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In this article we will look at the geometry of sea creatures in more detail.  In the previous article we saw the astonishing Platonic solid shapes of diatoms (phytoplankton) and radiolarian (zooplankton).

We will now take a visual journey through some of the other sea creature realms from small to large.

 

Sea Anemones

Sea anemones are water-dwelling, predatory animals.

They are related to corals, jellyfish, tube-dwelling anemones and Hydra.

Visually they resemble a flowering plant on land.  They are quite colorful and vary in their patterning.

Many show radial symmetry.

Below is an anemone with spiral tendrils.

Here is the Venus Fly-trap sea anemone:

 

And here are some close-up images of sea anemones:

 

We will now look at one plate from Ernst Haeckel’s Art Forms in Nature.

Here we see Plate 49: Actiniae.  These specimens include beautiful sea anemones that resemble mushrooms and flowers.

  1. Heliactis bellis = Cereus pedunculatus
  2. Mesacmaea stellata
  3. Aiptasia Couchii = Aiptasia mutabilis
  4. Cylista impatiens = Choriactis impatiens
  5. Bunodes thallia = Anthopleura thallia
  6. Metridium praetextum = Actinostella flosculifera
  7. Heliactis troglodytes = Sagartia troglodytes
  8. Anthea cereus = Anemonia sulcata
  9. Aiptasia undata = Aiptasia diaphana
  10. Aiptasia diaphana
  11. Bunodes monilifera = Paractis monilifera
  12. Corynactis viridis
  13. Metridium concinnatum = Oulactis concinnata
  14. Sagartia chrysoplenium = Chrysoela chrysosplenium
  15. Actinoloba dianthus = Metridium senile

 

 

 

Glass Sponges

Hexactinellida sponges (glass sponges) are sponges with a skeleton made from 4, 6 and/or 8-pointed siliceous spicules.  They are relatively uncommon and are found at depths of 1480 to 2950 feet under the sea.

They are more or less cup-shaped animals and are made of sturdy lattice-like internal skeletons made up of fused spicules of silica.

One example is the beautiful Venus Flower Basket (Euplectella aspergillum) shown below.

8-fold geometry can be found among glass sponges when looking from below.  Credit: Michael Schnieder – A Beginner’s Guide to Constructing the Universe

 

Ernst Haeckel’s Art Forms in Nature, from 1904, has beautiful illustrations of these and other sea creatures.

See, for instance, Plate 5 : Calcispongiae – Cellular sponges from animal phylum Porifera

  1. Ascandra pinus = Leucosolenia complicata, habitus
  2. Ascandra sertularia = Leucosolenia sertularia, habitus
  3. Ascilla gracilis = Guancha gracilis, habitus
  4. Syculmis synapta = Amphoriscus synapta, rooting spicule
  5. Syculmis synapta = Amphoriscus synapta, rooting spicule
  6. Sycurus primitivus = Sycettaga primitiva, habitus (cut open)
  7. Sycodendron ampulla = Sycon ampulla, habitus
  8. Sycarium elegans = Sycon elegans, habitus (cut open)
  9. Sycortis quadrangulata = Sycon quadrangulatum, lengthwise section of body wall
  10. Sycandra compressa = Grantia compressa, skeleton of wall tube
  11. Sycarium elegans = Sycon elegans, lengthwise section of body wall
  12. Sycaltis perforata = Amphoriscus perforatus, lengthwise section of body wall
  13. Sycetta strobilus = Grantia strobilus, habitus

 

And Plate 35: Hexactinellae – Glass sponges

  1. Farrea Haeckelii = Farrea occa, habitus
  2. Farrea Haeckelii = Farrea occa, mesh of skeleton
  3. Euplectella aspergillum, habitus
  4. Euplectella aspergillum, mesh of skeleton
  5. Holtenia crateromorpha, habitus
  6. Sclerothamnus spiralis, habitus
  7. Sclerothamnus spiralis, branch
  8. Polyopogon amadu = Poliopogon amadou, cross-section
  9. Pheronema rhaphanus, needle spicule
  10. Hyalonema indicum, amphidiscus spicule
  11. Hyalonema conus, amphidiscus spicule
  12. Regadrella phoenix, floricome spicule
  13. Saccocalyx pedunculata = Saccocalyx pedunculatus, discohexaster spicule
  14. Crateromorpha Meyeri, discohexaster spicule
  15. Hyalostylus dives, hexaster spicule
  16. Polylophus philippinensis = Lophocalyx philippinensis, plumicome spicule
  17. Stylocalyx tenera = Hyalonema tenerum, amphidiscus spicule

 

The images illustrate Haeckel’s fundamental monistic notion of the ‘unity of all living things’.

 

 

Corals

Corals are marine invertebrates that typically live in compact colonies of many identical individual polyps.  Their structure is one of cellular tessellation.

 

Some interesting examples include:

Star coral – hexagonal skeleton

 

Tube coral – cylindrical skeletons packed in hexagonal groups

Credit: John Connell

 

Fungia – live animal and skeleton pictured below

 

Eusmilia fastigiata

 

Convexastrea sexradiata

 

Montastraea cavernosa

 

Pillar coral

 

Brain coral

 

Staghorn coral

 

Fan coral

 

Fire coral

 

Here is an interesting soft coral with fractal branching:

 

 

Now we will take a look at several examples from Ernst Haeckel’s Art Forms in Nature.  These include:

Plate 29: Tetracoralla (Rugosa) – These species are an extinct order of solitary colonial corals from approximately 250 to 500 million years ago.  Much beautiful geometry can be seen in these specimens.

