In this article we will look at the geometry of moss, algae and fungi, noticing many of the same geometric forms that we see in other plant structures.

Geometry of Moss

Mosses are small flowerless plants that typically grow in dense green clumps or mats, often in damp or shady locations.

Notice the tessellating geometry in Bryum capillare and other moss leaves.

Geometry can be seen in other moss structures as well:

Azolla (Fairy Moss)

Moss spore capsules reveal beautiful radial symmetry

Moss spore capsule (Plagiomnium cuspidatum)

Peristome of water moss (Funaria hygrometrica)

Credit: George Shepherd

Now take a look at Plate 72 from Ernst Haeckel’s Art Forms in Nature.

This plate shows various mosses including:

Thamnium alopecurum
Eurhynchium praelongu = Eurhynchium pumilum
Polytrichum commune
Sphagnum cymbifolium = Sphagnum palustre
Splachnum luteum
Mnium undulatum = Hypnum ligulatum
Rhodobryum roseum
Physcomitrium acuminatum = Physcomitrium eurystomum
Physcomitrium ericetorum
Physcomitrium sphaericum
Sphagnum medium = Sphagnum magellanicum
Andreaea Thedenii = Andreaea obovata
1. Note the 4-pointed geometry upon which this moss opens.
Hypnum castrense = Hypnum crista-castrensis
Tetraplodon urceolatus
Dissodon Hornschuchii = Tayloria hornschuchii
Dissodon Froelichii = Tayloria froelichiana

Algae

Algae is an informal term for a large, diverse group of photosynthetic organisms. The most recent estimate suggests there are 72,500 algal species worldwide.

Water-net algae (hydrodictyon reticulatum) have a hexagonal or pentagonal mesh structure that makes up their colonies.

In Article 122 we will look specifically at two types of algae: Diatoms (phytoplankton) and Radiolaria (zooplankton). In this article we will show other examples from Ernst Haeckel’s Art Forms in Nature.

Plate 15: Fucoideae – Brown Algae

(top left): Nereocystis Lütkeana
(top center): Cutleria multifida
(top right): Cystosira erica = Cystoseira spinosa
(bottom center): Thalassophyllum clathrus = Agarum clathrus
(center): Scaberia Agardhi = Scaberia agardhii
1. Note the vortex streets and spiral structures
(bottom left): Zonaria pavonia = Padina pavonica
(bottom right): Turbinaria gracilis
1. Note the 5-pointed star shaped flowers

Plate 24: Desmidiea – Algae: These specimens are full of amazing geometry.

Staurastrum furcatum, single cell from above
Staurastrum vestitum, mating
Staurastrum aculeatum (3a: mating, from the side 3b: single cell, from above)
Staurastrum paradoxum (4a: mating, from the side 4b: mating, from above)
Staurastrum spinosum = Staurastrum furcatum, spore immediately after mating
Micrasterias denticulata, dividing cell from above
Micrasterias trigemina, dividing cell from above
Micrasterias melitensis = Micrasterias furcata, single cell from above
Spirotaenia condensata , single cell from the side
Closterium costatum, single cell from above
Euastrum pecten, single cell from above
Euastrum agalma, single cell from above
Euastrum apiculatum = Micrasterias apiculata, single cell from above

Plate 34: Melethallia – Algae: These specimens are distinctly geometric and resemble precise geometric mandalas as seen in Islamic art and Christian Rose windows.

Pediastrum tetras = Stauridium tetras
Pediastrum rotula
Pediastrum granulatum = Pediastrum boryanum
Pediastrum octonum
Pediastrum cruciatum
Pediastrum selenaea = Pediastrum duplex
Pediastrum pertusum = Pediastrum duplex
Pediastrum elegans
Pediastrum lunatum
Pediastrum furcatum
Pediastrum braunii
Pediastrum ellipticum
Pediastrum darwinii
Pediastrum trochiscus
Pediastrum solare

Plate 52: Filicinae – vascular plant; reproduces through spores; no seeds or flowers: These specimens are full of fractal branching structures.

