The Geometry of Fruits
We will now continue our visual journey, this time focusing on the geometry inherent in fruits. We will examine the cross-section of various fruits in order to see the geometric structure.
Always keep in mind, when looking at the cross-section of the structure of any life-form, how similar cross-sectional patterns are to radially symmetric Cymatic patterns.
If everything in reality is a complex symphony of wave-patterns, then cross-sections can be seen as pulsations of a specific wave-form as it moves longitudinally through space.
Keith Critchlow tells us, “As the flower can be considered nourishment to the soul in its beauty, perfume and colors, so the fruit is nourishment to the other members of the ecological community such as the insects, the birds, the animals and the human family.”
Ken Arnold writes in the preface to Fruits: Edible, Inedible, Incredible, “As a foodstuff, a topic of scientific investigation, a metaphor for sin or hard work, and much else, fruits occupy an especially fecund place in our material and psychological responses to the living world. There is a lot to them and they matter much to us.
Say the word ‘fruit’ and chances are we will first picture something juicy and filling. People have for millennia consumed literally thousands of their varieties, either cooked, dried, preserved, or just raw. Say it again, and we might instead ponder their biological function: a vehicle for genes packaged into seeds to be passed on to the next generation. Say it a third time, and we might now start thinking in metaphorical terms of our own physical efforts, or the realms of the forbidden – bits of knowledge at once delicious and dangerous.
So while we ponder technical questions of scale, form, structure, variety, mechanism, and purpose, let’s not forget that we are also leafing through a body of artwork that conjures up worlds within worlds, where beauty lurks within imperfection and where perfection can still dramatically reappear out of order.”1
Cross-Section of Fruits & Vegetables
- cross-section resembles plant cellular structure
- Delauney Triangulation & Voronoi tessellations are present
Plant cellular structure and convection cells show Delauney Triangulation & Voronoi Tessellation
The Garlic Flower. A Vesica-shaped spherical bulb opens into a beautiful geometric florescence containing many tiny flowers that each open in turn. The last image shows how each flower creates a tiny fruiting body that resembles a miniature version of a garlic bulb.
Monad/Dyad – 1 & 2
Onion – concentric circles, sometimes surrounding two central buds
Leek cross-sections – concentric circles
Beets – concentric circles
Avocado – 1 central pit
“The oil-rich avocado is one of the most nutritious and fiber-rich fruits. Its extremely large seed in combination with the green skin-color and high oil content suggest that the avocadoes adapted to dispersal by the large mammals that were part of the extinct Pleistocene megafauna that disappeared some 13,000 years go.. Today, in their native America, avocados are relished by wild cats and jaguars.”2
Coconut – 1 central pit
Stone Fruits: peach (1), nectarine, plum (2), cherry (3) – 1 central pit
Breadfruit – single central seed; note the hexagonal tessellations on the skin.
Durian – Dyad (2) geometry with the pit; note again the hexagonal tessellations of the pointy skin.
Triad – 3
Many fruits and vegetables display three-corner structure. Below is a bell pepper, melon and cucumber.
Bell peppers (sometimes 4-point geometry, but mostly 3)
Zucchini & summer squash
Tomato – 5 sepals; cross-section: 3-pointed; 4-pointed & 5-pointed
Passionfruit – the ovary is formed by three joined carpels
Melons: cantaloupe (1) & horned melon (2) & watermelon
The triad (triangle) gives the rose’s thorn its efficacy.
Tetrad – 4
Cranberries – A 4-petaled flower produces a 4-fold fruit.
Pentad – 5
Credit: page 126 The Hidden Geometry of Flowers by Keith Critchlow
Currants – 10 point radial geometry from a 5-petaled flower
Pear – The pear has the same 5-pointed star arrangement of its seeds as does the apple.
Papaya – a ripe papaya shows a 5-point geometric arrangement of seeds (1). A slice off the end of an unripe papaya shows beautiful 5-point geometry (2).
Pomegranate (sometimes 5 & sometimes 6)
Okra (sometimes 8 – 5 & 8 are both Fibonacci numbers)
Hexad – 6
Carrot slices – these cross-sections range from nearly perfect hexagons to rough estimates of hexagons.
Eggplant – eggplants are one of the few vegetables with 6-petaled flowers.
Kiwi – sometimes 5 – like the carrot, cross-sections reveal nearly perfect hexagons ranging to rough approximations of hexagons.
Octad – 8
Persimmon – A 4-petal flower yields 8-point geometry.
