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In this article we will continue our visual journey through the geometry of animal life forms.  In the last article we saw the geometry of sea creatures great and small.  Now we will look at the geometry of insects, land and air animals.

Once again, we will continue to see the proportions of bodies formed on the golden ratio.



Geometry of Insects & Animals

The Golden Proportion (1:1.618) can be found in the structure of most insect and animal bodies.

Some are based upon square root 2 and square root 3 proportions as well.

Of course some are more exact than others but they are all based upon the same geometry we have been investigating throughout Cosmic Core: the five Platonic solids and the regular polygons the square, equilateral triangle, pentagon and hexagon.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe



Geometry of Insects

“The insect kingdom offers numerous examples proving the power of geometry on proportional form, such for instance as may be found in butterflies.  These creatures are always distinctly geometric, even to the casual observer, and while each species has its own peculiar form still the angles of 45° and 30°/60° are always in evidence and the influence of the ideal angle is generally present.”1

There are three parts of an insect: the head, thorax and abdomen.

Insects have six legs.  They lift alternate legs in a stable tetrahedral fashion.

Each have two antennae.


Many insects have spirals in their forms:

Snail shells


Centipede and millipede bodies


Caterpillars and other larvae


Moth Proboscis


Scorpion tails


Egg laying in a spiral pattern




The body sections of an ant are defined by the golden sections of its length.

Its leg sections are also golden sections of its length.

Ant eyes are composed of tessellating hexagons.

“The ant has long been a symbol of work and industry.  Their wisdom and intellect in their endeavors is often acclaimed.  There are many types of ants, some of which are solitary, but most of which are part of a larger community…Ants are social.  Much of their work centers on community activity.  The primary activities are gathering, hunting and growing..there is discipline and order within most ant communities, and everyone knows his/her place.  Predominantly, there are three castes: 1) the queens who found new colonies, 2) the winged males who fertilize the queens for life, and 3) the sterile females who serve as babysitters and laborers…

The worker ants are skilled architects.  They build complex homes, galleries, and even vaulted ceilings.  Their skill and undaunted efforts reflect much about what this totem can awaken.  The ant is the teacher of how to build, how to become the architect of your own life.  It can show you how to construct your dreams into a reality.  It wills how you that the greatest success occurs with persistence…ant can teach you how to harness your own power to design and recreate your life and its circumstances from the ground up.  Ant can show you how best to work with others for the good of everyone.  Ant teaches us that regardless of circumstances, if the effort is true, the rewards will follow – int he most beneficial time and manner.  ant is the promise of success through effort.” ~ From Animal Speak: The Spiritual & Magical Powers of Creatures Great & Small by Ted Andrews



Flying insects

Most flying insects have golden mean proportions in bodies & wings.  They all have bilateral symmetry.




Butterflies have beautiful Li symmetry patterning.


“All butterflies have four wings, two on either side of the body, and their separation is usually along one of the angles of the plan…the ideal angle usually measures the length of the body”2 and they can clearly be seen to be proportioned by equilateral triangles.

Pages 153-155 in Nature’s Harmonic Unity show the proportions of three different butterflies.  Credit: Samuel Colman


Butterflies are also structured by the phi ratio.

“The proportioning of butterfly wings varies from species to species, yet there is a unity in these diversities, a unity created by the same proportional limitations we observe [in other species], with the 1:2 = 0.5 proportion added, corresponding to the octave-diapason.”

Below is the Clodius parmassius butterfly – “It shows octave-diapason relationships between body length and total wing height (2), and between width and length of the front wings (5).  Fifth-diapente harmony is approximated by the relationship of body length to half wing width (3), while fourth-diatessaraon appears in three relationships: half the wing width to wing height (1), length of hind wing to length of front wing (4), and width to length of hind wing (6).”

Credit: Gyorgy Doczi – The Power of Limits, 1981


Next is the Clouded swallowtail butterfly.  “We find the equivalent of diapason in the relationship of body length to half the wing width (2), while diatessaron relates half the wing width to the wing height (1).  There are three diapente relationships in this specimen: one is between body length and wing height (3), the other two are between the width and length of each of the wings (5 and 6).”

Credit: Gyorgy Doczi – The Power of Limits, 1981


Last is the Zebra butterfly.  “It has elongated front wings which fit into the √5 rectangle, composed of two reciprocal golden rectangles.  The proportions of the hind wing approximate 3:4 = 0.75, the visual equivalent of diatessaron.”

Credit: Gyorgy Doczi – The Power of Limits, 1981



See the work of Linden Gledhill to see close-up images of butterfly wings.  The scales have a cellular tessellation that resemble the structure of tessellating plant cells.3


The scales themselves resemble the scales on fish, reptiles and sharks as well.  Below we have an image of butterfly scales (1 – Credit: Anatoly Mikhaltsov), iguana scales (2), peacock feathers (3), fish scales (4), snake scales (5) & shark skin (6).


Many butterflies, like moths, have beautiful long spiral tongues to reach nectar in flowers.


“Perhaps no animal or insect has come to represent the process of transformation and shape-shifting more than that of the butterfly.  With butterflies and moths there are always four distinct stages of change (egg, larva, pupa, adult)…change is inevitable in life, and life only becomes more difficult and dangerous on many levels when we resit its natural flow.  change ensured growth.  We have to shed the old before we can come into the new.  Metamorphosis is the magic inherent within insects.  It is the magic of life that they can teach.

“In the egg stage, there is the fertilization process.  We give birth – to an idea, an activity, a new quality, or something.  From the egg stage comes the larva.  In insects, the egg becomes a caterpillar in this stage.  The caterpillar feeds and works to strengthen itself and achieve its foundation.  Ideas and creativity need to be worked with, shaped, formed, developed, and honed.

When this is accomplished, when the caterpillar has laid itself a new foundation, the stage of chrysalis begins.  A cocoon is woven around itself by the caterpillar, and a mummy-like pupa forms within.  In this stage there is a reorganization of the caterpillar’s cells even though there is the appearance of lifelessness.  Sometimes individuals need to back off, after laying a foundation, and then go deeply within so that creation will be able to come froth strong and in new light.  There are points in the creative process where we must be passive and let things take a natural course.  We do what we must, and then let it move on its own.

