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In this article we will focus on the geometry of diatoms and radiolarians.

We will see the Platonic solids show up in astonishing ways in these strange and interesting life forms.

We will also continue to explore images from German biologist Ernst Haeckel’s (1834-1919) Art Forms in Nature, from 1904.

These images (and their common geometry) illustrate Haeckel’s fundamental monistic notion of the ‘unity of all living things’.



Geometry of Phytoplankton

Phytoplankton are self-feeding components of the plankton community.

They photosynthesize and are key part of oceans, seas, and freshwater ecosystems.




Diatoms are one of the most common types of phytoplankton.

They are a unicellular type of algae.

There are an estimated 100,000 extant species.

Most have radial symmetry.

“Diatoms are microscopic algae living in both fresh and salt water.  They are unicellular organisms with heavily silica impregnated cell walls.  Living diatoms are amongst the most abundant forms of plankton and represent an essential part of the food chain in the ocean.  Once dead, their shells accumulate on the seabed and eventually form siliceous sediment deposits.”1


How do they Grow and Reproduce?

“Diatoms are almost all photosynthetic. Diatoms represent a large fraction of the marine biomass and thus are responsible for a large proportion of the total energy production of the oceans, possibly as much as half.  The few diatoms that don’t photosynthesize live on dissolved nutrients from rich organic matter.  Under the right conditions, diatoms can reproduce very rapidly and a population of a small species can double each day. Diatoms normally reproduce by binary fission, where one ‘mother cell’ splits into two daughter cells.”2

Diatom Reproduction. Image from page 205 of “An introduction to the structure and reproduction of plants” (1920)


“With the combination of microscope and camera these tiny objects may be enlarged and all of their parts accurately measured and divided, and from such examinations we learn that Nature never changes her binary progressions, especially when her forms are enclosed in definite geometric plans, and no change caused by accident can be considered as of moment in the judgment of her law.”3


The Ocean Data Center Phytoplankton Identification Guide4 shows all the various structures of different phytoplankton.


“The beauty of form and intricacy of pattern in diatoms have frequently been noted, but nothing has been stated in an adequate way of the perfection of proportion to be discovered in this microscopic cryptogram.  It offers some of the clearest revelations of Nature’s methods of establishing her proportional spaces and these expressed in no doubtful or vague way.

Other families of plants and shells disclose equally remarkable presentations, but none show a more direct influence of the octagon, the hexagon, and the angles of polar force of 45° and 60° than is displayed in the composition of these marvelous creations; and owning to the fact that their cell-walls are strongly impregnated with silica, these flinty forms preserve their perfect beauty even under the action of acid or intense heat.  Their fixed and enduring shape thus lends itself to analysis and when is added to this the fact that by placing them in a coloring matter the various details become more strongly marked, their uses as examples for study will be appreciated.”5


The regular polygons (the pentagon, octagon and hexagon are most prominent), spirals and Platonic solids show up with great frequency in these species.

Below are six images of diatoms from Samuel Colman’s Nature’s Harmonic Unity.

“All of these are to be found in a single drop from a salt or fresh water pool where they abound.  Under the microscope they reveal an immense variety of beautiful decorations in a perfect balance of parts, many of them covered with almost innumerable lines.  Some idea of their extreme minuteness may be gathered from the fact that one of them measures fifty one-thousandths of an inch in diameter, and this larger than the average size.”6


Some examples include:

Surirella spiralis – shaped like an infinity spiral

Credit: Microscopy


Actinoptychus senarious – circular with 6 divisions

Credit: Microscopy


Surirella fastuosa – ovoid/vesica shaped

Credit: Microscopy


Triceratium favum – triangular with hexagonal lattice

Credit: Microscopy


Biddulphia anetdiluviana – four-sided rounded star shape

Credit: Microscopy


Pleurosigma angulatum – long thin rhomboid/disc shaped


Azpeitia nodulifera – Fibonacci spiral lattice on a circle


Arachnooidiscus – spider’s web diatom – kaleidoscope interior

Credit: Microscopy


Licmophora – triangular


Licmophora slendida

Credit: Microscopy


Triceratium antediluvian – 5-pointed star

Triceratium favus – hexagons and heptagons

Asterionella formosa – 8-spiked star

Credit: Aimar Rakko

Lingulodinium polyhedrum – octahedral polyhedron

Credit: Ocean Data Center


Credit: Ocean Data Center



Credit: Ocean Data Center



Now we will take a look at several examples of diatoms from Ernst Haeckel’s Art Forms in Nature.  These specimens show beautiful, astonishing and precise geometry.