Decad – 10
Citrus fruits – grapefruit, lemon, orange & lime
Pineapples – baby pineapple from the top; Fibonacci Spirals & Pentagonal segments of a mature pineapple
Artichokes – artichoke flower & mature fruits
Cabbage – Purple cabbage, ornamental cabbage, Chinese cabbage & Savoy cabbage
Spiraling cross-section of a celery bunch & cellular structure of celery by Christine Baek
Broccoli & cauliflower
Blackberry & raspberry drupelets
Banana bunches & stalks
Prickly pear cactus
Magnolias, tulip tree– carpels arranged spirally around a central column
“Whether they are tropical, subtropical or temperate, we enjoy fruits in a great many ways, either fresh, dried, cooked or preserved, in yoghurts, ice creams, jams and biscuits, as juices or alcoholic drinks. Some serve as spices in the form of peppercorns, cardamom and chili peppers. The most valuable of all, the fermented pods (‘beans’) of the vanilla orchid are traded as a highly priced ingredient in chocolate, ice cream and many other sweet dishes. Others, such as the fruits of the West African oil palm and olive are pressed for their valuable oils. Countless other fruits are important to humans as a source of natural raw materials such as fibers, dyes and medicine, or simply ornaments.”4
Fruit classification is extremely complex. “Classifying fruits according to their structure is a very difficult task. For nearly every carefully crafted scientific definition the angiosperms have produced a puzzling range of exceptions with which they have managed to foil all attempts to create a classification system of their fruits that is both logical and practical…
Until recently the last comprehensive reviews of fruit terminology were those of Bischoff (1833), Dumortier (1835), and Lindley (1832, 1848). Hence, even today, little has changed. Confused by the tangled mass of terms and left without a standardized fruit classification system, botanists have invented their own simpler solutions in a desperate attempt to undo the Gordian knot of fruit classification…
Surprisingly, most twenty-first-century botanical authors still subscribe either to Tournefort’s or Gaertner’s definition of a fruit as a mature flower or a mature ovary, respectively. But this still leaves anthocarpous and compound fruits to be sorted out…
Finally, relief came in 1994 when Richard Spjut published his Systematic Treatment of Fruit Types, the most recent and most significant carpological classification of the twentieth century. Spjut’s attempt to bring some order into the 300 year-old chaos of fruit classification was admirable in itself, but he also provided a precise scientific definition of the term ‘fruit’ that effectively reinstated de Candolle’s liberating definition, allowing botanists for the first time in more than a hundred years to address strawberries, pineapples and figs, as well as the cones of conifers, cycads and Welwitschia simply as ‘fruits’. In Spjut’s (re)-enlightening classification system, gymnosperms and angiosperms are treated as equal, and Procrustean terms such as ‘pseudocarp’, ‘false fruit’ and ‘accessory fruit’ – once again – become superfluous. They are replaced by the more appropriate expressions ‘anthocarpous fruit’ and ‘compound fruit’.”5
Live seed-bearing plants (spermatophytes) fall within two categories:
- Gymnosperms – no flowers or fruits; unenclosed or ‘naked’ seeds on the surface of scales or leaves; seeds are often configured as cones
- About 1000 extant species
- Angiosperms – flowering plants; seeds are enclosed within an ovary (usually a fruit)
- About 422,000 extant species
The closest (living or extinct) relatives of angiosperms are still unknown “and so are any intermediate forms that could help us document the evolutionary steps that led from gymnospermous to angiospermous organization. What is even more puzzling is that the angiosperms seem to have appeared abruptly out of nowhere, undergoing a remarkably rapid evolution.”
Charles Darwin described the sudden appearance of the angiospersms in the fossil record as a ‘most perplexing phenomenon.’ To the present day, the question of the evolutionary origin of the angiosperms has remained unanswered.
Scientists have gathered evidence indicating that the angiosperms underwent a whole-genome duplication event (i.e. a doubling of all their genes) early in their evolutionary history.”6
What this indicates is that there seems to have been a very sudden and relatively spontaneous upgrade of the DNA of species upon the Earth – transforming the existing species into newer, more evolved forms.
This data aligns very nicely with what we learn about cycles of extinction and biodiversity in Article 256.