  1. Omphyma turbinata, polyp skeleton from the side
  2. Cyathophyllum marmini = Marisastrum marmini, coral head in cross-section
  3. Pachyphyllum devoniense = Phillipsastrea devoniensis, coral head in cross-section
  4. Goniophyllum pyramidale, polyp skeleton from above
  5. Menophyllum tenuimarginum, polyp skeleton from above
  6. Zaphrentis cornicula = Zaphrentis phrygia, polyp skeleton from the side with part of wall removed
  7. Cyathophyllum expansum = Palaeosmilia murchisoni stutchburyi, polyp skeleton from the side
  8. Cyathaxonia cynodon, polyp skeleton from the side with part of wall removed
  9. Lithostrotion irregulare = Siphonodendron irregulare, polyp skeleton in lengthwise section
  10. Alveolites battersbyi = Caliapora battersbyi, polyp skeleton in lengthwise section
  11. Hadrophyllum multiradiatum, polyp skeleton from above
  12. Clisiophyllum turbinatum = Dibunophyllum bipartitum turbinatum, polyp skeleton in cross-section
  13. Acervularia ananas, coral head from above
  14. Syringophyllum organum = Sarcinula organum, coral head from above
  15. Cyathophyllum articulatum = Entelophyllum articulatum, coral head from the side

 

 

Here we see Plate 9: Hexacorallia – Beautiful geometry can be seen among these specimens of corals & gorgonians.

  1. Lophohelia prolifera, skeleton of colony from the side
  2. Leptocyathus elegans, skeleton of polyp from above (2a: from the side)
  3. Cyathina cylindrica = Caryophyllia cylindrica, skeleton of polyp from above
  4. Balanophyllia floridana, skeleton of polyp from the side
  5. Rhizotrochus fragilis = Polymyces fragilis, skeleton of polyp from the side
  6. Stephanophyllia elegans, skeleton of polyp from above
  7. Astrocyathus paradoxus = Caryophyllia paradoxus, skeleton of polyp from above
  8. Maeandrina filograna = Meandrina filograna, skeleton of coral head from above
  9. Madrepora fruticosa = Acropora humilis, skeleton of colony from the side
  10. Flabellum australe = Flabellum australe, skeleton of polyp from above
  11. Flabellum alabastrum, skeleton of polyp from above
  12. Thamnastraea arachnoides, part of skeleton of coral head from above
  13. Porites furcata, part of skeleton of coral head from above
  14. Stephanophyllia complicata, skeleton of polyp from above (14a: from the side)
  15. Leptopenus discus, skeleton of polyp from above

 

 

Plate 69: Hexacorallia – These include approximately 4,300 species of aquatic organisms formed of polyps, generally with 6-fold symmetry.  These include all the stony corals, sea anemones and zoanthids.

  1. (center, above): Turbinaria transformis
  2. (center left, below): Turbinaria robusta
  3. (center left, above): Lophoseris frondifera = Pavona frondifera / Madrepora ficoides
  4. (top center): Lophoseris divaricata = Pavona divaricata
  5. (top right): Hydnophora racemosa = Hydnophora exesa
  6. (top left): Tridacophyllia lactuca = Pectinia lactuca
  7. (center right, above): Manicina crispata
  8. (center right, below): Lithophyllia lacera = Scolymia lacera / Scolymia cubensis
  9. (center, below): Astraea magnifica = Goniastrea aspera
  10. (bottom center): Astraea expansa
  11. (bottom left): Pterogyra laxa
  12. (bottom right): Euphyllia striata = Euphyllia glabrescens

 

Plate 39: Gorgonida (Alcyonacea) – Soft corals – These species do not produce calcium carbonate skeletons.  They contain minute spiny skeletal elements called sclerites that give their flesh a spiky, grainy texture that deters predators.  We see interesting geometry, fractal branching and vortex streets among these species.

  1. Gorgonia verrucosa = Eunicella verrucosa, polyp
  2. Platycaulos Danielsseni, skeleton spicule
  3. Euplexaura parciclados, skeleton spicule
  4. Primnoella biserialis = Primnoella chilensis, two 8-polyp whorls of a branch
  5. Primnoella murrayi = Convexella murrayi, two 6-polyp whorls of a branch
  6. Stenella spinosa = Parastenella spinosa, 2-polyp whorl of a branch
  7. Juncella juncea, skeleton spicule
  8. Calyptrophora japonica, three 3-5-polyp whorls of a branch
  9. Gorgonia verrucosa = Eunicella verrucosa, branch with polyps
  10. Acanthogorgia longiflora, branch with polyp
  11. Primnoella australasiae, three 8-polyp whorls of a branch
  12. Calypterinus allmani = Narella allmani, three 5-7-polyp whorls of a branch
  13. Paramuricea spinosa, polyp
  14. Juncea barbadensis = Ctenocella barbadensis, skeleton spicule
  15. Anthomuricea argentea, polyp
  16. Calyptrophora wyvillei, three 4-polyp whorls of a branch

 

Now we will take a look at two other marine colonial organisms and the Ascidiae, solitary organisms.

 

Plate 17: Siphonophorae – This order of marine animals contains 188 species.  These appear to be a single organism but are actually a colonial organism composed of small individual animals called zooids that have their own special function for survival.  Some, such as the Portuguese man o’war, resemble jellyfish.  We see beautiful flower-like geometry among these species.

  1. (top center): Porpema medusa = Porpema prunella, adult colony, side view
  2. (top left, upper): Porpema medusa = Porpema prunella, adult colony, group of sexual polyps
  3. (top right, upper): Porpema medusa = Porpema prunella, adult colony, gas bladder
  4. (center right): Porpema medusa = Porpema prunella, adult colony, side view without tentacles
  5. (bottom right): Porpalia prunella, adult, top view
  6. (top left, lower): Discalia medusina = Porpema prunella, young colony, bottom view without tentacles
  7. (center left): Discalia medusina = Porpema prunella, young colony, side view
  8. (center): Disconalia gastroblasta, young colony, bottom view
  9. (top right, lower): Disconalia gastroblasta, very young colony, bottom view without tentacles
  10. (bottom left, left half): Disconalia gastroblasta, young colony, horizontal section through upper bell
  11. (bottom left, right half): Disconalia gastroblasta, young colony, horizontal section through lower bell
  12. (bottom center): Disconalia gastroblasta, young colony, sexual polyp

 

 

Plate 33: Bryozoa – “Moss animals” – These aquatic invertebrate animals are filter feeders.  One genus is solitary and the rest are colonial organisms.  We see beautiful mandala-like geometry among these species.