Plate 4: Diatoms

  1. Triceratium digitale, top view
  2. Navicula lyra = Lyrella lyra, top view
  3. Navicula excavata = Lyrella excavata, top view
  4. Triceratium mirificum, top view
  5. Triceratium pentacrinus, top view
  6. Actinoptychus constellatus, top view
  7. Aulacodiscus mammosus, top view
  8. Navicula Wrightii, top view
  9. Auliscus crucifer, top view
  10. Biddulphia pulchella = Biddulphia biddulphiana, top view
  11. Auliscus craterifer = Auliscus johnsonianus craterifer, top view
  12. Auliscus mirabilis, top view
  13. Aulacodiscus Grevilleanus, top view
  14. Surirella Macraeana, top view
  15. Denticella regia = Odontella regia, protoplast
  16. Asterolampra eximia, top view
  17. Actinoptychus heliopelta, top view
  18. Plagiogramma barbadense = Rutilaria barbadensis, top view
  19. Pinnularia Mülleri, side view
  20. Biddulphia granulata = Odontella granulata, side view
  21. Triceratium pentacrinus, side view
  22. Triceratium moronense, top view


Plate 84: Diatoms

  1. Pyrgodiscus armatus, top view
  2. Rutilaria monile = Pseudorutilaria monile, side view
  3. Auliscus elegans, top view
  4. Cocconema cistula = Cymbella cistula, colony
  5. Campyloneis grevillei, top view
  6. Asteromphalus imbricatus, top view
  7. Odontella aurita, colony
  8. Grovea pedalis = Biddulphia pedalis, top view
  9. Biddulphia pulchella = Biddulphia biddulphiana, colony
  10. Navicula bullata, top view
  11. Navicula didyma = Diploneis didyma, top view
  12. Campylodiscus bicruciatus = Campylodiscus simulans
  13. Surirella pulcherrima, top view
  14. Licmophora flabellata, colony
  15. Triceratium robertsianum, top view
  16. Gephyria constrict, side view
  17. Amphithetras elegans = Biddulphia elegans, top view



Coccolithophore (unicellular phytoplankton)

See for images.

Credit: Nannotax

Some examples include:

Braarudosphaera bigelowii – perfect dodecahedron

Credit: Nannotax


Coccolithus pelagicus – circular discs covering icosahedral shape

Credit: Nannotax

Braarudosphaera perampla – 5-pointed rounded star

Credit: Nannotax

Braarudosphaera bigelowii

Credit: Nannotax


Braarudosphaera pentagonica

Credit: Nannotax

Braarudosphaera perampla

Credit: Nannotax



Dictyochophyceae – Silicoflagellates

These have an icosahedral/dodecahedral basal ring with protruding radial spines.




Radiolaria are tiny protozoa (0.1 – 0.2mm) that produce intricate mineral skeletons made of silica.  They are found as zooplankton and exist solitary and in colonies.

“Despite being single-celled protozoans Radiolaria are quite complex, sophisticated organisms.  The body is divided into a central capsule which contains the endoplasm and nuclues 9or nucleii) and the extracapsulum which contains peripheral cytoplasm composed of a frothy bubble-like envelope of alveoli and a corona of ray-like axopodia and rhizopodia.  They feed on other zooplankton, phytoplankton and detritus using their axopodia and rhizopodia in a similar fashion to foraminifera, except that Radiolaria seldom possess pseudopdia and their rhizopodia are not as branching or anastomosing as in foraminifera.”7

Radiolaria reproduce by simple asexual fission.  Sexual reproduction has not been observed but it is assumed to happen.  The average lifespan is about two weeks, ranging from a few days to a few weeks.