From the chrysalis comes yet a new and final form – a winged insect.  The fact that it has wings in its final stages in very significant, and all of the symbolism associate with wings apply to insects as well as birds.  The adult only comes out of the cocoon in the warmth of spring – again reflecting the ability to go within and determine the best time to set the new creations in motion on a higher level…

Butterflies bring color and joy with them.  When butterflies come into your life look at how much or how little joy is within your life.  Lighten up.  Look for change.  Don’t forget that all change is good.  Butterfly medicine reminds us to make changes when the opportunities present themselves.  Transformation is inevitable, but butterfly will help teach you that growth and change does not have to be traumatic.  It will tech you that it can occur as gently, as sweetly, and as joyfully as we wish.” ~ From Animal Speak: The Spiritual & Magical Powers of Creatures Great & Small by Ted Andrews




The dragonfly bodies are formed on the golden ratio.


“Though dragonflies, damselflies, darners, and skimmers (order of Odonata) don’t look like flies, they nevertheless share the same basic proportional limitations.  The common skimmer’s overall shape, for instance, neatly fits into a single golden rectangle.

The common skimmer’s hind wing has the same proportions as the mydas fly’s wing, its narrower part fitting into a golden rectangle, while the broader portion is contained in a square equal to the wing width.  The elongated front wing fits into two golden rectangles and their reciprocal.”

Credit: Gyorgy Doczi – The Power of Limits, 1981


See page 160 in Nature’s Harmonic Unity (pictured below) to see the proportions of a dragon fly and how they are developed by the angles of 45° and 30°/60° connected with the ideal angle.

Their wings are composed of tessellating polygons – many are hexagonal and pentagonal.  This is the same hexagonal structure that we see all throughout Nature.  Remember, the hexad – 6 – is used to create boundaries like molecular and cellular structure.  We see these hexagonal boundaries in molecules, minerals, insects, plants, animals and humans.  The heptad – 7 – represents the transcending of boundaries.

In the article Nature by Numbers4 it says, Voronoi Tessellations “are based on a distribution pattern that are easily recognizable in many natural structures, like the wings of some insects and plant leaves.”

The article goes on to show how Voronoi tessellations and Delaunay triangulation can form the structure of a dragonfly wing.

We have discussed Voronoi tessellations and Delaunay triangulation numerous times in Cosmic Core.

Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation.  They tend to avoid sliver triangles (extremely acute triangles that look like slivers).

“Dragonflies and damselflies are very ancient with estimates of having been around for over 180 million years…Both are known for their fast flight and their dazzling aerial feats, as if imitating how light itself can be moved and directed…Both are excellent hungers of flying insects.  They can fly at speeds up to 30 miles an hour, and their eyes help spot flying insects.  They can spot a movement forty feet away…

Dragonflies and damselflies inhabit two realms – water and air.  In their early life – as a nymph – they live within the water.  As they mature and go through their metamorphosis, they move to the realm of air…Their realm is the realm of light…

Dragonflies remind us that we are light and can reflect the light in powerful ways if we choose to do so.  Life is never quite the way it appear, but it is always filled with light and color.  Dragonfly can help you to see through your illusion and thus allow your own light to shine forth.  Dragonfly brings the brightness of transformation and the wonder of colorful new vision.”  From Animal Speak: The Spiritual & Magical Powers of Creatures Great & Small by Ted Andrews




The body segments are based upon the golden ratio.  The head is to the thorax as the thorax is to the abdomen.  The abdomen is a golden ratio to the entire body length.

Credit: Designa by Wooden Books


“The largest insect order is Coleoptera, the order of beetles.  The figure below shows a random selection of species, belonging to different families and subfamilies, all having different shapes and yet all united by the same kind of proportional relationships between their interconnected parts.  This drawing shows that these relationships range between the 0.618 ratio of the golden section and the 0.75 ratio of the Pythagorean triangle.

As a result of these shared relationships, the overall proportions as well as the detailed articulations of every one of these beetle shapes approximate the visual equivalents of musical root harmonies of fifth-diapente and fourth-diatessaron.”

Credit: Gyorgy Doczi – The Power of Limits, 1981


Beetles have a fascinating and beautiful array of colors and patterns on their bodies as well.

Pages 160-161 in Nature’s Harmonic Unity show the proportions of the water beetle.  It states, “This insect is the most beautiful in form of all beetles, and as it is designed to pass swiftly through the water its proportions are decided by the Vesica Piscis, while the curves dividing the upper part of its shell and the arcs from thence to the top of the head are those developed by the angles of 60° and 45°.  The whole form is just enclosed in the rectangle of the hexagon.  The Vesica Piscis which produces its great curves is rendered by taking the semi-diameter of the primary circle as a radius and then, with the points C as centers, the arcs may be described which form the body.”5

Credit: Samuel Colman


Beetles, as well as butterflies, have long been a symbol of spiritual transformation.

“The beetle is the mot varied of all insects.  There may be as many as 280,000 kinds of beetles.  To put this into perspective, the number of vertebrate animals – including fish, reptiles, amphibians, birds and mammals – may be around 44,000.

In Egypt the scarab, or sacred beetle, had great significance…as with many insects, the beetle goes through a tremendous metamorphosis from the grub stage to the winged.  Because of this, it is associated with resurrection and change.  If the beetle has shown up in your life examine the need for metamorphosis.  Are you in the process?  If so, what stage?  Do you need a change?  Are you needing new sunshine?  Is it time to resurrect some aspect for your life?   Is it time to leave the past behind?  The beetle can show you how to do this with the greatest success.”  From Animal Speak: The Spiritual & Magical Powers of Creatures Great & Small by Ted Andrews




Like butterflies, moth bodies and markings conform to golden ratios.

The eye-like markings of some moths fall at golden sections of the lines that mark its width and length.

Like butterflies, many moths have beautiful long spiral tongues for reaching nectar in slender flowers.


See Plate 58: Tineida – Moths from Ernst Haeckel’s Art Forms in Nature.  These moth species include:

  1. Alucita hexadactyla / Orneodes hexadactyla
  2. Pterophorus pentadactylus / Aciptilia pentadactyla (2a, 2b: wing scales)
  3. Pterophorus rhododactylus / Cnaemidophorus rhododactylus
  4. Lithocolletis populifolia / Gracilaria populifolia = Phyllonorycter populifoliella
  5. Plutella xylostella / Cerostoma xylostella (5a, 5b: wing scales)
  6. Harpella geoffroyella / Geoffroyella gruneriella = Alabonia geoffrella




Mark Booth writes, “The bee is one of the most important symbols in the secret tradition. Bees understand how to build their hives with a sort of pre-conscious genius.  Bee-hives incorporate exceptionally difficult and precise data in their construction.  For example, all hives build into them the angle of the earth’s rotation.”