  1. Lepralia spinifera = Phaeostachys spinifera, partial skeleton of colony
  2. Cribrilina punctata, skeleton of individual
  3. Umbonula verrucosa = Umbonula ovicellata, skeleton of individual
  4. Cribrilina radiata) / Colletosia radiata, skeleton of individual
  5. Lepralia alata = Umbonula alvareziana, partial skeleton of colony
  6. Bugula flabellata = Bugulina angustiloba, partial skeleton of colony
  7. Cupularia stellata = Cupuladria canariensis, skeleton of young colony
  8. Farciminaria aculeata, partial skeleton of colony
  9. Umbonula reticulate, skeleton of individual
  10. Cribrilina costata = Beania costata, skeleton of individual
  11. Smittia Landsborovii = Smittina landsborovii, skeleton of individual
  12. Smittia reticulata = Smittoidea reticulata, skeleton of individual
  13. Lepralia annulata = Cribrilina annulata , skeleton of individual
  14. Diachoris magellanica = Beania magellanica, partial skeleton of colony
  15. Diachoris crotali = Beania crotali, partial skeleton of colony
  16. Flustra Gayi = Gemelliporina glabra, skeleton of colony
  17. Flustra Gayii = Gemelliporina glabra, partial skeleton of colony
  18. Schizoporella hyalina = Celleporella hyalina, partial skeleton of colony
  19. Lepralia variolosa = Escharella variolosa, skeleton of individual
  20. Chorizopora Brongniartii, partial skeleton of colony
  21. Flustra Aragoi = Membraniporella aragoi = Klugerella aragoi, partial skeleton of colony

 

 

Plate 85: Ascidiae – These are sac-like marine invertebrate filter feeder animals.  We see astonishingly beautiful geometry among these species that calls to mind beautifully decorated Faberge eggs.

  1. Cynthia melocactus = Boltenia echinata, individual from above
  2. Cynthia melocactus = Boltenia echinata, individual from the front
  3. Cynthia melocactus = Boltenia echinata, individual in lengthwise section
  4. Molgula tubulosa = Eugyra arenosa, mouth region
  5. Fragarium elegans = Aplidium elegans, colony
  6. Polyclinum constellatum, colony
  7. Polyclinum constellatum, part of colony
  8. Synoecum turgens = Synoicum turgens, part of colony
  9. Botryllus polycyclus = Botryllus schlosseri, part of colony
  10. Botryllus rubigo = Botryllus schlosseri, part of colony
  11. Botryllus Marionis = Botryllus schlosseri, part of colony
  12. Botryllus helleborus = Botryllus schlosseri, part of colony
  13. Polycyclus cyaneus = Botryllus schlosseri, colony
  14. Botrylloides purpureus = Botrylloides leachii, part of colony

 

 

Plate 95: Amphoridea – These are fossil echinoderms, without arms and ambulacra, and are considered ancestors of all Echinoderms.  Much 5-fold and 6-fold geometry can be seen.

  1. Placocystis crustacea = Enoploura balanoides (1a: from above, 1b: from below)
  2. Pleurocystis filitexta = Pleurocystites filitextus (2a: from above, 2b: from below)
  3. Orocystis Helmhackeri (3b: single armour plate)
  4. Deutocystis modesta = Echinosphaerites, juvenile
  5. Citrocystis citrus / Echinosphaera citrus = Echinosphaerites kloedeni (5b,5c: single armour plates)
  6. Acanthocystis briareus
  7. Aristocystis bohemica, single armour plate
  8. Ophiothrix fragilis, juvenile
  9. Pluteus bimaculatus / Ophiura filiformis, pluteus larva
  10. Plutellus aequituberculatus / Echinocidaris aequituberculata = Arbacia lixula, pluteus larva
  11. Bipinnaria asterigera / Luidia sarsi = Luidia sarsi, bipinnaria larva
  12. Auricularia nudibranchiata = Protankyra brychia, auricularia larva

 

 

Echinoderms

There are about 950 species of echinoderms.

They are recognized by their 5-point symmetry.

Examples include:

Sea stars – Sea stars have 5 legs, 7 legs, 9 legs, 10 legs, 18 legs and even up to 50 legs!

 

A starfish mouth also has 5-pointed symmetry:

 

Here is the Labidiaster annulatus – a starfish with 40-50 legs – from the Antarctic Peninsula.

 

Sea urchins

 

Sand dollars

 

Sea cucumbers – sea cucumber and cross-section of sea cucumber shown below.

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

 

Sea lilies

 

 

Other examples we will see are from Haeckel’s Art Forms in Nature.  These include:

Plate 10: Ophiodea – Brittle stars

  1. (center): Ophiothrix capillaris = Macrophiothrix capillaris, adult, underside
  2. (top center): Ophiotholia supplicans, adult, underside
  3. (top right): Ophiocoma rosula = Ophiothrix fragilis, adult, underside
  4. (top left): Astroschema brachiatum, adult, upperside
  5. (center left): Astroschema horridum, adult, underside
  6. (bottom center): Astroschema rubrum, adult, underside
  7. (center right): Ophiocreas oedipus, adult, upperside
  8. (bottom left): Pluteus paradoxus = Ophiura albida, young pluteus larva
  9. (bottom right): Pluteus paradoxus = Ophiura albida, older pluteus larva

 

 

