Return to Free Library

Return to Science Menu

Previous Article                                                                         Next Article


In this article we will look at the geometry of viruses, bacteria and several species of single-celled protozoans including ciliates, flagellates, nematodes and Foraminifera.

We will start with viruses in which the Platonic solids, especially the icosahedron, can very clearly be seen.



Viruses lie at the borderline between living and nonliving creatures.

A virus is a small infectious agent that replicates only inside the living cells of other organisms.

Around 5000 species have been described in detail.

There are millions of types.

They are the most abundant type of biological entity.

They are found wherever there is life.

They have existed since living cells first evolved.


When not inside an infected cell or in the process of infecting a cell, viruses exist in the form of independent particles called virions.

They are acellular and do not grow through cell division.

They use the machinery and metabolism of a host cell to produce multiple copies of themselves.



Virus Classification & Platonic Solids in Viruses

The Platonic solids can clearly be seen in the structure of viruses.  Icosahedral symmetry is quite common.


Helical viruses are composed of a single type of capsomere stacked around a central axis to form a helical structure.  A helical structure is essentially a coil.

An example is the tobacco mosaic virus.



Most animal viruses are icosahedral or near-spherical with chiral icosahedral symmetry.

Prolate refers to an icosahedron elongated along the fivefold axis.  This is a common arrangement of the heads of bacteriophages.

Examples include:


Hepatitis C



HIV virus

Herpes Simplex – Type I

Infectious Bursal Disease Virus

Canine Parvovirus



Cowpea Mosaic Virus

Rhinovirus – Common cold





Examples of spherical viruses include:

Hepatitis B Virus



Dengue Virus

Symian Virus


Human Papilloma virus



West Nile Virus


Zika Virus




Complex viruses possess a capsid that is not purely helical nor purely icosahedral.  They may possess extra structures such as protein tails or a complex outer wall.


Examples include:

Bacteriophage (pictured below), Pox viruses & Mimivirus



Visit Virus World for more great images of viruses.


Now we will explore the geometry of bacteria.  With bacteria we see similar shapes as we saw with structures in plants.  Bacteria morphology also shares similar components with viruses.



Geometry of Bacteria (Morphology)

  • Cocci – spheres
  • Ovoid Cocci – ovals
  • Cocobacilli – square or rectangular with rounded ends
  • Bacilli – Rods
  • Long Rods
  • Vibrios – curved rods
  • Diphtheroids – cone-like with rounded ends
  • Spirilli – fatter, wider spirals
  • Filaments – thin threads
  • Spirochetes – thin, tighter spirals



Arrangement of Bacterial Cells

The arrangement of bacterial cells most often involve clusters of cocci (spheres or ovals) or rods.  Keep in mind that all Platonic solids can be formed from close-packed spheres.  Platonic solids can be represented in nature with spheres or straight lines, or often, a combination of both.


Diplococci – two spheres joined in center (Dyad – Vesica Piscis)


Ovoid Diplococci – two ovals joined in center


Lanceolate Diplococci – two ovals joined – flattened at central joining


Streptococci – string of spheres like beads


Tetrads – four spherical cells joined


Sarcinae – eight spherical cells joined – octahedra or cubes


Staphylococci – Cocci in Clusters.  Possibly icosahedral and cuboctahedral clusters.


Diplobacilli – two rods joined together


Streptobacilli – string of rods


Palisades – string of rods folding together



Bacterial Flagellum

The bacterial flagellum is a 20-nanometer thick hollow tube.  “It is helical and has a sharp bend just outside the outer membrane.”1  The hook allows the axis of the helix to point directly away from the cell.

The flagellar filament is the long, helical screw that propels the bacterium when rotated by the motor through the hook.

Different species of bacteria have different numbers and arrangements of flagella.

Most bacteria filaments are made up of 11 protofilaments.  Some however have seven.

See below a physical model of a bacterial flagellum imaged and modeled at Brandeis University in DeRosier lab and printed at the University of Wisconsin, Madison.

Notice the intricate geometry and spirals involved in its structure.




Actinomycetes are a type of bacteria, yet they share some characteristics with fungi.

They form aerial mycelium representing fractal-branching of filamentous growth.