Ernst Haeckel described over 4000 different species alone.

“The delicately perforated enclosing shell made of silica, and often protected by spicules, occurs in an inconceivable diversity of forms.”8

Credit: Marc Perkins


Their skeletal remains make up a large part of the ocean floor (siliceous ooze).

“The exquisite skeletal forms are not works of art, but instead serve as protection, while the long, ray-like projections act as flotation devices, increasing the body surface areas, and thus the buoyancy of these tiny creatures.  Groups of contractile molecules, which attach themselves to the spicules, allow the surface area of the internal soft body to be varied, enabling the buoyancy to be actively regulated.

As observers, we find radiolarians beautiful.  However, there are reasons for this, which have nothing to do with these organisms, but, instead, are based solely on the manner in which our perceptual mechanism functions.”9


Platonic solid geometry can be very clearly seen in radiolaria.

Some examples include:

Aulonia hexagona – spherical hexagonal lattice

Credit: On Growth and Form by Darcy Thompson


Stylatractus – pores are five-petaled flowers


Circogonia icosahedra – perfect regular icosahedron

Credit: Ernst Haeckel


Ovulus coralli – Octahedral

Credit: Bernotat & Co.


Tentaculus minimus – Dodecahedral (with spheres on corners)

Credit: Bernotat & Co.


Hexapodus inflatus – Octahedral

Credit: Bernotat & Co.


Metamorphosus lucidus – Icosahedral

Credit: Bernotat & Co.


Pseudoglobulus footballi – Dodecahedral

Credit: Bernotat & Co.


Tentaculus gigantiucs – Dodecahedral (spheres on corners)

Credit: Bernotat & Co.


Astrosphera curvata Icosahedral (stellated)

Credit: Bernotat & Co.


Rhizoplegma boreale – Icosahedral with spikes radiating outwards



We see many other examples presented to us by Ernst Haeckel in Art Forms in Nature.  Exquisite and detailed geometry are seen in all of these specimens.  These include:


Plate 1: Phaeodaria – amoeboid Cercozoa, single-celled Eukaryotes

  1. Circogonia icosahedra: Icosahedron structure
  2. 12-pointed mouth structure (opening of shell)
  3. Circostephanus coronarius: six-pointed star structure; 12-pointed arms; hexagonal body
  4. Haeckeliana porcellana: spherical structure; five-pointed circular pattern; tessellating circle pattern on spherical body
  5. five-pointed circular geometry mouth structure (pore circle without spines)
  6. Cortinetta tripodiscus: tessellating hexagonal pattern (fish/reptile scale/plant cells); arms with 3-pronged tetrahedral structure; total structure – 3 arms with 4th pointed ‘hat’ structure with cell structure inside
  7. Medusetta tetranema: 4 spiked arms with 5th pointed ‘hat’ structure; hexagonal tessellating pattern (scales/plant cells)
  8. Protocystis murrayi: spherical structure with strange 6-pointed ‘hat’ (points look like shark’s teeth); tessellating circle pattern



Plate 11: Discoidea – These specimens show beautiful 4-point, 5-point and 6-point geometry as well as intricate hexagonal cellular structure.

  1. Histiastrum Boseanum
  2. Stephanastrum quadratum
  3. Dicranastrum furcatum
  4. Rhopalastrum trispinosum = Dictyastrum trispinosum
  5. Chitonastrum lyra = Amphirhopalum virchowii lyra
  6. Euchitonia carcinus
  7. Myelastrum dodecaceros
  8. Myelastrum papilio
  9. Pentinastrum asteriscus
  10. Hexinastrum geryonidum
  11. Heliodrymus dendrocyclus
  12. Heliodiscus glyphodon



Plate 14: Peridinea – Dinoflagellates – These specimens show remarkable detail and sculpturing bringing to mind medieval armor and weaponry.

