the Vesica Piscis are evident all around us. And of course, whenever we see the Vesica shape, the √3 is embedded within it. Thus, the growth of leaves and the √3 are closely related.
The square root of 3 = the height of the Vesica Piscis.
The Square Root 3 Spiral
The Square Root 3 Spiral and the Begonia. Bottom image credit: Samuel Colman
Some leaves form vortex street structures – one of the spirals found in nature we will discuss in Article 119.
One of the key spirals of fluid dynamics – The Vortex street.
Plants that show vortex streets: ferns, Heliconia sp., wheat raceme & cherry tomatoes
In Keith Critchlow’s The Hidden Geometry of Flowers, he shows some of the various underlying geometry of leaf forms.
On page 156 he shows the decagonal star as the controlling archetype of a set of growing petal leaf or petal forms.
Critchlow writes, “Each side of this leaf-form is based on a tenfold sequence from vertical axis back to vertical axis…Three different sizes of this leaf-form are shown, all generated from points of the bottom central star decagon or ten-pointed star.”
On page 157 he shows the collaborative effect of two √3 spirals moving to the left and right. “A metaphor for petals and leaf forms is evident.”
He says, “Thus it is only the polarity of a single progression that creates the effects of potential leaf forms.”7
Fractal Geometry of Leaf Veins
Keith Critchlow reminds us that, “An important feature of leaves is their energy-distributing and structurally-strengthening ‘veins’.”
Not only do leaf shapes follow geometric harmony, but the placement of veins do as well.
The diagram below by Gyorgy Doczi shows that the distances between the starting points of the veins along the midrib group themselves in a harmonic order like organ pipes.
Doczi writes, “These relationships reveal growth patterns that are harmonious and dinergic in the sense that all the minors and majors (large and small veins and branches) unite with their neighbors in proportions limited to ratios of the same small whole numbers which create the root harmonies of music. Similarly dinergic and harmonious growth processes can be observed in the shaping of leaves other than the lilac.”
Credit: Gyorgy Doczi – The Power of Limits, 1981
A close examination of the veins of a leaf reveals a branching pattern similar to the whole tree. Also, the overall profile of a given tree resembles the overall profile of one of its leaves. This is yet another way in which fractals show up in the structure and function of plants.
A leaf is a “self-similar” component of the tree.
- Arcuate – secondary arcing toward the apex (Silky dogwood)
- Dichotomous – veins splitting in two (Gingko biloba)
- Palmate – three or more primary veins arising near the base of the blade and spreading out like a fan (Red maples, sweetgum)
- Parallel – all veins parallel and not intersecting (Dwarf palmetto; yucca; bamboo)
- Pinnate – secondary veins borne from midrib (Flowering dogwood)
- Reticulate – all veins branching repeatedly – net veined (Hibiscus)
- Rotate – veins coming from the center of the leaf and radiating toward the edges (taro)
- Transverse – tertiary veins running perpendicular to axis of main vein, connecting secondary veins (Ficus sinuate)
“An ‘angle’ is a ‘gon’, a ‘generator’ in the original Greek. So we assume that, according to the viewpoint of Proclus and the sages of the Greek tradition, each angle (or portion of 360) is an ‘intelligence’ in itself and thereby a generator. In the green world of plant life we find particular angles recurring, the most persistent of these being 90º, 45º, and 30º on the spatulate form of the leaf.”8
See page 369 in The Hidden Geometry of Flowers to see the geometry behind the Nasturtium leaf.
Credit: Keith Critchlow
Leaf arrangement refers to the ways that leaves are arranged along a stem.
Opposite refers to two leaves that arise from the stem at the same level (at the same node) on opposite sides of the stem.
This can be thought of as a whorl of two leaves.
Examples: Ash trees (Fraxinus spp.); Fringe tree; Maple trees; Olive trees
Decussate refers to leaves in an opposite pattern, if successive leaf pairs are 90 degrees apart.
Examples: family Crassulaceae; Aizoaceae
Alternate = spiral.
Each leaf arises at a different point.
This includes 80% of the 250,000 higher plant species.
Examples: Barberry, Black walnut, Japanese zelkova, ninebark, Smoke bush, Sweetgum
Disticious refers to “two-ranked leaf arrangement”.
This is a special case of either opposite or alternate leaf arrangement where the leaves on a stem are arranged in two vertical columns on opposite sides of the stem.
Examples: various bulbous plants such as Boophone; Gasteria; Aloe seedlings; corn
Whorled refers to when several leaves arise, or appear to arise, from the same level (at the same node) on a stem.
This is fairly unusual on plants except those with particularly short internodes.
Examples: mint; Brabejum stellatifolium; related Macadamia genus
Rosette arrangement refers to a basal whorl (leaves attached at the base of the shoot) with a large number of leaves spread out in a circle.
Examples: agave Americana; dandelion, liverwort (Ricciocarpos natans);
Repeating spirals can be represented by a fraction describing the angle of windings leaf per leaf.
The numerator and denominator usually consist of Fibonacci numbers.
- alternate distichous leaves – ½
- beech and hazel leaves – 1/3
- oak & apricot leaves – 2/5
- sunflower, poplar and pear leaves – 3/8
“Leaves are silences around which flowers are their words.” Rabindranath Tagore
With that we will move into the next article about flowers.
- Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
- Let the Leaves Fall, https://www.awakeningself.com/writing/let-the-leaves-fall/
- Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011