Credit:  page 63 of “The actinomycetes” (1961)




“Protozoans are neither animal nor plant, but single-celled eukaryotes that commonly show animal characteristics such as motility and heterotrophy; some groups are able to form cysts. Most are about 50-100 |m in size and are very common in aquatic environments and in the soil.”2




Ciliates are a group of porotzoans characterized by the presence of hair-like organelles called cilia.

Here we see Plate 3: Ciliata from Ernst Haeckel’s Art Forms in Nature.  We can see a variety of interesting geometry in these specimens.

They include:

  1. Codonella campanella = Tintinnopsis campanula, shell with individual inside
  2. Dictyocysta tiara = Dictyocysta elegans, shell
  3. Dictyocysta templum = Dictyocysta elegans, shell
  4. Tintinnopsis campanula, shell
  5. Cyttarocylis cistellula = Codonaria cistellula, shell
  6. Petalotricha galea = Codonella aspera, shell
  7. Stentor polymorphus, individual
  8. Stentor polymorphus, group
  9. Freia ampulla = Ascobius claparedi, individual
  10. Vorticella convallaria , group
  11. Carchesium polypinum, colony, relaxed
  12. Carchesium polypinum, colony, contracted
  13. Epistylis flavicans = Campanella umbellaria, colony
  14. Zoothamnium arbuscula, young colony
  15. Zoothamnium arbuscula, old colony




A flagellate is a cell or organism with one or more whip-like appendage called flagella which they use for locomotion.

Here we see Plate 13: Flagellata from Art Forms in Nature.  Once again much interesting geometry and fractal branching can be seen in these single-celled protozoa.

These specimens include:

  1. Anthophysa vegetans
  2. Cephalothamnium cyclopum
  3. Codonocladium candelabrum
  4. Trichomonas intestinalis
  5. Tetramitus rostratus
  6. Rhipidodendron splendidum
  7. Codonosiga botrytis
  8. Phalansterium digitatum
  9. Dinobryon sertularia
  10. Poteriodendron petiolatum = Bicoeca petiolata
  11. Uvella glaucoma




Nematodes – Roundworms

Nematodes are roundworms.  Over 25,000 species have been described.  Over half are parasitic.

They are long and thin like worms and often form spiral or coil shapes.

The Nippostrongylus brasiliensis is a particularly spiraled form.




Flatworms are a phylum of relatively simple bilaterian, unsegmented, soft-bodied invertebrates.  They have no body cavity which restricts them to a flattened shape.  There are both parasitic and non-parasitic varieties.  The non-parasitic varieties live free in water or in shaded, humid terrestrial environments.

Dugesia subtentaculata


Here we see Plate 75: Platodes from Art Forms in Nature.  Even in these simple flatworms astonishing detail and geometry can be seen.

These non-parasitic specimens include:

  1. Cercaria dichotoma / Distoma = Gymnophallus choledochus / Gymnophallus rebecqui Bartoli, cercaria larva from below
  2. Cercaria spinifera = Echinostoma echinatum, cercaria larva from below
  3. Cercaria bucephalus / Gasterostomum fimbriatum = Bucephalus polymorphus, cercaria larva from above
  4. Polystomum integerrimum, adult from below
  5. Polystomum integerrimum, miracidium larva
  6. Gyrodactylus elegans, adult from below
  7. Diplozoon paradoxum, mated pair from below
  8. Tristomum coccineum, adult from below
  9. Callicotyle Kroyeri, adult from below
  10. Caryophyllaeus mutabilis = Caryophyllaeus laticeps, adult
  11. Tetrarhynchus longicollis(/ Halysiorhynchus longicollis, young adult
  12. Phyllobothryon gracile, head from the front
  13. Taenia solium, mature proglottid
  14. Taenia solium, head from the front




Foraminifera are yet another group of protozoans.  They are characterized by a complex network of granular pseudopodia.  Most have a shell comprising chambers, interconnected through holes or foramina.  Often these chambers or foramina are in a spiral pattern.  Some can be quite elaborate in structure.

Most foraminifera are marine species.  Fewer numbers live in freshwater or brackish conditions.  A very few number have been been found to live in soil.

Once again, there are similarities in protozoa structure and plant structure.  Most have radial symmetry.