  1. Ceratium tripos = Neoceratium tripos
  2. Ornithocercus magnificus
  3. Ceratocorys horrida
  4. Goniodoma acuminatum = Triadinium polyedricum
  5. Dinophysis homunculus = Dinophysis caudata
  6. Dinophysis sphaerica
  7. Ceratium cornutum
  8. Ceratium macroceros = Ceratium hirundinella
  9. Pyrgidium pyriforme
  10. Peridinium divergens = Protoperidinium divergens
  11. Histioneis remora



Plate 21: Acanthometra – Yet more exquisite and precise 4-point and 8-point geometry can be seen in these specimens.

  1. Xiphacantha ciliata
  2. Xiphacantha spinulosa = Stauracantha spinulosa
  3. Stauracantha quadrifurca
  4. Pristacantha polyodon
  5. Lithoptera dodecaptera
  6. Acantholonche peripolaris, core spicule
  7. Acantholonche favosa, core spicule



Plate 22: Spyroidea – These specimens show exquisite and detailed sculpturing and beautiful hexagonal cellular structure.  These also remind one of medieval armor and weaponry.

  1. Triceraspyris gazelle, skeleton in side view
  2. Clathrospyris pyramidalis, skeleton in side view
  3. Pylospyris canariensis, skeleton in side view
  4. Anthospyris mammillata, skeleton in side view
  5. Dendrospyris polyrrhiza, skeleton in side view
  6. Sepalospyris pagoda , skeleton in side view
  7. Elaphospyris cervicornis, skeleton in side view
  8. Tholospyris cupola = Androspyris ramosa, skeleton in side view
  9. Dictyospyris stalactites, skeleton in side view
  10. Dictyospyris anthophora = Tholospyris anthophora, skeleton in side view
  11. Dorcadospyris dinoceras, skeleton in side view
  12. Triceraspyris damaecornis = Clathrocircus stapedius, skeleton in top view
  13. Ceratospyris Strasburgeri = Lophospyris pentagona pentagona, skeleton in side view



Plate 31: Cyrtoidea – These specimens also show detailed sculpturing and beautiful hexagonal cellular structure.  These also bring to mind medieval armor and weaponry.

  1. Cyrtophormis spiralis, shell
  2. Clathrocanium reginae, shell
  3. Anthocyrtium campanula, shell
  4. Pterocorys rhinoceros = Lipmanella bombus?, shell
  5. Lithornithium falco, shell
  6. Alacorys Bismarckii, shell
  7. Calocyclas monumentum, shell with living animal
  8. Pterocanium trilobum, shell
  9. Stichophaena Ritteriana, shell
  10. Dictyocodon Annasethe, shell
  11. Artopilium elegans = Pterocanium elegans), shell



Plate 41: Acanthophracta – These specimens show yet more precise and intricate geometry.

  1. Dorataspis typica
  2. Diporaspis nephrophora
  3. Lychnaspis miranda
  4. Lychnaspis polyancistra, spicule
  5. Echinaspis echinoides, spicule
  6. Diplocolpus costatatus
  7. Diploconus hexaphyllus
  8. Icosaspis elegans, shell plate
  9. Hexaconus serratus
  10. Hexacolpus nivalis



Plate 51: Polycyttaria – These specimens show yet more beautiful geometry and exquisite fractal branching.

  1. Collosphaera primoridalis;colony with symbiotic zooxanthellae
  2. Thalassoxanthium medusinum
  3. Sphaerozoum ivodimare
  4. Thalassoxanthium cervicorne
  5. Spaherozoum spinosissimum
  6. Coronophaera diadema
  7. Trypnophaera trepanata
  8. Acrosphaera inflata
  9. Mazosphaera lagotis
  10. Caminosphaera dendrophora
  11. Coronosphaera calycina
  12. Solenosphaera familiaris



Plate 61: Phaeodaria – Cercozoa – These specimens show more exquisite geometry particularly #9 which shows astonishingly precise and intricate geometry.