Plate 30: Echinidea – Sand dollars

  1. Clypeaster rosaceus, shell without spines from above
  2. Clypeaster rosaceus, internal organs from above
  3. Encope emarginata, shell without spines from above
  4. Encope emarginata, shell without spines from below
  5. Echinocyamus pusillus, 48-hour-old plutellus larva from below
  6. Echinocyamus pusillus, 10-day-old plutellus larva from below
  7. Echinocyamus pusillus, 45-day-old juvenile from below
  8. Echinocyamus pusillus, 50-day-old juvenile from below
  9. Echinocyamus pusillus, 60-day-old juvenile from above

 

 

Plate 40: Asteridea – Sea Stars

  1. Asterias rubens, immature
  2. Asterias, scaphularia larva from below
  3. Asterias, bipinnaria larva from below
  4. Asterias, young brachiolaria larva from the side
  5. Asterias, older brachiolaria larva from above
  6. Asterias, older brachiolaria larva from below
  7. Asterias, pentapalmar larva from below
  8. Asterias, pentapalmar larva from above
  9. Hymenaster echinulatus, mouth region
  10. Pteraster stellifer, 1882, mouth region
  11. Hymenaster echinulatus, from above (top half) and below (bottom half)
  12. Pteraster stellifer, from above (top half) and below (bottom half)

 

 

Plate 60: Echinidea

  1. Cidaris tribuloïdes, from below
  2. Cidaris baculosa = Prionocidaris baculosa, mouth region of shell from below (spines removed)
  3. Cidaris baculosa = Prionocidaris baculosa, periproct region of shell from above (spines removed)
  4. Dorocidaris papillata = Cidaris cidaris, pedicellaria
  5. Strongylocentrus nudus, pedicellaria in lengthwise section
  6. Phyllacanthus annulifera = Prionocidaris baculosa, spine
  7. Phyllacanthus baculosa, spine
  8. Psammechinus miliaris, endplate of tube-foot
  9. Centrostephanus longispinus, spine in cross-section
  10. Sphaerechinus esculentus = Sphaerechinus granularis, pedicellaria in lengthwise section
  11. Sphaerechinus esculentus = Sphaerechinus granularis, mouthparts (“Aristotle’s Lantern”) from the side

 

 

Plate 70: Ophiodea – Brittle Stars

  1. Astrophyton darwinium, bottom view
  2. Astrophyton darwinium, center without arms, top view
  3. Ophiopholis japonica, center without arms, partial top view
  4. Ophiotholia supplicans, lower center without arms, side view
  5. Ophiohelus umbella, skeleton of an arm section near the arm’s end
  6. Ophioglypha minuta = Aspidophiura minuta, center without arms, partial top view
  7. Hemipholis cordifera = Hemipholis elongata, mouth, bottom view

 

 

Plate 80: Blastoidea – Extinct Echinoderms

  1. Pentremites pyriformis, from the side
  2. Pentremites orbignyanus, from above
  3. Pentremites, from above; brachioles extended (top), folded back (center), removed (below)
  4. Zygocrinus cruciatus = Astrocrinus cruciatus, from above
  5. Orophocrinus stelliformis, from above
  6. Phaenoschisma acutum, from above
  7. Elaeacrinus olivanites, from the backside
  8. Elaeacrinus Verneuili = Nucleocrinus verneuili (8a: from the backside, 8b: from below, stalk removed)
  9. Codonaster trilobatus = Codaster acutus, from above
  10. Eleutherocrinus Cassedayi (10a: from above, 10b: from below, 10c: from the side)
  11. Asteroblastus stellatus (11a: from above, 11b: from below, 11c: from the backside)
  12. Asteroblastus Volborthi, from the side with brachioles shown

 

 

Sea Snails

Sea snails are marine gastropods with shells that demonstrate logarithmic spiral patterns.

“The shape of shells,” writes Gyorgy Doczi, “have been the subject of many studies which show that their harmonious shapes unfold in logarithmic spirals characterized by the golden section’s proportions.”

There are many varieties.  Most species are spirally coiled.

Some have conical shells such as limpets.

 

As Samuel Colman writes, “Among all the enchanting things in the realm of beauty, the shell is one of the most perfect as a representative form of spiral construction and the one where the spiral is most obvious to the casual eye, at once announcing itself a geometric product.”

He then says, “A plant is numbered by the geometric figure which renders its general form, influencing the division of its petals, stamens and pistils; shells receive their distinguishing number from the turns in their spirals, generally six, but sometimes running from three to fifteen according to the family.  The number of ribs and horns which decorate many shells is also decided by the intersecting lines of their spiral arcs and the geometric plan…The spirals of Nature are few in number, generally produced by the angles of 36°, 40° and 38° 30’.”

 

Samuel Colman shows the geometric analysis of many of these spirals found naturally in shells.  These are on pages 116-140 of Nature’s Harmonic Unity.  A few examples are shown below.

 

 

Other examples of these beautiful creations include:

Buccinum undatum

 

Littorina

 

Natica

 

Wentletrap

 

Cochlespira radiata

 

Terebra dislocata

 

Conus marmoreus

 

Harpa davidis

 

Mitra stictica

 

Oliva mantichora

 

Vexillum ornatum

 

Babelomurex finchii

 

Syrinx aruanus 


 

Take a look at Plate 53: Prosobranchia – Sea snails from Art Forms in Nature.

  1. Calcar triumphans = Guildfordia triumphans, from above
  2. Conus imperialis
  3. Harpa ventricosa = Harpa davidis
  4. Murex tenuispinus = Murex pecten pecten
  5. Murex inflatus = Chicoreus ramosus
  6. Fusus longicauda = Fusinus colus, cut open
  7. Astralium imperiale = Astraea heliotropium, from below
  8. Astralium imperiale = Astraea heliotropium, from above

 

“That even widely differing shell shapes share the proportions of golden dinergy can be seen by coparing, for instance, the Atlantic sundial (Architectonica nobilis) – which is almost perfectly circular – with a delicate variety of abalone (Haliotis asinina) which is elongated like a donkey’s ear; (asininum means “donkey” in Latin).  Successive stages of growth in the abalone – measured along neighboring, equidistant radii – are Fibonacci numbers.