Some examples include:

Ammonia beccarii – globular chambers in a spiral (Rotallida)


Baculogypsina sphaerulata – pointed star shape from Japan


Heterostegina depressa – intricate spiral lines


Globigerinida – globular


Elphidium – spiral patterns


Xenophyphore – soft and lumpy shapes to fans and complex structures


We will now look at several other examples all from Ernst Haeckel’s Art Forms in Nature.  These include:

Plate 2: Foraminifera – Incredibly beautiful spiraling geometry can be found in these specimens.

  1. Nodosaria spinicosta = Amphicoryna spinicosta side view (1a: top view)
  2. Uvigerina aculeata = Euuvigerina aculeata
  3. Bolivina alata = Brizalina alata
  4. Cristellaria echinata = Spincterules anaglyptus, side view (4a: top view)
  5. Cristellaria siddalliana = Planularia siddalliana
  6. Cristellaria compressa = Planularia magnifica falciformis
  7. Polystomella aculeata = Elphidium aculeatum, side view (7a: top view)
  8. Polystomella venusta = Elphidium?
  9. Nummulites orbiculatus = Nummulites millecaput
  10. Globigerina bulloides
  11. Pavonina flabelliformis
  12. Bulimina inflata
  13. Frondicularia alata
  14. Calcarina clavigera
  15. Tinoporus baculatus = Baculogypsina sp?
  16. Orbulina universa
  17. Lagena alata = Fissurina laevigata
  18. Lagena interrupta, side view (18a: top view)
  19. Lagena acuticosta
  20. Lagena spiralis



Plate 12: Thalamophora – sea-dwelling Foramina – Again, beautiful spiraling geometry can be found in these specimens.

They include:

  1. Miliola parkeri = Quinqueloculina parkeri, side view
  2. Miliola reticulata = Quinqueloculina, side view
  3. Miliola striolata Miliolinella fichteliana / Triloculina fichteliana, side view
  4. Cornuspira planorbis, side view
  5. Articulina sagra, side view
  6. Spiroloculina nitida, side view (6a: view on opening)
  7. Alveolina melo = Borelis melo, single tube in cross-section
  8. Peneroplis planata = Peneroplis planatus, side view with pseudopods
  9. Hauerina circinata / Polysegmentina circinata, side view (9a: view on opening)
  10. Hauerina ornatissima, side view (10a: view on opening)
  11. Vertebralina mucronata = Articulina mucronata, side view
  12. Vertebralina insignis, side view (12a: view on opening)
  13. Vertebralina catena = Miliolina?, side view
  14. Vertebralina furcata = Miliolina?, side view
  15. Biloculina comata = Pyrgo comata, top view
  16. Orbiculina adunca = Archaias angulatus, side view
  17. Orbitolites laciniata = Marginopora vertebralis laciniata, top view



Plate 81: Thalamophora – Here we have yet more beautiful spiraling geometry.

These specimens include:

  1. Lagena formosa = Fissurina formosa / Solenina formosa, side view
  2. Lagena auriculata = Fissurina auriculata auriculata, side view
  3. Lagena pannosa = Lagena?, side view
  4. Lagena torquata = Cushmanina torquata, 4a: side view, 4b: top view
  5. Lagena squamosa = Oolina melo, top view
  6. Lagena Milletti = Oolina?, top view
  7. Lagena Walleriana = Fissurina walleriana, top view
  8. Lagena castrensis = Buchnerina castrensis, top view
  9. Lagena semistriata, side view
  10. Lagena plumigera = Cushmanina plumigera, side view
  11. Bulimina spinulosa = Reussella spinulosa, side view
  12. Bulimina marginata, side view
  13. Bolivina Durrandii, side view
  14. Bolivina convallaria = Sagrinella convallaria, side view
  15. Uvigerina porrecta = Siphouvigerina porrecta, side view
  16. Truncatulina ungeriana = Cibicides pachyderma, side view
  17. Rotalia calcar, side view
  18. Polystomella imperatrix = Parrellina imperatrix, side view
  19. Cristellaria calcar = Lenticulina calcar, side view
  20. Bifarina Mackinnonii = Valvobifarina mackinnoni, side view
  21. Lingulina pagoda = Nodosariidae sp?, side view
  22. Mimosina hystrix, side view



In this article we have seen many examples of beautiful and amazing geometry in single-celled creatures of the world.  In the next article we will specifically focus on diatoms and radiolaria.  Four different diatoms are pictured below.


Return to Free Library

Return to Science Menu

Previous Article                                                                         Next Article