  1. Aulographis candelabrum = Aulographonium bicorne, end of spicule
  2. Aulographis pulvinata, end of spicule
  3. Aulographis verticillata, end of spicule
  4. Aulographis asteriscus, end of spicule
  5. Aulographis furcula, end of spicule
  6. Aulographis triglochin, end of spicule
  7. Aulographis bovicornis, end of spicule
  8. Aulographis ancorata, end of spicule
  9. Sagenoscena stellata, animal in shell
  10. Sagenoscena stellata, end of spicule
  11. Sagenoscena ornata, end of spicule
  12. Auloscena mirabilis, end of spicule
  13. Conchoceras cornutum, shell
  14. Conchonia quadricornis, shell
  15. Coelographis regina, spicule from shell apex
  16. Coelospathis ancorata, spicule from shell apex



Plate 71: Stephoidea – These specimens show yet more amazing geometry.

  1. (top center): Lithocircus magnificus, living animal
  2. (top left): Semantis sigillum = Tholospyris procera, skeleton of juvenile
  3. (bottom left, upper): Acanthodesmia corona, skeleton
  4. (center right, upper): Tristephanium dimensivum, skeleton
  5. (center): Trissocyclus sphaeridium, living animal
  6. (center right, lower): Octotympanum cervicorne = Acanthodesmia viniculata, skeleton
  7. (bottom right, upper): Microcubus zonarius = Amphispyris zonarius, skeleton
  8. (top right): Tympaniscus tripodiscus, skeleton
  9. (center left, upper): Tympaniscus quadrupes, skeleton
  10. (bottom center): Tympanidium foliosum = Tympanidium foliosum, living animal
  11. (bottom left, lower): Lithotympanum tuberosum, living animal
  12. (center left, lower): Circotympanum octogonium, skeleton
  13. (bottom right, lower): Lithocubus astragalus, living animal



Plate 91: Spumellaria – These specimens show yet more intricate and beautiful geometry.

  1. Astrosphaera stellata
  2. Hexancistra quadricuspis
  3. Cannartidium mammiferum = Didymocyrtis mammifera
  4. Cannartidium mastophorum
  5. Cannartiscus amphiconiscus = Cannartus violina
  6. Cyphinus amphilophus
  7. Panartus diploconus
  8. Peripanartus amphiconus
  9. Panicium coronatum
  10. Peripanicium amphicorona
  11. Trochodiscus stellaris
  12. Dicranastrum bifurcatum = Tetracranastrum bifurcatum
  13. Archidiscus pyloniscus
  14. Pylodiscus triangularis = Hexapyle cf. dodecantha
  15. Tholoma metallasson = Cubotholus octoceras




We have seen a huge variety of shapes of diatoms and radiolaria in this article.  It is astonishing to see how perfect Platonic solids – especially the icosahedron and dodecahedron – show up in these forms.

After viewing these it would be a wonder if one thought the universe was a random, chaotic accident.  How can the same forms keep showing up over and over again if the universe is not based upon harmony, order and an underlying geometric intelligence?


We see these Platonic solids in every scale of the universe.  We even see them in the geometry of musical ratios and light.  We see them on the tiniest scale and we see them on the galactic clustering scale, and everywhere in between.

“Other forms of minute vegetable and animal life disclosed by the microscope are to be found in countless numbers among the various families of Asteroida, Echinoderma, Crinoids, etc., revealing the fact that their proportions are produced by similar geometric principles, many of them developing a beautiful mosaic work.”10

We will see many of these in the next article where we will continue our investigation into the geometry of sea creatures.

It is fascinating that the same geometry keeps coming up over and over and over again…some are crude approximations and combinations and some are absolutely perfect.  The natural world is truly amazing.



  2. ibid.
  3. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on Its Relation to Proportional Form, Forgotten Books, 2017
  5. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on Its Relation to Proportional Form, Forgotten Books, 2017
  6. ibid.
  8. Eibl-Eibesfeldt, Irenaus, Ernst Haeckel – The Artist in the Scientist, Art Forms in Nature, Prestel-Verlag, 2017 reprint
  9. ibid.
  10. Colman, Samuel, Nature’s Harmonic Unity: A Treatise on Its Relation to Proportional Form, Forgotten Books, 2017

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