Credit: Gyorgy Doczi – The Power of Limits, 1981

“In the Atlantic sundial there is a proportion of 20:1 between radial and rotational growth.  The abalone takes only four to five radii or squares to move from one circle to the next: a proportion of 4 or 5:1.  Construction of the golden section at the right the figure below prove that the proportions of neighboring whorl-widths are the golden ones:  all the shaded rectangles – width and length of which correspond to the width of neighboring whorls – are golden rectangles.  The harmony created by this relationship is illustrated also by the organ-pipe like bars, indicating the cumulative length of successively widening whorls.  This sharing of the same proportional limitations throughout the entire growth process is also expressed by a series of equations all equaling phi.”

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

The diagram below shows the dilated whelk (penion dilatus).

The wave diagram at the left indicates how all neighboring whorls share the same golden relationship.  Credit: Gyorgy Doczi – The Power of Limits, 1981

 

“A side view of the bear’s paw clam illustrates the marvelous precision of dinergic sharing between the two opposite valves interlocking and complementing each other.  Reconstructions of the basic outlines reveal that the harmony of this lovely shape shares the same golden dinergies as so many earlier examples of organic growth.”

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

 

Nautilus

The nautilus is the familiar mollusk that has a chambered nautilus shell in a Logarithmic/Fibonacci spiral. It is one of the oldest species of cephalopods on Earth.

 

Ammonite (extinct mollusk)

 

Take a look at Plate 44: Ammonitida – Ammonites from Art Forms in Nature.

  1. Ammonites (Cardioceras) cordatus, from the side
  2. Ammonites (Cardioceras) cordatus, from below
  3. Ammonites (Schloenbachia) Coupei, from the side
  4. Ammonites (Schloenbachia) Coupei, from below
  5. Ammonites (Ptychites) opulentus, from the side (outer shell layer removed)
  6. Ammonites (Ptychites) opulentus, from below
  7. Ammonites (ornatus) mammillaris = Douvilleiceras mammillatum, from the side
  8. Ammonites (planulatus) cavernosus = Puzosia planulatus  cross-section
  9. Ammonites (amaltheus) rotula = Amaltheus margaritatus, from the side
  10. Ammonites (stephanoceras) Humphryi = Stephanoceras humphriesianum, from the side

 

Bivalve (clam) shells grow as mirror spirals.

 

See Plate 55: Acephala – Bi-valve shells (Mollusks)

  1. Cytherea Dione = Pitar dione = Hysteroconcha dione, opening view
  2. Cytherea Dione = Pitar dione = Hysteroconcha dione, valve view
  3. Cytherea Dione = Pitar dione = Hysteroconcha dione, hinge view
  4. Cardium aculeatum = Acanthocardia aculeata, valve from outside
  5. Cardium aculeatum = Acanthocardia aculeata, side view
  6. Hemicardium cardissa = Corculum cardissa, valve from outside
  7. Hemicardium cardissa = Corculum cardissa, opening view
  8. Hemicardium cardissa = Corculum cardissa, frontside view
  9. Hemicardium cardissa = Corculum cardissa, backside view
  10. Tridacna squamosa, hinge view
  11. Tridacna squamosa, opening view
  12. Tridacna squamosa, valve from outside
  13. Tridacna squamosa, valve from inside
  14. Hippopus maculatus = Hippopus hippopus, hinge view

 

Samuel Colman tells us, “All of the above examples go far toward the justification of the hypothesis that the three simple figures of geometry, the square, pentagon, and hexagon, with their progressions, constitute the principle by which Nature coordinates her design, revealing that the study of the shell alone may teach man how to reach a similar perfection in his architectonic compositions.”

 

 

Sea Slugs

The Spanish Dancer nudibranch Hexabranchus sanguineus lay egg ribbons that look like a red rose.

Credit: Asbjorn Hansen

 

Sea Worms

Here is the Spirobranchus giganteus – the Christmas tree worm.  These creatures are tube-building polychaete worms belonging to the family Serpulidae.

 

 

Jellyfish

Jellyfish are soft-bodied free-swimming aquatic animals with a gelatinous umbrella-shaped bell and trailing tentacles.  These are well known for their poisonous effects on humans.

 

Examples include:

Pacific sea nettle

 

Flower hat jelly

 

Purple-striped jelly

Credit: Mike Baird

Moon jelly

Credit: Heather Paul

 

White-spotted jellyfish

 

Umbrella jelly

 

Lion’s mane jellyfish

Credit: Derek Keats

Aequorea victoria – hydromedusa

 

Box jellyfish

 

Other examples from Art Forms in Nature include:

Plate 8: Discomedusae

  1. Desmonema Annasethe = Cyanea annasethe, adult
  2. Desmonema Annasethe = Cyanea annasethe, bell of adult from below, tentacles and most other appendages removed, retaining gonads (top left) and mouth veils (bottom); top right: cut open to show stomach
  3. Floscula Promethea, juvenile
  4. Chrysaora mediterranea = Chrysaora hysoscella, adult

 

 

Plate 16: Narcomedusae

  1. (center): Pegantha pantheon, side view
  2. (bottom center): Pegantha pantheon, single bell lobe
  3. (top left): Aeginura myosura = Aeginura grimaldii, bottom view
  4. (top right): Solmaris Godeffroyi = Pegantha godeffroyi, bottom view
  5. (center left): Cunarcha aeginoides = Aegina citrea, side view
  6. (bottom left): Cunarcha aeginoides = Aegina citrea, top view
  7. (top center): Cunarcha aeginoides = Aegina citrea, auditory tentacle
  8. (center right): Cunantha primigenia = Aegina, young medusa, side view
  9. (bottom right): Cunoctantha discoidalis = Cunina octonaria, top view

 

 