Furthermore, “Little is known from the fossil record about how the close relationships between fruit-eating animals and plants began to evolve millions of years ago…just how did plants come to produce nutritious fruits in exchange for the dispersal of their seeds?”7
Two essential components of a ripe fruit:
- Presence of seeds
- Seeds contain the embryo
- The fruit wall or pericarp containing the seeds
Fruit classification is based on three things:
- the underlying ovary type (apocarpous or syncarpous)
- the consistency of the fruit wall (soft or hard)
- whether or not a fruit opens up at maturity to release its seeds
Three Basic Types of Fruits
- Simple fruits – develop from one flower with only one pistil
- Examples: cherries, tomatoes, oranges, cucumbers
- Multiple fruits – develop from flowers with several separate pistils
- Examples: Annonaceae, Magnoliaceae, Winteraceae; tulip tree, Winter’s bark tree, custard apple, blackberries, raspberries
- Compound fruits – Each bead or fruitlet (such as the mulberry) was once a tiny flower.
- Examples – mulberry, pineapple, figs, breadfruit (Artocarpus altilis), jackfruit (Artocarpus heteropyllus)
Fleshy vs. Dry Fruits
Fleshy fruits – drupes and berries
- Drupes – the ovary wall (pericarp) is differentiated into three layers: the thin outer epicarp (skin of fruit); the fleshy mesocarp (pulp of fruit); and a hard inner layer (endocarp). Epi and Meso are edible. Endo is not.
- Examples are cherries, plums, peaches, mangoes
- Berries – the pericarp is soft and juicy; mostly contain several or many seeds, rather than just one.
- Examples – tomatoes, cucumbers, grapes, blueberries; the avocado is a rare one-seeded berry.
Dry Fruits – Nuts
The fruit wall of a nut remains closed, making the independent distribution of several seeds impossible. Nuts are mostly single-seeded like drupes.
“All types of nuts or nutlets have one feature in common: functionally they act in the same way as seeds.”8
“In a botanical sense, a nut is only a nut if it consists of nothing but the mature ovary of an indehiscent simple fruit with a hard, dry pericarp, usually harboring a single seed. This is true of hazelnuts, sweet chestnuts, walnuts, pecan nuts, beechnuts, acorns, and unshelled peanuts. Other culinary ‘nuts’ such as unshelled almonds, pistachios, and at least technically, cashew nuts are, in fact, the stones of drupes, whereas Brazil nuts, macadamias, ginkgo nuts and pine nuts simply represent seeds.”9
Samuel Colman shows us on page 113 of Nature’s Harmonic Unity that the nuts generally fall into the same rectangular proportions as the leaves did that we saw in Article 184.
These are rectangles formed from the:
- Pentagon: 36° (peanut), 54° (hazelnut, brazil nut) and 72°
- Equilateral triangle: 60° diagonal (hazelnut side view; pecan side view)
- Square: 45° (almond) and 63° diagonal
- Egyptian triangle: 51°30’ (walnut) and 38°30’ and 58° diagonal
- Ideal Angle: 42° (walnut side view), 48° (brazil nut side view), 66° diagonal and 61° diagonal
“Many of these nuts of the same species,” Colman tells us, “somewhat vary in their proportions, some being a little wider in relation to their length while others are the reverse. These, however, are the exceptions which ‘prove the rule’ for the large majority are like those in the diagrams. It must be remembered that such variations occur in everything in nature, the majority conforming to the law.”
Now we will take a look at the geometry of other fruits:
Valerianella coronata – hexagram stars packed onto a sphere
Krameria erecta – barbed spines covering single-seeded indehiscent fruit of this small shrub
Credit: Don Davis
Cornus Cousa – Asian Dogwood Blossom & Fruit
Rattan palm fruit – spiraling cone-like form
Credit: Richard Parker
Rumbia – Fibonacci spirals
Lychee – fruit surface tessellating hexagonal structure
Durio zibethinus – durian – tessellating hexagonal structure
Eucalyptus Globulus (1); Eucalyptus Obstans (2); Eucalyptus Robusta (3); Eucalyptus Rudis (4); Eucalyptus Tereticornis (5)
Anagallis arvensis – scarlet pimpernel – the fruit capsule opens with a lid
Medicago orbicularis – blackdisk medick; Archimedean spiral
Medicago scutellata (Snail Medick)
Credit: Giancarlo Pasquali
Medicago polymorpha (Toothed Medick)
Citrus medica var. sarcodactylis – Buddha’s hand, fingered citron
Esenbeckia pumila– capsular fruit w/ 5-pointed geometry
Kiwano – Horned Melon
Strelitzia reginae – Bird-of-paradise – 3-pointed geometry capsule that opens; black seeds w/ feathery orange wings
Ravenala madagascariensis – Traveler’s palm from Madagascar – 3-pointed geometry capsule w/ bright blue feathery seeds
Flindersia austalis – Australian crow ash – spiny 5-pointed geometry capsule that opens like a star