Page 156 in Nature’s Harmonic Unity shows the proportions of the humble bee and the honey bee.

The proportions of the humble bee reveal “the same influences of the triangle of 60° and the square, while the spaces are verified or others given by the ideal angle as drawn in dotted lines.  If the quadrant chords of 42° and 48° are started from the pole of the primary circle at 0° they will oscillate around the form intersecting many important spaces in the proportion of the bee.

This [humble] bee is a distinguished object in coat of yellow and black as it forages among the flowers, while the honeybee worker proceeds with an absorbing industry, for as she gathers sweets, she is storing material to build her house.  This tiny confectioner was one of the earliest architects to employ geometric principles for use as well as beauty, considerations which man so continually ignores.”6


Not only is the anatomy of bees based upon harmonic geometric principles, but the dance language of bees is as well.

In the figure below by Gyorgy Doczi (The Power of Limits, 1981) you see:  A. Round dance; three bees are receiving message. B. Transition from round dance to tail-wagging dance.  Schematic: upper via figure eight forms; lower, via sickle-shaped form.  C.  Tail wagging dance; four bees are receiving message.

Doczi writes, “From the researches of von Frisch and other we know that bees share detailed communications about food sources by suing the body language of dance.  A round dance performed to convey that food is nearby (A).  If the food is farther away, the dance is in a figure-eight pattern (B), and vigorous tail-wagging is added (C).  Such dancing not only shares information: the dancer also shares energy and excitement, which induces fellow bees to swarm out and get the food, upon which their life depends.”

Note how the dances resemble simple harmongraph patterns.


Honey bees are thus the most common insect associated with geometry due to their familiar hexagonal honeycomb.

The use of 120° joints allows for the least wax to hold the most honey.

“A mere 1 & ½ oz. of wax holds 4 pounds of honey!”


French entomologists and writer Rene Antoine Ferchault de Reaumur (1683-1757) wrote of the phenomenon of the honeycomb, “[The honeybees] have solved a very difficult problem of geometry, for only a limited material or wax is at their disposal for the construction of their house, the rooms or cells to which must be of a determined capacity of the largest size, and with the strongest walls in proportion to the amount of matter to be employed…The cylindrical form would seem to be the best adapted to the shape of the bee’s body, but this would leave vacant or waste room between the contiguous cells; on the other hand, had the cells been square or triangular, they might have been constructed without so many unnecessary vacancies, still they would have required more material and then not fitted the body of the insect…the six-sided form of the hexagon fulfills the problem. In every particular; the base of each cell, instead of forming a plane, is composed of three diamond-shaped pieces placed in such a manner as to produce a shallow pyramid, which structure imparts greater strength while giving a larger capacity with the smallest expenditure of time and material.  The angles of these cells on the longest space measure 109° 28’ and on the smallest 78° 32’.”


Now take a look at the amazing hive of the tetragonula carbonaria species of bee, a sting-less variety from Australia.

Credit:Brito, Rute M., Schaerf, Timothy M., Oldrovd, Benjamin P., Brood comb construction by the stingless bees Tetragonula hockingsi and Tetragonula carbonaria, Swarm Intelligence, 2010, DOI:10.1007/s11721-012-0068-1, Link


Bee colony numbers also conform to the golden ratio.  When you divide the number of females by the number of males, the answer approximates the phi ratio 1.618.

Credit: David A. Cushman


Bee wings are similar to dragonfly wings and can be formed with the same Voronoi cell/Delaunay triangulation process.  Honey bee wings, however, are covered with tiny hair-like projections.

Bee eyes are composed of tiny tessellating hexagons.


In D’Arcy Thompson’s book On Growth and Form we can find a lot of information about honeycombs.

He started the chapter The Bee’s Cell‘ with these words:

“The most famous of all hexagonal conformations, and one of the most beautiful, is the bee’s cell.   As in the basalt or the coral, we have to deal with an assemblage of co-equal cylinders, of circular section, compressed into regular hexagonal prisms; but in this case we have two layers of such cylinders or prisms, one facing one way and one the other, and a new problem arises in connection with their inner ends.  We may suppose the original cylinders to have spherical ends, which is their normal and symmetrical way of terminating; then, for closest packing, it is obvious that the end of any one cylinder in the one layer will touch, and fit in between, the ends of three cylinders in the other. It is just as when we pile round-shot in a heap; we begin with three, a fourth fits into its nest between the three others, and the four form a “tetrad”, or regular tetragonal arrangement.

Just as it was obvious, then, that by mutual pressure from the sides of six adjacent cells any one cell would be squeezed into a hexagonal prism, so is it also obvious that, by mutual pressure against the ends of three opposite neighbors, the end of each and every cell will be compressed into a trihedral pyramid.”7

Credit: Matematicas Visuales


This construction with rhombi is related to the rhombic dodecahedron (pictured below) – a Catalan solid, dual of the Archimedean solid the cuboctahedron, or vector equilibrium.8  The rhombic dodecahedron is one of the few polyhedra among the Platonic, Archimedean and Catalan solids that tessellates.


“Bees have been mythical symbols throughout the world…Bees are also long-time symbols for accomplishing the impossible.  For many years, scientists were unable to determine how bees were able to fly.  Aerodynamically, the body was too large for the wings.  It has only been in more recent times that science has determined that bees move their wings at such a high rate of speed that it makes flight possible…

The bee is the reminder to extract the honey of life and to make our lives fertile while the sun shines.  The bee reminds us that no matter how great the dream there is the promise of fulfillment if we pursue it.  The elixir of life is as sweet as honey, and the bee is a symbol that promises us that the opportunity to drink of it is ours if we but pursue our dreams.”  From Animal Speak: The Spiritual & Magical Powers of Creatures Great & Small by Ted Andrews



Wasps & Hornets

Wasp and hornet bodies conform to the golden ratio.

They also make hexagonal nests.