Plate 18: Discomedusae

  1. center): Linantha lunulata, adult medusa from above
  2. (right): Linantha lunulata, adult medusa from the side
  3. (top left): Palephyra primigenia = Palephyra antiqua, half-grown medusa from the side, swimming
  4. (left): Palephyra primigenia = Palephyra antiqua, half-grown medusa from the side, descending
  5. (bottom center): Palephyra primigenia = Palephyra antiqua, half-grown medusa from the side, resting on bottom
  6. (top right): Zonephyra zonaria = Palephyra pelagica, young medusa from the side
  7. (top center): Strobila monodisca = Coronatae, polyp budding off medusa from the side
  8. (bottom left): Nauphanta Challengeri = Nausithoe challengeri, adult medusa from above
  9. (bottom right): Atolla Wyvillei, adult medusa from below

 

 

Plate 26: Trachomedusae

  1. (center): Carmaris Giltschi = Geryonia proboscidalis, adult medusa, underside view
  2. (top left): Carmaris Giltschi = Geryonia proboscidalis, adult medusa, side view
  3. (bottom center): Carmaris Giltschi = Geryonia proboscidalis, gonads and mouth, underside view
  4. (top right): Carmarina hastata = Geryonia proboscidalis, adult medusa, side view
  5. (top center): Carmarina hastata = Geryonia proboscidalis, young medusa, side view
  6. (bottom left): Carmarina hastata = Geryonia proboscidalis, half-grown medusa, side view
  7. (bottom right): Geryones elephas = Geryonia proboscidalis, adult medusa, side view

 

 

Plate 28: Discomedusae

  1. (top center): Toreuma bellagemma, medusa from the side
  2. (center): Toreuma bellagemma, medusa from
  3. (bottom right): Toreuma bellagemma, medusa, mouth from below
  4. (bottom left): Toreuma bellagemma, medusa, gonads from above
  5. (top left): Toreuma thamnostoma, medusa, bell from above
  6. (top right): Cassiopeja cyclobalia = Cassiopea andromeda, medusa, bell from above

 

Plate 36: Leptomedusae

  1. (top left): Aequorea discus = Aequorea pensilis, bottom view
  2. (top right): Zygocanna diploconus, side view
  3. (bottom left): Zygocanna diploconus, bottom view
  4. (center right): Polycanna germanica = Aequorea forskalea, side view
  5. (center left): Zygocannula diploconus, side view
  6. (bottom right): Orchistoma elegans, side view

 

 

Plate 38: Peromedusae

  1. (center): Periphylla mirabilis = Periphylla periphylla, bottom view
  2. (bottom left): Periphylla mirabilis = Periphylla periphylla, side view
  3. (bottom right): Periphylla Peronii = Periphylla periphylla, side view
  4. (top left): Periphylla hyacinthina = Periphylla periphylla, mouth tube
  5. (top right): Periphylla mirabilis = Periphylla periphylla, schematic section
  6. (top center): Periphylla mirabilis = Periphylla periphylla, sensory bulb
  7. (bottom center): Periphylla mirabilis = Periphylla periphylla, cross-section through tentacle root

 

 

Plate 46: Anthomedusae

  1. Gemmaria sagittaria = Zanclea gemmosa, young medusa from the side
  2. Rathkea fasciculata = Koellikerina fasciculata, medusa from above
  3. Tiara pileata = Neoturris pileata, medusa from below
  4. Tiara pileata) = Neoturris pileata, internal organs from the side
  5. Stomotoca pterophylla = Larsonia pterophylla, internal organs from the side
  6. Thamnostylus dinema = Sarsiella dinema, medusa from the side

 

 

Plate 78: Cubomedusae

  1. Chirodropus palmatus, from the side
  2. Chiropsalmus quadrigatus = Chiropsoides quadrigatus, from the side
  3. Charybdea obeliscus / Alatina obeliscus, from the side
  4. Charybdea obeliscus / Alatina obeliscus, alimentary tract
  5. Charybdea murrayana, from below
  6. Charybdea murrayana, from above
  7. Procharybdis tetraptera = Alatina tetraptera, from the side
  8. Tamoya prismatica = Tamoya haplonema, from the side

 

 

Coleoidea

Coleoidea includes soft-bodied or shell-less cephalopods including the octopus, squid and cuttlefish.

 

Octopus

These are the most intelligent of all invertebrates.

They have two eyes; 8 arms; bilateral symmetry.

 

The pattern of the vortex street appears in the alternating placement of gripping suckers on an octopus’s spiral tentacle.

 

 

Squid

There are about 304 species.

They have 1 head; 8 arms; and 2 tentacles.

 

Plate 54: Gamochonia from Art Forms in Nature show several species of squid.

  1. Chiroteuthis Veranyi, from below
  2. Histioteuthis Rüppellii = Histioteuthis bonnellii, from above
  3. Pinnoctopus cordiformis, from above
  4. Octopus vulgaris, from above
  5. Octopus granulatus = Octopus vulgaris, from below

 

 

Cuttlefish

Cuttlefish have large W-shaped pupils; 8 arms; and 2 tentacles.

They have the largest brain-to-body size ratios of all invertebrates.

 

 

Arthropods

Arthropods are invertebrate animals that have an exoskeleton, a segmented body, and paired jointed appendages.  These include some insects, arachnids, myriapods, horseshoe crabs and crustaceans.

 

Horseshoe Crab

Horseshoe crabs are marine and brackish water arthropods that originated 450 million years ago.  Because of this they are considered ‘living fossils’.

 

These are illustrated in Plate 47: Aspidonia in Art Forms in Nature.