Swientenia mahagoni – West Indian mahogany – 5-pointed geometry capsule that splits open
Datura ferox – fierce thornapple – spiny fruit that splits open along a 4-pointed geometric axis
Lecythis Pisonis – monkey pot; fruit that resembles a pot with a lid
Acacia vittata – Lake Logue wattle – very curvy fruit that splits in two
Credit: Rob Kesseler
Entada sp. – Monkey ladder vine – spiral staircase vines; extremely large pods that spiral or are straight; each contain 10-20 segments with a single large brown seed in each segment
Credit: Project Noah
Hippocrepis unisiliquosa – horse-shoe vetch; curving fruit wall
Couroupita guianesis – cannonball tree; hard fruit shell w/ 6-pointed geometrical structure
Sedum spp. – stonecrops – 4, 5 or 6-pointed flower fruits
Paeonia – peonies – fruit capsules that open – 3 or 5-pointed geometry
Caltha palustris – marsh marigold – flower-like fruits
Delphinium spp. – larkspurs – 3-pointed geometry capsule
Aquilegia spp. – columbines – 5-pointed geometry capsule
Annona cherimola – cherimoya – spiraling scales around the fruit
A. reticulate – bullock’s heart – spiraling nodules
A. muricata – soursop – spiraling spines; cross-section w/ nice geometry
A. Squamosa – sweetsop – spiraling nodules
Rollinia mucosa – biriba – spiraling pointy nodules
Annona glabra – alligator apple – skin reminiscent of reptile skin
Illicium simonsii – star-like fruit w/ 10, 11, 12 or 13-pointed geometry
Ochna spp. – between 3 and 12 carpels that develop into black oily druplets
Phytolacca acinosa – Indian pokeweed – 8-pointed geometry
Pandanus odorifer – fragrant screwpine
Credit: Dinesh Valke
Ficus dammaropsis – dinner-plate fig
Liquidambar styraciflua – sweet gum; has compound fruits resembling a morning star
Banskia menziesii – firewood banksia – spiraling follicles on the cones
Allocasuarina tesselata – she-oak; beautiful tessellating geometric design of the cone-like fruit
Credit: Rob Kesseler
Broussonetia papyrifera – paper mulberry
Paliurus spina–christi – Christ’s thorn; round disc
Gossypiumhirsutum ‘Bravo’ – upland cotton – 5-pointed geometry
Cardiospermum halicababum – love-in-a-puff; balloon vine
Nypa fruticans – nipa palm; the large football-sized compound fruits break up into obvoid angular fruitlets
Rumia crithmifolia – spongy, sinuous ridges of the fruit wall create a brain-like pattern
Credit: Rob Kesseler
Tribulus terrestris – puncture vine; devil’s thorn; 5-pointed geometry
Arctium lappa – greater burdock; spiraling inflorescences
Viscum album – European mistletoe; beautiful white berries
Macrozamia lucida – spiraling structure
Blighia sapida – ackee – 3-pointed geometry
Clerodendrum trichotomum – harlequin glory bower
Euonymus europaeus – spindle tree
Maclura pomifera – Osage orange
Tordylium apulum – Roman pimpernel
As we saw with seeds in Article 181 fruits have many methods of dispersal. These include:
- Wind dispersal: wings, monoplanes, flying discs, spinning cylinders, shuttlecocks, woolly travelers, balloon travellers
- Indirect wind dispersal (anemobiallism) – wind can cause certain fruits to sway so that they scatter or eject their seeds
- Water dispersal – floatability and buoyancy
- Indirect water dispersal by raindrops – kinetic energy of dripping rainwater or dew drops can cause the expulsion of seeds from their fruits
- Self-dispersal – plants eject their seeds through ballistic dispersal caused by tissues drying out or by high hydraulic pressure in living cells
- Animal dispersal – fruits can cling to fur or be dropped by birds; they can also be excreted through the digestion system of an animal
- Dispersal by scatter-hoarders such as squirrels
- Dispersal by ants
- Combinations of the above
“The dispersal of fruits and seeds through vertebrate animals (i.e. fish, amphibians, reptiles, birds and mammals) is a common feature of many gymnosperms and modern angiosperms. Of all dispersal strategies, endozoochorty is the most efficient. Apart from affording much more reliable transport of potentially long distances, the seeds of many endozoochorously-dispersed species germinate much better after passing through the gut of an animal.”9
To end with a poetic representation of the ‘fruit’ of our labors:
“In most ancient traditions there is an agricultural goddess for all ‘husbandmen’, farmers and those who ‘plow’, ‘sow’, and ‘reap’ on the earthly plane. But according to Vedic tradition she is Goddess of the Field of Life, the Kshetra or “Field” of the Bhagavad-Gita. The Kshetra is the entire field of the inner psyche. The ‘dweller’ in the field is the human soul. In this field the continuous Battle of Life between the Divine Will and the Titanic-Ahankaric diabolic forces is always present.