Their eyes are also composed of hexagonal tessellations.




Like many insects, fly bodies are proportioned on triangles.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe


“The body of the mydas fly (order of Diptera) fits into two golden rectangles.  Growth seems to have proceeded in this fly from the thorax upwards and downwards.

The shape of the wings shares the basic proportional relations of the body.  A golden rectangle contains the narrower part of the wing, while a square corresponding to the width of the wing fits the broader part, the relation between square and rectangle being the reciprocal ‘relation of neighbors’.”

Credit: Gyorgy Doczi – The Power of Limits, 1981


Close-packed hexagonal facets of a fly’s eye are similar to bees and wasps.

A fly approaches an object along a spiral path due to the structure of its eyes and nervous system.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe



Geometry of Spiders

“Many other examples of a like exact geometric construction might be produced in the habitations of insects, but none are more remarkable than the web of the spider.  These creatures resemble the human family in so far that some are careless and slovenly in their work, while others even in the same species appear to take pride in the perfection of their spinning.  Many seem to become discouraged by repeated accidents to their webs until finally any makeshift suffices if only it may succeed in catching a fly now and then.  They family of spiders building hexagonal or nearly circular webs appears to be especially endowed with a sense of proportion and with mathematical instincts so strongly developed that they employ their legs as measuring rods by means of which they obtain an exact ratio, using them as compasses with which to strike correct arcs.  It is by this means that they obtain that perfection of construction which gives man so much cause for wonder, admiration, and thought.  These webs are most perfect in the early morning and at the same time most beautiful when they are covered with minute dewdrops that sparkle in the sun like so many diamonds, and their form is then clearly distinguished.”9

Spiders have five pairs of extremities.  There are five parts to each extremity.

They also have a belly divided into 8 segments, with 8 legs, 0 antennae.

Spider webs, especially orbs, are beautiful and fascinating examples of geometry in nature.

Orb weavers weave the most beautiful webs consisting of radial symmetry and Archimedean spirals.

“Of course this result cannot be always exact as the spider is so frequently interrupted, but every passage of continuous work is quite perfect, the resultant circles progressing in harmonic order.”10

Here is a particularly striking spiral example from Octonoba spp. of spiders.

There are also triangular webs spun by Uloboridae, funnel web spiders, cobweb spiders, mesh web spiders and sheet web spiders.


“Imagine a multidimensional spider’s web in the early morning covered with dew drops.  And every dew drop contains the reflection of all the other dew drops.  – And so ad infinitum.

That is the Buddhist conception of the universe in an image.”  ~ Alan Watts, Following the Middle Way


Incidentally, the Octad (eight-legged creature) is an ancient symbol of the “Great Mother Goddess”.

There is the Grandmother Spider of Native American mythology; spider emblems found in Crete, Germany, Mexico and Peru; and weaver and spinner goddesses such as the Egyptian Neith, Nutet and Isis, the Greek Athena and the Three Fates, and the Mayan Ixchel.

From the ancient  Moche people of Peru, circa 300 CE

Spider depicted on a shell gorget by the Stone Grave people, from a mound on Fain’s Island, Tennessee.

Pre-Columbian spider image from a conch shell gorget at the Great Mound at Spiro, Oklahoma

Other Pre-Columbian spider amulets found in the Midwest mound country of the United States.


“What might we see, how might we act, if we saw with a webbed vision?  The world seen through a web of relationships…as delicate as spider’s silk, yet strong enough to hang a bridge on.” ~ Catherine Keller, From a Broken Web, 1989

Michael Schneider writes in A Beginner’s Guide to Constructing the Universe, “In her terrible aspect, the spider goddess is the ensnaring feminine dangling from the heavens to trap the unwary in her web.  Foreboding doom, she ultimately provides her catch with a fate of spiritual transformation and renewal.”

In its archetypal form, the spider represents the weaver.  The weaver represents the foundational force of oscillation from time/space to space/time (warp and weft), or the continuous weaving back and forth of consciousness and matter.

Weaving represents oscillation, and everything in the universe continually oscillates.


See Plate 66: Arachnida – Spiders in Art Forms in Nature.

These interesting species include:

  1. Tegeocranus hericius = Protocepheus hericius
  2. Tegeocranus latus = Cepheus latus
  3. Tegeocranus cepheiformis = Cepheus cepheiformis
  4. Leiosoma palmicinctum = Tereticepheus palmicinctum
  5. Phrynus reniformis = Phrynichus reniformis / Phrynichus ceylonicus
  6. Arkys cordiformis = Gnolus cordiformis
  7. Gasteracantha cancriformis
  8. Gasteracantha acrosomoïdes) = Acrosomoides acrosomoides
  9. Gasteracantha geminata
  10. Gasteracantha arcuata = Macracantha arcuata
  11. Acrosoma hexacanthum = Gasteracantha cancriformis
  12. Acrosoma spinosum = Micrathena schreibersi
  13. Acrosoma bifurcatum = Micrathena furcata
  14. Oxyopes variegatus = Oxyopes ramosus
  15. Epeira diadema = Araneus diadematus


“Spider teaches you to maintain a balance – between past and future, physical and spiritual, male and female.  Spider teaches you that everything you now do is weaving what you will encounter in the future…The spider awakens creative sensibilities.  It weaves a web of intricate and subtle fabric, as if to remind us that the past always subtly influences the present and future.”  From Animal Speak: The Spiritual & Magical Powers of Creatures Great & Small by Ted Andrews




Geometry of Animals



Fibonacci Numbers & Animals

Like plants, animals often display certain structures based on the Fibonacci sequence.


For instance, there are:

13 horn plates on a turtle shell; 5 centered, 8 on the edges, 5 paw pins, 34 backbone segments


144 vertebrae in a Gabon snake


34 teeth of a hyena


233 teeth of a dolphin

However, when looking at animals we see far more of the golden ratio (1.618) than we do of actual Fibonacci numbers.




Pentadactylism is the fivefold symmetry of hands, paws, and claws.  Each can fit inside a pentagon, either stretched or compressed, much like leaf structures of many plants.

In plants and animals the five-fold nature expresses the Pentad, the symbol of life.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe


The Golden Ratio in Animals

The golden ratio is found in the structure of most animals.  Sure there will be divergences, but overall, the golden ratio is a foundational ratio that structures all life.