  1. Limulus moluccanus = Tachypleus gigas, adult male from above
  2. Limulus moluccanus = Tachypleus gigas, adult male from below
  3. Limulus moluccanus = Tachypleus gigas, larva
  4. Eurypterus Fischeri = Eurypterus tetragonophthalmus / Baltoeurypterus tetragonophthalmus, from above
  5. Pterygotus anglicus, from above
  6. Trinucleus Goldfussi = Onnia goldfussi, from above
  7. Deiphon Forbesi, from above
  8. Phacops latifrons, curled-up individual (8a: from forward, 8b: from the side)
  9. Dalmania punctata = Asteropyge punctata, from above
  10. Ampyx Rouaulti = Raphiophorus rouaulti, from above
  11. Paradoxides bohemicus, from above
  12. Cheirurus insignis, from above
  13. Acidaspis Dufresnoyi = Selenopeltis buchi, from above
  14. Megalaspis extenuatus = Megistaspidella extenuata, from above
  15. Harpes ungula = Bohemoharpes ungula (15a: from above, 15b: from the side)
  16. Agnostus pisiformis, from above
  17. Lichas palmata = Trochurus speciosus, from above
  18. Hydrocephalus saturnoides = Eccaparadoxides pusillus / Phanoptes pusillus, from above
  19. Sphaerexochus mirus, from the side
  20. Triarthrus Becki (20a: from above, 20b: from below)

 

 

Crustaceans

Crustaceans are large, diverse arthropods.

Examples include:

Crabs – 20 body segments; 2 claws, 8 legs

 

Gyorgy Doczi tells us: “In crustaceans we again find golden dinergic relations shared by all neighboring parts of the anatomy.  The carapace of a typical Dungeness crab, from the Northwest Pacific  coast of the U.S. fits into a golden rectangle as drawing A of the figure below shows.  The length of the neighboring parts of the pincers and legs (C) in relation to each other (table B) fluctuates between visual equivalents of the three musical root harmonies of 0.75, fourth-diatessaron; 0.618, fifth-diapente; and 0.5, octave-diapason (D).  The wave diagrams of pincer and legs € show the harmonious rhythm created between all these differently sized members by their shared proportional relationships.

Proportional harmonies of Dungeness crab (Cancer magister)  Credit: Gyorgy Doczi, The Power of Limits, 1981

 

Lobsters – 20 body segments; 10 legs with claws on the ends; front pair much larger claws

 

Crayfish – 20 body segments; 10 walking legs


Shrimp – 20 body segments

 

Woodlice (roly-poly)

 

Barnacles

 

Other examples are found in Plate 76: Thoracostraca from Art Forms in Nature.

  1. (top center): Lucifer typus, leucifer embryo (gastrula)
  2. (bottom left, above): Penaeus Mülleri, nauplius larva
  3. (bottom left, below): Mastigopus dorsipinalis = (Sergestes dorsispinalis nomen dubium), mastigopus postlarva
  4. (top right): Elaphocaris Dohrnii = Sergestes (non ortmanni group; S. elaphrocaris?), elaphocaris larva
  5. (top left): Phyllosoma palinuri = Palinuridae (Palinurus elephas?), phyllosoma larva
  6. (bottom right, above): Zoea Carcini = Carcinus maenas, zoea larva
  7. (bottom right, below): Gonerichthus chiragra = Gonodactylus chiragra, gonerichthus larva
  8. (left of center): Alima gracilis = Alima neptuni, alima postlarva
  9. (right of center): Alima gracilis = Alima neptuni, alima postlarva
  10. (center): Alima bidens = Alima neptuni, old postlarva

 

Others are seen in Plate 86: Decapoda – Crustaceans

  1. Parthenope horrida = Daldorfia horrida
  2. Podophthalmus vigil
  3. Pisa armata
  4. Gonoplax rhomboides
  5. Pisolambrus nitidus = Solenolambrus tenellus
  6. Stenopus hispidus
  7. Palaemon serratus
  8. Albunea symnista = Albunea symmysta
  9. Lissa chiragra
  10. Birgus latro

 

 

Fish

There is much geometry to be found in the structure of fish.

 

 

The phi ratio is commonly found to structure different fish bodies.

Gyorgy Doczi writes, “A study of a variety of fish shapes reveals similar rhythmic harmonies resulting from similarly shared proportional limits.  The proportional analysis of ten different, randomly chosen fish from the Pacific waters of Canada demonstrates that their basic outlines – and frequently also the detailed articulations of their bodies – share in a variety of ways both the proportions of the golden section and of the 3-4-5 triangle.  The pictures of each of the ten fish in the figure below illustrate how the outlines fit into golden rectangles, their multiples and reciprocals, at times combined with squares.  In many instances, the mouth is at the golden section point of the body’s height, as in the Coho salmon (1), quillback rockfish (2), calico surfperch (7), and opah (10).  The lower row of fish shows how a series of double  3-4-5 triangles  can be fitted into the outlines from mouth to tail.”

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

Many are structured according to the golden ratio of tail length to body length.

Credit: Wooden Books – Designa – page 315

 

The porcupine fish fits into two reciprocal golden rectangles, while the sunfish fits into two 3-4-5 triangles.

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

“The combined three-unit length of such 3-4-5 triangulation diagrams, representing vertical body height, constitute true harmonic progressions, shown in the figure below for the Coho salmon, curlfin sole, Pacific pompano, calico surfperch, and ribbon barracudina.  The harmonies of these rhythms likewise approximate the root harmonies of music, as wave diagrams and graphs show.”

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

Even in the third dimension of thickness, fish tend to partake of the golden sections proportions, as three typical transverse sections in the figure below illustrates.

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

 

Fish scales have a hexagonal lattice structure.

Fish scales have growth rings like trees:

 

Many other fish patterns are Li symmetries such as the giant puffer (Tetraodontidae).

The Puffer Fish also creates beautiful Cymatic looking geometric designs in the sand to attract females:

 

Many fish bodies are Vesica shaped.

The phrase Vesica Piscis means ‘bladder of the fish’.

 

The angelfish:  Every key body feature of the angelfish falls at golden sections of its width and length.  The nose, tail section and centers of the fins fall at first golden sections.

The second golden section defines the indents on the dorsal and tail fins as well as the top of the body.

The marking around the eye and the eye fall on golden section lines as well.

