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
Viruses lie at the borderline between living and nonliving creatures.
A virus is a small entity that replicates only inside the living cells of other organisms.
Around 5000 species have been described in detail though there are millions of types. They are, after all, the most abundant type of biological entity and are found wherever there is life. They have existed since living cells first evolved and they play an important, though misunderstood, function in regards to being carriers of genetic information. A separate Article will go into all the understood and misunderstood aspects of viruses. For now you can see Article 240.
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
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.
Polyhedral
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.
Note how the structure of viruses mirrors the structure of pollen in Article 182. This is because the prime function of both of these types of entities is that of a genetic carrier. Examples include:
Adenovirus
Hepatitis C
Measles
HIV virus
Herpes Simplex – Type I
Infectious Bursal Disease Virus
Canine Parvovirus
Credit: http://www.virology.wisc.edu/virusworld/viruslist.php
Cowpea Mosaic Virus
Rhinovirus – Common cold
Credit: http://www.virology.wisc.edu/virusworld/viruslist.php
Spherical
Examples of spherical viruses include:
Hepatitis B Virus
Credit: http://www.virology.wisc.edu/virusworld/viruslist.php
Dengue Virus
Symian Virus
Human Papilloma virus
Credit: http://www.virology.wisc.edu/virusworld/viruslist.php
West Nile Virus
Zika Virus
Complex
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
Bacteriophage
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
Actinomycetes are a type of bacteria, yet they share some characteristics with fungi.
They form aerial mycelium representing fractal-branching of filamentous growth.
Protozoa
“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 Amoebas are protozoans and exhibit both plant and animal characteristics. It is almost as if they are an intermediary species between the evolution of plant consciousness to animal consciousness.
Kenneth Boulding, in The Image, writes of the fascinating behavior of certain amoebas and the plant-like nature of single-celled organisms: “As long as the food supply is abundant they eat, grow, divide and so multiply. If, however, food becomes scarce, an extraordinary change in behavior occurs. Thousands of separate cells move together to form a worm-like object. By means of concerted movements along the cells, this object moves forward somewhat on the principle of the inchworm. After it has moved a certain distance, it begins to erect itself into a plant-like object. Differentiation takes place within the separate cells, depending on their position within the object. Those in the stalks become hard and rigid and die. Those in the flower-like portion eventually transform themselves into a seed-like spore, which is then scattered and may then remain dormant for a long time, until conditions become favorable again. Here is exhibited in most dramatic form the mystery of the plant, that is, the cell society. Can it be doubted that each single-celled amoeba possesses in some sense an image of its function in the social organization and that certain messages that it receives – for instance the frequency of food ingestion – are interpreted to mean that the drama must now begin.”
Ciliata
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:
- Codonella campanella = Tintinnopsis campanula, shell with individual inside
- Dictyocysta tiara = Dictyocysta elegans, shell
- Dictyocysta templum = Dictyocysta elegans, shell
- Tintinnopsis campanula, shell
- Cyttarocylis cistellula = Codonaria cistellula, shell
- Petalotricha galea = Codonella aspera, shell
- Stentor polymorphus, individual
- Stentor polymorphus, group
- Freia ampulla = Ascobius claparedi, individual
- Vorticella convallaria , group
- Carchesium polypinum, colony, relaxed
- Carchesium polypinum, colony, contracted
- Epistylis flavicans = Campanella umbellaria, colony
- Zoothamnium arbuscula, young colony
- Zoothamnium arbuscula, old colony
Flagellata
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:
- Anthophysa vegetans
- Cephalothamnium cyclopum
- Codonocladium candelabrum
- Trichomonas intestinalis
- Tetramitus rostratus
- Rhipidodendron splendidum
- Codonosiga botrytis
- Phalansterium digitatum
- Dinobryon sertularia
- Poteriodendron petiolatum = Bicoeca petiolata
- 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
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:
- Cercaria dichotoma / Distoma = Gymnophallus choledochus / Gymnophallus rebecqui Bartoli, cercaria larva from below
- Cercaria spinifera = Echinostoma echinatum, cercaria larva from below
- Cercaria bucephalus / Gasterostomum fimbriatum = Bucephalus polymorphus, cercaria larva from above
- Polystomum integerrimum, adult from below
- Polystomum integerrimum, miracidium larva
- Gyrodactylus elegans, adult from below
- Diplozoon paradoxum, mated pair from below
- Tristomum coccineum, adult from below
- Callicotyle Kroyeri, adult from below
- Caryophyllaeus mutabilis = Caryophyllaeus laticeps, adult
- Tetrarhynchus longicollis(/ Halysiorhynchus longicollis, young adult
- Phyllobothryon gracile, head from the front
- Taenia solium, mature proglottid
- Taenia solium, head from the front
Foraminifera
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.