Platycerium grande, habitus of young plant
Platycerium grande, young basal frond
Platycerium grande, older basal frond
Platycerium grande, old basal frond
Platycerium stemmaria, habitus of young plant
Platycerium stemmaria, habitus of young plant
Platycerium Hilli, habitus of young plant

Plate 64: Siphoneae – Algae: These specimens are full of many beautiful geometries.

(top left): Caulerpa racemosa, habitus
(top right): Caulerpa uvifera = Caulerpa racemosa uvifera, habitus
(left): Caulerpa pinnata = Caulerpa mexicana / Caulerpa pinnata, habitus
(right center): Caulerpa peltata, habitus
(bottom left): Caulerpa paspaloïdes, habitus
(bottom right): Caulerpa macrodisca = Caulerpa peltata macrodisca, habitus
(right): Struvea plumosa, habitus
(center): Neomeris kelleri = Neomeris annulata, cross-section
(left center): Neomeris kelleri = Neomeris annulata, habitus
(top center): Acetabularia mediterranea = Acetabularia acetabulum, habitus
(bottom center): Bornetella capitata, habitus with lengthwise section through lower “head”

Plate 65: Florideae – Red Algae: These specimens are full of beautiful fractal branching structures.

(top center): Chondrus crispus
(top left): Amansia glomera = Amansia glomerata / Melanamansia glomerata
(top right): Constantinea rosa-marina
(left): Ptilota serrata
(center): Ptilota densa = Neoptilota densa
(right): Rissonella verruculosa
(bottom left, upper): Delesseria involvens = Hypoglossum involvens
(bottom right, upper): Delesseria sanguinea
(bottom center): Nemastoma cervicorne = Platoma cyclocolpum
(bottom left, lower): Solieria chordalis, cystocarp in cross-section
(bottom right, lower): Binderella neglecta = Amphiplexia hymenocladioides, cystocarp in cross-section

Lichens

Lichens are defined as “an association of a fungus and a photosynthetic symbiotic resulting in a stable vegetative body having a specific structure.”

Here we see Plate 83: Lichens from Ernst Haeckel’s Art Forms in Nature.

These specimens are full of fractal branching structures and include:

Cladonia retipora
Cladonia perfoliata
Cladonia verticillata = Cladonia cervicornis verticillata
Cladonia squamosa
Cladonia fimbriata
Cladonia cornucopiae = Cladoniaceae?
Sticta pulmonaria = Lobaria pulmonaria
Parmelia stellaris= Physcia stellaris or Physcia aipolia
Parmelia olivacea = Melanohalea olivacea
Parmelia caperata = Flavoparmelia caperata
Hagenia crinalis = Anaptychia crinalis

Geometry of Fungi

“A fungus is any member of the group of eukaryotic organisms that includes microorganisms such as yeasts and molds, as well as the more familiar mushrooms. [They are] separate from the other eukaryotic life kingdoms of plants and animals.”1

Fungi acquire food by absorbing dissolved molecules like animals. They do not photosynthesize.

Fungi are the principal decomposers in ecological systems.

Many mushrooms adopt radial symmetry.

Many are composed of fractal branching structures and some use clear hexagonal/pentagonal structures such as the basket fungus.

Notice how the familiar geometry of polygons shows up in the structure of fungi below:

Ileodictyon cibarium – Basket fungus

Clathrus ruber – Red Cage Fungus

Phallus indusiatus

Phallus hadriani

Morels

Favolaschia cyatheae (White pore fungi)

Favolaschia calocera (Orange pore fungi)

Earthstar (Myriostoma coliforme)

Starfish fungus (Aseroe rubra)

Stinky Squid Fungus (Pseudocolus fusiformis)

Basket stinkhorn (Dictyophora indusiata)

Stinkhorn Fungus

Credit: National Park Service

Lycoperdon echinatum

Chorioactis geaster (Devil’s cigar)

Rhodotus palmatus

Coral Fungi

Coprinus Comatus

Fan-shaped Fungi

Hexagonia sp Homobasidiomycetes

Vascellum curtisii

Marasmius rotula & Marasmius fulvoferrugineus

Four unidentified species of fungi

Four different species of colonies of mushrooms

Take a look now at Plate 63 from Ernst Haeckel’s Art Forms in Nature. There are clear geometric structures at work including the familiar hexagonal cellular structure we see in so many other places.