The golden mean proportion is found in birds: bodies to wings.


Page161 in Nature’s Harmonic Unity shows the geometric proportions of a hawk.  “The measurements of the head, tail, and wings are decided by the ideal angle drawn in the various progressions of the square.  A similar principle controls the proportions and spacing of the feathers in the various families of birds: all being related to the size of the body in a like manner.”11

Credit: Samuel Colman


Bird Flight Patterns

When a hawk approaches its prey, its sharpest view is at an angle to their direction of flight – an angle the same as the spiral’s pitch.

“The fastest animal on earth is the Peregrine Falcon, which dives at over 200 miles per hour to capture its prey.  It plummets not straight down but in a spiral, widening by a constant factor. In a paper published in 2000, Dr. Vance A. Tucker of Duke University points out that to maintain constant vision of its prey, a straight-diving bird would have to cock its head to the side.

He observes:

Although the spiral path is longer than the straight path, a mathematical model for an ‘ideal falcon’ shows that the falcon could reach the prey more quickly along the spiral path because the speed advantage of a straight head more than compensates for the longer path.

By employing a logarithmic spiral, the bird maintains top flying speed, shortest time of descent and the ability to constantly view its target.”12



V-Formations of Birds

“The moving wings of a jet plane split the still air and leave a vortex street of spiral turbulence behind them.  Designers know that turbulence is a cause of drag on the vehicle, and they strive to reduce it.

Flocks of flying birds take advantage of vortex streets in their familiar V formations.  Only the lead bird must really work at flapping its wings; the others latch onto the undulating spiral wake of turbulence trailing behind it.  They simply relax their wings and let the rolling waves move them up and down and forward.  When the lead bird becomes tired she falls back while another moves ahead to work at splitting the breeze for others.”13

This has now been proven.  Henri Weimerskirch, in 2001, fitted pelicans with heart rate monitors and found that birds at the back of the V had slower heart rates than those in front, and flapped less often.

Johannes Fritz fitted northern bald ibis with equipment to record the birds position, speed and heading, several times a second.

“The recordings revealed that the birds fly exactly where the theoretical simulations predicted: around a meter behind the bird in front, and another meter off to the side. Some ibises preferred to fly on the right of the V, or on the left. Some preferred the center, and others the edges. But on the whole, the birds swapped around a lot and the flock had no constant leader.”14

It was also found that they were tracking the air throughout the flap cycle, when it was thought to be only 20% of the time.  They are able to instantly respond to the wake that hits them.


“The migrations of animals are immense shared enterprises,” Gyorgy Doczi reminds us.  “One rarely appreciates the magnitude of these shared animal enterprises…

Sharing is not limited to the proportional relationships of physical anatomy, but extends also to the social relationships of animals, thus supporting our view that sharing is indeed one of the basic pattern-forming processes of nature.

There is the well-known sharing of parental care.  Mating dances of cranes and other birds are also well known.  But birds have also been reported to dance and sing in concert for no other apparent reason than shared delight.  The male cricket shares his mating urge with distant females whom he attracts with his mating call.  Birds to sing to attract mates, and they also share all kinds of other information with each other, for instance warning other birds about approaching enemies, threatening intruders upon their territory, or calling fellow birds together.  Even whales are known to communicate with each other by singing.  Sharing in each other’s distress and rescuing endangered fellow animals is reported among many species.  Friendship and loyalty abounds among animals.

Sharing in the form of ‘mutual aid and cooperation’ was not only the ‘prehuman origin’ of all moral behavior but was also a basic condition of survival and a critical factor of evolution.”



Let’s not forgot how many flocks of birds and schools of fish resemble harmonograph patterns.

Harmonograph Patterns


“Birds have an ancient mythology and mysticism.  In most societies, animals were visible signs of invisible forces, and people realized that you could only understand the Divine through its creations.  This was especially true of birds…Birds were often thought of as deities or the thoughts of deities.  In Norse mythology, the god Odin had two ravens as messengers, Hugin (Thought) and Munin (Memory).  In Central American mythology, the god of the air, Quetzlcoatl, is most often depicted as a feathered serpent.  In Native American tradition, the Thunder Bird s a spirit creature of great creative power and might.  The Egyptian god Horus is usually depicted with a hawk’s head, while Maat, the Egyptian goddess of truth, is often shown with a vulture feather.  To the Hindus, birds represented a higher state of being.

Birds are one of the most ancient forms of life upon the planet.  Many scientists believe that birds evolved from reptiles over 140,000,000 years ago.  The oldest bird known to humanity is the archaeopteryx, Greek for ‘ancient wing’.  The fossil of this bird, dated from the late Jurassic period, shows that it had a lizard-like tail, jaws with sharp teeth, and claws on the outer joint of its wings.

Flight is a means of leaving the earth and rising to the heavens.  It is a means of descending out of the sky to land upon the Earth.  Birds are the bridge between humans and the divine, the Earth and Heaven.  They are the symbols of transcendence, the rising above lower natures.  They reflect a taming or rising above a juvenile nature.  Often times liberation from any state of being that is to fixed, final or immature is reflected through bird symbology and appearances.  They are the ultimate symbols of transcendence and release from any patter of existence to a more superior one.

Birds reflect a union of the conscious mind with the unconscious.  They reflect the achievement of full realization.  Because of the ability to fly, they are associated with aspiration, flights of intuition, beauty, and levitation.  Birds are a source of creative imagination, and they have the ability to awaken within us our own flights of magic.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



Peacock feathers are stunning Fibonacci spirals.

“The peacock is called the most beautiful of birds and its geometric analysis is [shown on page 161 in Nature’s Harmonic Unity.]  Marvelous color is one of its principal charms, but the form of its spreading tail fills one with quite as much wonder and admiration.  The iridescent ‘eyes’ that come on the extremity of each feather are all placed in proportional ratios, in gradually diminishing size, based on the progressions of the pentagon, and if the quadrants of the circle enclosing the spreading tail be divided into ten equal parts each, and the spiral of 3 + 5 placed on the radiating lines to correspond to each angle, the resultant arcs will intersect each other on these angles on the progression of the pentagon, and these intersecting points will be the centers for the so-called ‘eyes’ of each feather.