- Nodosaria spinicosta = Amphicoryna spinicosta side view (1a: top view)
- Uvigerina aculeata = Euuvigerina aculeata
- Bolivina alata = Brizalina alata
- Cristellaria echinata = Spincterules anaglyptus, side view (4a: top view)
- Cristellaria siddalliana = Planularia siddalliana
- Cristellaria compressa = Planularia magnifica falciformis
- Polystomella aculeata = Elphidium aculeatum, side view (7a: top view)
- Polystomella venusta = Elphidium?
- Nummulites orbiculatus = Nummulites millecaput
- Globigerina bulloides
- Pavonina flabelliformis
- Bulimina inflata
- Frondicularia alata
- Calcarina clavigera
- Tinoporus baculatus = Baculogypsina sp?
- Orbulina universa
- Lagena alata = Fissurina laevigata
- Lagena interrupta, side view (18a: top view)
- Lagena acuticosta
- Lagena spiralis
Plate 12: Thalamophora – sea-dwelling Foramina – Again, beautiful spiraling geometry can be found in these specimens.
They include:
- Miliola parkeri = Quinqueloculina parkeri, side view
- Miliola reticulata = Quinqueloculina, side view
- Miliola striolata Miliolinella fichteliana / Triloculina fichteliana, side view
- Cornuspira planorbis, side view
- Articulina sagra, side view
- Spiroloculina nitida, side view (6a: view on opening)
- Alveolina melo = Borelis melo, single tube in cross-section
- Peneroplis planata = Peneroplis planatus, side view with pseudopods
- Hauerina circinata / Polysegmentina circinata, side view (9a: view on opening)
- Hauerina ornatissima, side view (10a: view on opening)
- Vertebralina mucronata = Articulina mucronata, side view
- Vertebralina insignis, side view (12a: view on opening)
- Vertebralina catena = Miliolina?, side view
- Vertebralina furcata = Miliolina?, side view
- Biloculina comata = Pyrgo comata, top view
- Orbiculina adunca = Archaias angulatus, side view
- Orbitolites laciniata = Marginopora vertebralis laciniata, top view
Plate 81: Thalamophora – Here we have yet more beautiful spiraling geometry.
These specimens include:
- Lagena formosa = Fissurina formosa / Solenina formosa, side view
- Lagena auriculata = Fissurina auriculata auriculata, side view
- Lagena pannosa = Lagena?, side view
- Lagena torquata = Cushmanina torquata, 4a: side view, 4b: top view
- Lagena squamosa = Oolina melo, top view
- Lagena Milletti = Oolina?, top view
- Lagena Walleriana = Fissurina walleriana, top view
- Lagena castrensis = Buchnerina castrensis, top view
- Lagena semistriata, side view
- Lagena plumigera = Cushmanina plumigera, side view
- Bulimina spinulosa = Reussella spinulosa, side view
- Bulimina marginata, side view
- Bolivina Durrandii, side view
- Bolivina convallaria = Sagrinella convallaria, side view
- Uvigerina porrecta = Siphouvigerina porrecta, side view
- Truncatulina ungeriana = Cibicides pachyderma, side view
- Rotalia calcar, side view
- Polystomella imperatrix = Parrellina imperatrix, side view
- Cristellaria calcar = Lenticulina calcar, side view
- Bifarina Mackinnonii = Valvobifarina mackinnoni, side view
- Lingulina pagoda = Nodosariidae sp?, side view
- 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.
Recent Comments