This plate represents several species of Basimycetes fungi including:

Dictyophora madonna = Phallus?, mature fruiting body
Phallus impudicus, mature fruiting body with lengthwise section of volva
Aseroë rubra, mature fruiting body
Clathrus cancellatus = Clathrus ruber, mature fruiting body
Clathrella crispa = Clathrus crispus Turpin, mature fruiting body
Clathrella pusilla = Clathrus pusillus, mature fruiting body
Calathiscus sepia = Lysurus?, mature fruiting body
Simblum sphaerocephalum = Category:Lysurus periphragmoides , mature fruiting body
Anthurus borealis = Lysurus borealis, young fruiting body in cross-section
Geaster multifidus = Geastrum, mature fruiting body
Coprinus comatus, mature fruiting body

Now we will look at Plate 73: Ascomycetes from Art Forms in Nature. There is much beautiful geometry to be found in this plate. It represents sac fungi.

Specimens include:

Erysiphe Berberidis = Microsphaera berberidis, fruiting body
Erysiphe Alni = Microsphaera penicillata, fruiting body
Erysiphe Salicis = Uncinula adunca , fruiting body
Erysiphe aceris = Sawadaea bicornis , fruiting body in lengthwise section
Cucurbitaria macrospora = Asteromassaria macrospora, fruiting body in lengthwise section
Hypomyces chrysospermus, habitus
Hypomyces asterophorus = Pyxidiophora asterophora, habitus
Melanconis umbonata = Pseudovalsa umbonata, fruiting bodies
Pleospora herbarum, fruiting bodies

Next is Plate 93: Mycetozoa from Art Forms in Nature that represent more species of mushrooms. Once again beautiful geometric structures can clearly be seen.

These specimens include the:

Arcyria punicea = Arcyria denudata, 4 active young amoebas and one hatching from spore
Trichia varia, 2 active and 2 mating mature amoebas
Physarum plumbeum = Badhamia panicea, active plasmodium
Badhamia panicea, sporangium
Didymium nigripes, sporangium from above
Didymium farinaceum = Didymium melanospermum, sporangium in lengthwise section
Lepidoderma tigrinum, sporangium from the side
Trichia fragilis = Trichia botrytis, fibre form sporangium head
Arcyria serpula = Hemitrichia serpula, fibre from sporangium head
Dictydium cernuum = Cribraria cancellata, sporangium from the side
Cribraria aurantiaca, sporangium from the side
Cribraria intricata, sporangium from the side
Cribraria pyriformis, sporangium from the side
Trichia verrucosa, sporangium discharging spores from the side
Arcyria cinerea, sporangium from the side
Stemonitis fusca, sporangium from the side without spores
Physarum didermoides, group of sporangia from the side
Arcyria incarnata, group of sporangia from the side, one discharging spores
Trichia botrytis, group of sporangia from the side, three discharging spores
Arcyria adnata = Arcyria incarnata, group of sporangia from the side, one discharging spores

There are also ‘fungus flowers’.2 These are called “mycotrophic wildflowers” by botanists. Mycotrophic means “fungus nutrition”.

Some of these interesting structures include:

Fringed pinesap (Pleuricospora fimbriolata)

California pinefoot (Pityopus californicus)

Pinedrops (Pterospora andromedea)

Credit: James St. John

Indian pipe (Monotropa uniflora)

Coral-root orchid (Corallorhiza maculata)

Newly emerging Coral-root orchid. Credit: brewbooks

Flowering coral-root orchid pictured below.

Candystick (Allotropa virgata) – newly emerging (Credit: Dan & Raymond) and flowering.