This drawing is a conventional one only, for in Nature the separate feathers bend in catenary curves instead of being in straight lines as in the diagram; the plan simply discloses the law of distribution of the feathers and various ‘eyes’.”15

Eyes of a peacock’s tail are at intersecting points of logarithmic spirals.  Interestingly, “a peacock’s unfolding plumage shares the dinergic pattern at the center of a daisy.  The dotted lines in the figures below connecting the eyes of the peacock’s plumage are identical with the logarithmic spirals used to reconstruct the daisy’s pattern.  When circles are inserted between the spiral lines, the peacock’s pattern is transformed into the central pattern of the daisy.  This is not magic:  it is the mana of sharing, which is the very nature of nature.”

Credit: Gyorgy Doczi – The Power of Limits, 1981


Visit: to see interesting artwork by Muhammad Hamza that uses circles in golden ratios to construct various bird faces and heads.


“Of all birds, the peacock most resembles the traditional descriptions of the phoenix.  The phoenix is the legendary bird of resurrection that is sacrificed in the fires of life and then rises from the flames out of its own ashes.” ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews

“In order to rise from its ashes, a phoenix first, must burn.” ~ Octavia E. Butler



The eye, throat, top of wing (shoulder), and ‘armpit’ all fall in golden section ratios.

“The penguin is a bird that doesn’t fly.  Its wings though, do serve a purpose when it is in the water.  The penguin is an excellent swimmer, and its movement in the water is as fluid and smooth as the flight of other birds.  The wings serve as fins, helping the penguin to propel itself and steer in the water.

The penguin can literally jump out of the water landing on its feet.  It can leap five to six feet.  This act, and its association with water, is very symbolic.  The water is the astral plane of life, the dream dimension.  The ability to maneuver so freely reflects an awakening of dream consciousness.  For anyone who has a penguin show up as a totem, you can expect to experience lucid dreams.  When you become aware in the course of the dream that you are dreaming, you can change the dream.  As you change the dream state, you also change those same energies that are playing upon you in your waking life as well.” ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



Reptiles & Amphibians

Snakes and lizards have a hexagonal skin structure.


Snakes & Lizards


Now see Plate 79: Lacertilia – Lizards in Art Forms in Nature.

These reptilian species include:

  1. Chamaeleon montium = Trioceros montium
  2. Lophyrus tigrinus = Gonocephalus chamaeleontinus
  3. Draconellus volans = Draco volans
  4. Phrynosoma cornutum
  5. Ptychozoon homalocephalum = Ptychozoon kuhli
  6. Basiliscus americanus = Basiliscus basiliscus
  7. Chlamydosaurus kingii
  8. Moloch horridus


  1. Cameroon Sailfin Chameleon
  2. Chameleon Forest Dragon
  3. Flying Dragon
  4. Texas Horned Lizard
  5. Kuhl’s Flying Gecko
  6. Common Basilisk
  7. Frill-necked Lizard
  8. Thorny Devil


“Lizard is an animal of great subtlety.  Its movements are quick.  It has four legs and can run with great speed…Most lizards have long tails which help them maintain balance and can also serve as a defense mechanism.  Most also have a crested back, ruffs, or spines.

Those with spines and crests along the spin usually reflect that the sensitivities of the chakras are heightened or about to be.  Are you being too sensitive or not sensitive enough?  Are you being too picky or are you missing the obvious?

There are a number of characteristics that distinguish a lizard from other reptiles.  They too, have a dry skin, and like many, they also have claws.  They are also sensitive to vibrations in the ground.  They feel it with their feet, tail and body.  Their eyes are sharp with an ability to detect the slightest movement around them.  They also have acute hearing.

All of these characteristics give it a symbolism associated with the psychic and the intuitive…Individuals with a lizard totem should listen to their own intuition over anyone else’s.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



“Of all the reptiles – and maybe even all animals – the snake has been the subject of great controversy and paradox.  Religious sources argue over whether it is the symbol of the higher or the lower.  Sometimes seen as a devil and sometimes as healer, it is an animal that truly has earned the mythical reputation.

The serpent and snake has long been a symbol of the sexual/creative life force within humans as is taught in Eastern traditions.  The kundalini or serpent force within humans as is taught in Eastern traditions.  The kundalini or serpent fire lies coiled past the base of the spine.  As we grow and develop, the primal energy is released, rising up the spine.  This in turn activates energy centers in the body and the mind, opening new dimensions and levels of awareness, health, and creativity.

Because it sheds its skin, the snake has long been a symbol of death and rebirth.  It sheds its skin as it outgrows the old.  This death and rebirth cycle is part of what snake represents.  This cycle of death and rebirth is often symbolized by the ouroborus, the ancient image of a snake swallowing its own tail.  It is the symbol of eternity.

The snake is sinuous and fast.  Although many people think of them as slimy, their skin is very dry.  In fact, humans are slimier than snakes.  If a human runs his or her hand on the floor, it will pick up dirt.  A snake’s skin will not which is why it is able to slide and move in the manner it does.

Snakes are symbols of change and healing.  They have speed and agility, so those who have snakes come into their life will usually find the changes and shifts occur quickly and are soon recognized and defined.  When snake comes into your life you can look for a rebirth into new powers of creativity and wisdom.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews




The phi ratio is found in a frog skeleton.  “When the skeleton of a frog (fig 117) is straightened out and the length of the bones are projected upon the central axis of the body, as shown in the figure below, the length of the bones show harmonically diminishing series as they move away from the pelvis.  The top of the pelvis is the center of a series of circles which widen at a harmoniously diminishing rate and all joints appear to be located upon these circles.

“The graph below shows how all shared proportional limitations approximate the three root harmonies of music.  The central wave diagram gives the relationships between the extremities and the trunk, while the diagram at the left reveals how these very same proportional limitations also exist between the length of the skull, the spine, the pelvis, and the greatly extended jumping legs and feet.”

Credit: Gyorgy Doczi – The Power of Limits, 1981


See Plate 68: Batrachia – Frogs in Ernst Haeckel’s Art Forms in Nature.