 

See Plate 42: Ostraciontes in Art Forms in Nature for more examples of the geometry of fish.

  1. Ostracion cornutus = Lactoria cornuta, body from above
  2. Ostracion cornutus = Lactoria cornuta, from the front
  3. Ostracion cornutus = Lactoria cornuta, mouth
  4. Ostracion cornutus = Lactoria cornuta, from the side
  5. Ostracion cornutus = Lactoria cornuta, armour plate
  6. Ostracion quadricornis = Acanthostracion quadricornis, from the side
  7. Ostracion quadricornis = Acanthostracion quadricornis, armour plate from flanks
  8. Ostracion quadricornis = Acanthostracion quadricornis, armour plate from belly
  9. Ostracion auritus = Aracana aurita, from the side
  10. Ostracion turritus = Tetrosomus gibbosus, from the side

English: 6-8: Scrawled Cowfish 9: Striped Cowfish 10: Humpback Turretfish

 

 

See also Plate 87: Teleostei – Ray-finned Fish for more examples.

  1. Pegasus chiropterus = Pegasus laternarius, side view
  2. Hippocampus antiquorum = Hippocampus hippocampus, side view
  3. Phyllopteryx eques = Phycodurus eques, top view
  4. Antennarius tridens = Antennarius striatus, side view
  5. Chrysophrys aurata = Sparus aurata, scale in top view
  6. Pagellus erythrinus, scale in top view
  7. Box vulgaris = Boops boops, scale in top view
  8. Anthias sacer = Anthias anthias, scale in top view
  9. Apogon imberbis, scale in top view
  10. Centriscus scolopax = Macroramphosus scolopax, scale in top view
  11. Hypostomum plecostomum, scale in top view
  12. Fistularia chinensis = Aulostomus chinensis, scale in top view
  13. Solea vulgaris = Solea solea, scale in top view
  14. Scarus enneacanthus = Chlorurus enneacanthus, scale in top view
  15. Haemulon elegans = Haemulon sciurus, scale in top view
  16. Cantharus vulgaris = Spondyliosoma cantharus, scale in top view

 

The seahorse is 1 of 54 species of small marine fishes in the genus Hippocampus.

They have spiraling tails that resemble many other Fibonacci spirals in nature (ferns, embryos, gecko tails, horns…etc.) and bodies proportioned by the golden ratio.

 

Stingrays

Stingray bodies are proportioned by the golden ratio.

Credit: Wooden Books – Designa – page 315

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

 

Gyorgy Doczi demonstrates that, “The main bodies of the deep-sea skate (1), black skate (2), longnose skate (3), and starry skate (4) are each encompassed by two golden rectangles, while the big skate’s body (5) fits into a single golden rectangle.  The greatest width of the bodies invariably tallies with the golden section point of the height.  The length of the tail shares the same proportional limits in a variety of ways outlined in the drawings.

The two stingrays shown display a preference for the 3-4-5 triangle’s proportional limits.”

Credit: Gyorgy Doczi – The Power of Limits, 1981

 

Sharks

Triangles give the shark’s tooth its bite.

The dorsal fins in sharks are also triangular.

Sharks do not have scales like fish do.  The skin of the shark is covered with large numbers of very small thorn-like structures or denticles.  These make sharkskin feel like sandpaper.

Shark skin viewed up close very closely resembles the scales of butterfly wings.

Shark skin close-up. 

Butterfly wings close-up.  Credit: https://spatialexperiments.wordpress.com/2016/09/19/7613/

 

 

Dolphins

There are 30 species of oceanic dolphins which include the orca and pilot whales.

The eye, fins and tail all fall at golden sections of the length of a dolphin’s body.

The dimensions of the dorsal fin are golden sections.

The thickness of the dolphin’s tail section corresponds to same golden section of the line from head to tail.

 

 

Whales

There are eight families of whales and 89 living species.  They are creatures of the open ocean.  They feed, mate, give birth, suckle and raise their young at sea.

They range in size from 8.5 feet to 98 feet long and range in weight from 298 lbs. to 210 tons.

Examples include the:

Sperm whale

 

Humpback whale

 

 

Sea Turtles

Sea turtles (except the leatherback) have hard shells to protect their soft bodies.  The shell is divided into two sections: the carapace (dorsal) and the plastron (ventral).  The shell is made up of smaller plates called scutes.  These are almost always in Fibonacci numbers.

There are seven existing species of sea turtles.  They are all endangered species.  These are the:

Green sea turtle

 

Loggerhead sea turtle

 

Kemp’s ridley

 

Olive ridley

 

Hawksbill sea turtle

 

Flatback sea turtle

 

Leatherback sea turtle – the largest measuring 6-9 feet in length; weighing up to 1500 lbs.

 

Plate 89: Chelonia in Art Forms in Nature illustrates several examples

  1. Dermatochelys coriacea
  2. Caretta imbricata = Eretmochelys imbricata
  3. Hydromeda tectifera
  4. Chelys fimbriata = Geochelone nigra
  5. Testudo geometrica = Psammobates geometricus
  6. Testudo elephantina  = Dipsochelys dussumieri
  7. Chelydra serpentina

 

 

Conclusions

We have seen ample evidence of the foundational geometric structure in plants, microscopic organisms, and larger sea creatures.  We will see even more evidence in the realm of insects, air and land animals.

We have particularly seen the golden ratio as the underlying structure in more complex life forms.

What this shows is that the world is one of order, harmony and symmetry and is based upon the sphere and the Platonic solids and associated geometry that comes out of the Platonic solids such as the regular polygons the square, equilateral triangle, pentagon and hexagon.

 

Geometry is the blueprint of reality, upon which all life and matter is formed.

 

At this point we have shown so much overwhelming evidence of this it should be undisputed by now in the scientific community.  Yet we will move on to the realm of insects, land animals, humans, planets, galaxies and galactic clusters to really drive the point home.

***

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