As the Ahankara (ego) sows its seed and thorns, so the ‘aspirants for realization’ exert themselves in weeding these negatives out. The aspirant proceeds in sowing the good seed and tending and reaping the fruit of immortality. Thus these fundamental activities reflect the different levels of the psyche.
Those engaged in the aspirant’s life, knowing that this immortal fruit may be won, are often criticized, even harassed by those who are ignorant or merely intolerant of the idea of a purpose to life.”10
The point is that the journey of life is meant to be journey of spiritual awakening and the expansion of consciousness. This process can be symbolized by the Seed of Life that is nourished by ethical actions, compassion and peace. This Seed of Life then sprouts and grows into the Flower of Life. The seven chakra system can be overlaid upon the Flower of Life to represent the journey of the expansion of our consciousness through spiraling octaves of growth. When our chakras are activated and balanced – that is, when our psychological and emotional states are balanced and at peace – then we begin to enjoy the fruits of our labors symbolized by the Fruit of Life.
The Fruit of Life represents the seed or potential of the power of our individual consciousness when used for the greatest good. This power of our consciousness can be used to create beautiful words, actions or things in life that can help inspire others upwards and onwards.
“Developmentally, we move through maturity into eldership as we enter the seventh chakra. The issue becomes one of surrender in that we must trust enough in the process of life to let go to a greater knowing. It allows us to live more fully in the present moment and to retrieve the right to have a greater perspective. It is also a time in our lives when we take our gifts and talents and give back to the community by sharing our wisdom and leaving a legacy. For this to happen, we must be aware of our life purpose or sacred path. It is the understanding and healing of past experiences and a reclamation of our strengths that provides us the fertile ground we need to bring forth these fruits of our labors.”11
We see how the same geometry continues to arise again and again as a matrix upon which all life is formed.
This geometric matrix, as we discuss in detail throughout Cosmic Core, is composed of simple geometric elements including the regular polygons (circle, triangle, square, pentagon…etc.) and the five Platonic solids, as well as the golden ratio and Fibonacci sequence (which are themselves embedded within the pentagon, dodecahedron, icosahedron and even a tetrahedron in a sphere, as well as an equilateral triangle in a circle).
This geometric matrix is the Aether itself. Remember, the Aether is a medium composed of fluid-crystals on an unimaginably small scale. These fluid-crystals are processes, not things. They are pulsating, oscillating toroidal wave vortexes that activate various geometries depending on the source wave.
This geometry is fractal-holographic in nature, so we see the same geometry – the same polygons and Platonic solids – on all scales.
However, it is important to note that in physical reality we do not see “perfect” geometry. This is the nature of our physical, material world. Everything is built off a geometric matrix, but when it crystallizes in the material realm, it is no longer perfect due to the myriad of forces at work (temperature, friction, wind, angle of sunlight, precipitation, population density, environment…etc.)
These ‘messy’ or ‘chaotic’ forces are what allow such variety to come from a few simple shapes and proportions.
The numbers of life forms that can be formed upon this geometric blueprint are literally infinite. And yet the geometry is very simple.
The plant world really gives us a good idea of just how varied the structures of nature can be, and yet, they are all built off the same geometry: circle, square, triangle, pentagon, hexagon, octagon, sphere, tetrahedron, cube, octahedron, icosahedron, and dodecahedron.
- Kesseler, Rob and Stuppy, Wolfgang, Fruit: Edible, Inedible, Incredible, Papadakis, 2011
- Kesseler, Rob and Stuppy, Wolfgang, Seeds: Time Capsules of Life, Papadakis, 2014
- Kesseler, Rob and Stuppy, Wolfgang, Fruit: Edible, Inedible, Incredible, Papadakis, 2011
- Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
- Pugh, Meagan J., The Spiral of Healing: A Journey through the Chakras to Awaken your Creativity and Body Wisdom, 2011