These amphibian species include:

  1. Notodelphys ovifera = Gastrotheca ovifera
  2. Hyla meridionalis
  3. Hyla tuberculosa = Ecnomiohyla tuberculosa
  4. Amphignathodon Güntheri = Gastrotheca guentheri
  5. Rhacophorus pardalis
  6. Hylodes lineatus = Lithodytes lineatus / Leptodactylus lineatus
  7. Limnodytes erythraeus = Hylarana erythraea
  8. Ceratobatrachus Güntheri
  9. Breviceps mossambicus
  10. Lithobates pipiens = Rana pipiens


“The frog is a totem of metamorphosis.  It is a symbol of coming into one’s own creative power.  It changes from an egg, to a polliwog, to a frog.  Even after it becomes a frog, it lives close to and spends much time in the water.

Emotions are often associated with water.  Individuals with frog totems are very sensitive to the emotional states of others, and seem to know instinctively how to act and what to say.  They know how to sincerely sympathetic.” ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



Tortoise shells have hexagonal plates.

“As a group, turtles are more ancient than any other vertebrate animal.  There are around 250 kinds, 48 of which are found in the United States.

A great deal of mythology exists in regard to the turtle.  In the Far East, the shell was a symbol of heaven, and the square underside was a symbol of earth.  The turtle was an animal whose magic could help you unite heaven and earth within your own life.

Because of its great age and its slow metabolism, the turtle is also associated with longevity.  Long life and groundedness within life is part of what is associated with the turtle.  It does not move fast.  It is as if, on some level, turtle knows it has all the time in the world.  Turtle medicine can teach new perceptions about time and or relationship with it.” ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



Alligators & Crocodiles

Alligators and crocodiles have a more cubical geometry to their skin, rather than hexagonal.  They are one of the few creatures that show this.

The armadillo has both cubical and hexagonal skin/armor.


“Alligators and crocodiles have had mixed symbology and imagery throughout the ages.  To the ancient Egyptians they have been associated with fury and ferocity – the same aspects often given within mythologies to the unbridled feminine/creative forces of the world…Inherent within this was the idea that there could not be death without life or life without death.

Birth and death in any form are initiations, events that mark the end of one period and the beginning of a new.  This indicates the culmination of knowledge on one level and the seeking for newer knowledge on others.  All of this is reflected within the essence of alligators and crocodiles.

They patrol the waters and shorelines that separate land from water – birth from death, etc. In this sense they can be seen as the keepers and protectors of all knowledge.  They are the primal mothers in whom all knowledge rests and waits to be born.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews




The phi ratio is found in the faces and bodies of dogs, cats, and horses, as well as many (if not all) other large mammals.

Like humans, the bone structure of large animals is based upon the golden ratio.



The Horse

The geometric analysis of the horse’s body is shown on page 165 in Nature’s Harmonic Unity.

“The main body of the diagram is enclosed in two rectangles of 42° and 48° of the ideal angle, while the diagonals of 36° of the pentagon coincide with the measurement producing the height of the top of the haunches.”16

Furthermore the angels of the pentagon mark the height of the top of the thigh, the elbow of the foreleg, the position of the tail where it leaves the body.


The horse is examined more closely by Gyorgy Doczi in The Power of Limits.

“The skeleton of a horse reveals some of the secrets of the delightful grace of its gallop – the proportional harmony that unites the diversities within the bone structure.

One aspect of this proportional relationship is between the skull and vertebral column on one side and the extremities on the other.  Lined up with their length-dimensions along their longitudinal axes, the parts of this anatomical structure show shared relationships between: skull and phalanges; neck and metatarsal-metacarpal bones; trunk and shinbone-forearm; and sacrum-tail and upperarm-thighbone.

The dimensioned wave diagrams further illustrate how these diverse articulations approximate musical root harmonies, as the graph demonstrates.”

Credit: Gyorgy Doczi – The Power of Limits, 1981


“The horse is rich in lore and mythology.  An entire book could be written on the significance of the horse alone, for no one single animal has contributed more to the spread of civilization than the horse.

The symbolism of the horse is complex.  It can represent movement and travel, or maybe it showed up to help you with movement.  It has been a symbol of desire – especially sexual.  the stallion was often used as a symbol of sexuality.  The taming of a stallion would then be the taming of sexuality and dangerous emoticons.

Horse brings with it new journeys.  It will teach you how to ride into new directions to awaken and discover your own freedom and power.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



The Lion & Tiger

All the key facial features of the tiger and house cat fall at golden section lines defining the length and width of its face.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe

“Tigers are magnificent and stir a feeling of awe in all who see them…If a tiger has entered your life, you can expect new adventures.  It will awaken new passion and power within life.  Examine what is going on in your life.  Do you need more passion for life?  Are you expressing your life passion in appropriately?  Has your energy been down?  If tiger has shown up, there will begin to manifest new adventures and renewed devotion and passion for life.” ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews


The face of the lion is shown on page 165 in Nature’s Harmonic Unity.  The same angles govern its proportional parts as we have seen throughout this article.

“The side view and whole body are not given for want of space, but their divisions are controlled in similar manner, as are those of other animals.”17

“The lion has been symbolic of a variety of energies through the years.  It is a symbol of the sun and of gold.  It was a symbol for the sun god Mithra…

The lion is unusual among members of the cat family in that it will live in groups called prides.  If a lion has shown up as a totem, you can expect lessons and issues dealing with community and groups to surface.  There may be a need to examine your own role in the group.

The idea of the young lion being associated with the rising sun is most significant…When a lion has shown up, there will be opportunity to awaken to a new sun.  Trust your feminine energies – creativity, intuition and imagination.  These will add new sunshine to your life.  Don’t be afraid to roar if you feel threatened or intruded upon.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



The Koala Bear

The facial features of a koala bear show golden ratio proportions in the dimensions and positions of the eyes, nose, and mouth in relation to the dimensions of the face.


Other Bears

“The bear is a powerful symbol and image in both myth and lore…Like birds, the bear is often considered among Native American peoples as kin to humans because, like birds, it can stand and walk upon two legs.

The bear can teach you to draw upon all of your inner stores of energy and essence even those which have never been tapped or accessed.   Meditating and working with bear will help you to go within your soul’s den your inner sanctum – to find your answers.

Bears are associated with trees…as bear teaches you to go in and awaken the potentials inherent, the tree serves as a reminder that we must bring what we awaken out into the world and apply it – make our marks with it.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews




A bat’s body is proportioned by the golden section.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe


See Plate 67: Chiroptera – Bats in Art Forms in Nature.

These species include:

  1. Plecotus auritus
  2. Plecotus auritus, head from front
  3. Nyctophilus australis = Nyctophilus geoffroyi geoffroyi, head from front
  4. Megaderma trifolium = Megaderma spasma trifolium, head from front
  5. Vampyrus auritus = Chrotopterus auritus, head from the side
  6. Lonchorhina aurita, head from front
  7. Lonchorhina aurita, head from backside
  8. Natalus stramineus, head from front
  9. Mormoops blainvillei, head from front
  10. Anthops ornatus, face from front
  11. Phyllostomus hastatum, head from front
  12. Furipterus coerulescens = Furipterus horrens, head from front
  13. Rhinolophus equinus = Rhinolophus ferrumequinum, head from front
  14. Centurio flavigularis = Centurio senex, head from front
  15. Vampyrus spectrum, head from front


“The bat is one of the most misunderstood mammals…most people fear transitions, holding onto a ‘better the devil you know than the one you don’t’ kind of attitude.  If a bat has flow into your life, then it is time to face your fears and prepare for change.  You are being challenged to let go of the old and create the new.

For many, change is always distressing.  When the bat comes into your life, you may see some part of our life begin to go from bad to worse.  That which worked before may no longer.  “This is not negative though!  And it will only be upsetting to the degree we are emotionally attached to the old way of life or to the degree we focus on the past rather than the infinite possibilities of the future.”  ~ From Animal Speak: The Spiritual and Magical Powers of Creatures Great & Small by Ted Andrews



“It is amazing to behold such unity in the manifold diversities of nature, each species developing freely its uniqueness, yet all united by sharing the same simple, dinergic and harmonious proportional limitations.  Butterflies flutter, dragonflies dart, and beetles bore.  The allosaurus must have moved with a heavy, lumbering gait, unlike the graceful gallop of the horse, the slow crawl of the crab, and the sudden leap of the frog.  How unique each is, and how different than winged maple seeds, daisies and sunflowers, and yet they all are united by sharing the proportional limitations of musical root harmonies.” ~ Gyorgy Doczi


Spirals in the Animal World

Spirals are seen in snail shells, elephant trunks, chameleon tails, snakes, monkey tails, pig tails, and sea horse tails.


The curves of bird beaks resemble the curves of tiger, lion, cat & bear claws.  These curves resembles the curves of tusks of the rhino, elephant, warthog and walrus.


Spirals are seen in ram horns, antelope horns, goat horns, black buck horns, boar tusks, elephant tusks and spiraling narwhal tusks.

“As the horn material of a ram accumulates, growing larger and more massive, its golden spiral maintains the same center of gravity.  Thus the ram need not adjust its posture throughout life to uphold the growing horns.”18


Take a look at Plate 100: Antilopina from Art Forms in Nature.

These horned antelope species include:

  1. Tetracerus quadricornis
  2. Catoblepas gnu = Connochaetes gnou
  3. Tragelaphus gratus = Tragelaphus spekii gratus
  4. Antilocapra americana / Antilope furcifera
  5. Antilope ellipsiprymna / Cervicapra ellipsiprymna = Kobus ellipsiprymnus
  6. Hippotragus niger
  7. Addax nasomaculatus
  8. Tragelaphus kudu / Antilope strepsiceros = Tragelaphus strepsiceros
  9. Tragelaphus scriptus / Antilope maculata


1: Four-horned Antelope
2: Black Wildebeest
3: Sitatunga
4: Pronghorn
5: Waterbuck
6: Sable Antelope
7: Addax
8: Greater Kudu
9: Bushbuck



The digestive system of a pig (like other animals) follows a whirlpool to transform food into energy.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe


Fish, tortoise, rabbit and other vertebrate embryos begin life as unfurling spirals.

The cochlea of a cow, guinea pig, mouse, and rat are spirals that register different levels of frequency.

Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe


Many more examples of the golden ratio can be found in the animal world as well.  A comprehensive analysis needs to be carried out to see just how much the golden ratio shows up in the structure of animals and to see if perhaps it shows up in all animals in one way or another.


“Examination proves that the same law of numbers governs the animate as well as the inanimate world, from the lowest to the highest form of life.

Learning thus that insects are divided or numbered by six, it would be a natural conclusion to suppose that their proportions would be interpreted by the square, the hexagon, and the equilateral triangle, as this we find to be precisely and emphatically the case.

The quadruped with his four legs, and the bird with two , each submits to the universal law, their proportions and the grouping of their organs being governed by the harmonic numbers of two, four, and eight, the various species producing each two eyes, two ears, two horns, if any, birds having two wings, each of two sections.

There can be no doubt that the numbers governing the necessary organs and members with which all creatures are variously endowed are chosen from the fact that thus endowed they can best walk, see, hear, climb, fly and perform whatever function is necessary for their existence, and thus the peculiar harmonic beauty which characterizes each family and species results from Nature’s application of her law of numbers.

To all of these rules the human family is no exception, and we find in it prominently established the number two, together with the continuing series of 1, 2, 3, 5 and 8.”19

This will segue us from the animal realm into the human realm.

In the next article we will discuss the consciousness of animals and then will move onto a 55-part series on humans and human life.  This will include the geometry of the human form as well as an extensive series on human nature, the human life cycle, chakras, sleep, dreams, death, reincarnation, health, illness and healing.


  1. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on its Relation to Proportional Form, Forgotten Books, 2017
  2. ibid.
  3. Gledhill, Linden, Butterfly wings,
  4. Vila Laguarta, Manuel Cristobal, Nature by Numbers,
  5. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on its Relation to Proportional Form, Forgotten Books, 2017
  6. ibid.
  7. Thompson, D’Arcy, On Growth And Form, Cambridge University Press, 2014
  9. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on its Relation to Proportional Form, Forgotten Books, 2017
  10. ibid.
  11. ibid.
  12. Wilson, Euan, Socionomics and Fibonacci: The Golden Ratio Governs Life, Beauty and the Universe,
  13. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, HarperPerennial, 1995
  14. Yong, Ed, Birds that Fly in a V Formation Use An Amazing Trick, 15 January 2014,
  15. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on its Relation to Proportional Form, Forgotten Books, 2017
  16. ibid.
  17. ibid.
  18. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, HarperPerennial, 1995
  19. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on its Relation to Proportional Form, Forgotten Books, 2017

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