Snow Plant (Sarcodes sanguinea) – newly emerging & flowering

Gnome Plant (Hemitomes conjestum)

Credit: Tab Tannery

Corynaea crassa – From Columbia (1) and Costa Rica (2)

Mycelium – Fractal Branching

Mycelium is the vegetative part of a fungus or fungus-like bacterial colony, consisting of a mass of branching, thread-like hyphae. Mycelium are extremely important for healthy forest ecosystems.

“Through the mycelium, a fungus absorbs nutrients from its environment…They contribute to the organic fraction of soil, and their growth releases carbon dioxide back into the atmosphere.”3

Credit: Kirill Ignatyev

“Under optimal conditions the hyphae often grow very rapidly, and it has been estimated that if all the hyphae produced in a day by a single soil fungus were laid end to end, they could extend for nearly a mile. Some mycorrhizal associations can be enormous, particularly in coniferous forest ecosystems. A single individual of Armillaria bulbosa has been discovered that permeates more than 30 acres of forest soil in northern Michigan and may be one of the world’s largest living organisms. Some scientists speculate that it was spawned by a single spore thousands of years ago. Another species of Armillaria (A. ostoyae) in the Blue Mountains of eastern Oregon was recently found to consist of a subterranean mycelial network with erect, above-ground mushrooms covering more than 2,000 acres of forest soil.”4

Mycelium represents one aspect of fractal branching. Fractal branching is commonly seen in a wide variety of ways involving the growth and structure of plants, including the way branches and roots grow. We have already seen many examples in this article alone. We discuss this in detail in the previous article.

Fractal branching is yet another aspect of gnomonic growth. The golden ratio is the fundamental example of self-similar gnomonic growth.

Gnomonic growth is ‘growth by accretion’ or accumulative increase. It is an additive process.

The Golden Spiral

“All forms of life which grow by gnomonic expansion expand at a repetitive mathematical increase of the prior growth range.”5

This allows for the organism to retain the aspect of self-similarity. Thus they can maintain their overall shape and proportions as they grow.

Square Root Spiral

Over and over we see how very simple geometric principles and ratios are used in Nature to create a myriad of complex forms.

The Vesica Piscis with the square roots of 2, 3 & 5. Along with the golden ratio these make up a family of ratios that life is built upon.

Mushroom Vortex Rings

Mushroom vortex rings can be seen in the traditional shape of mushrooms. This is covered in the previous article.

“A vortex ring, also called a toroidal vortex, is a torus-shaped vortex in a fluid or gas; that is, a region where the fluid mostly spins around an imaginary axis line that forms a closed loop. The dominant flow in a vortex ring is said to be toroidal.”6

“A vortex ring usually tends to move in a direction that is perpendicular to the plane of the ring and such that the inner edge of the ring moves faster forward than the outer edge.”7

They can be seen in the turbulent flows of liquids and gases, in smoke rings, in the firing of some artillery, in mushroom clouds and microbursts.

“One way a vortex ring may be formed is by injecting a compact mass of fast moving fluid into a mass of stationary fluid (which may be the same fluid.)”8

Think for a moment about the fluid-like Aether. Mushrooms are made of Aether and they grow in Aether. One part (the constituents of the mushroom) is moving much faster than the medium it is growing in (the Aether), forming the toroidal vortex we commonly see.

Again, like plants, fungi are simply slow-moving vortexes of ever-changing energy.

“You see [mushroom vortex rings] when you pour a moving liquid like milk into a motionless liquid like coffee or tea. The poured liquid meets resistance at its head and curls to the side. Each turn also meets resistance and curls further inward, repeating the cycle of meeting resistance and curling until the force dissipates and resistance disappears, leaving only the spiral path. The result appears in the form of a mushroom, which should be no surprise, since a living mushroom is mostly water, although it moves more slowly while displaying the same pattern as clashing liquids do.”9

The mushroom can also be a symbol for spiritual awakening and the kundalini experience. “It is thus a symbol of enlightenment with the stalk representing the spine and the little cap at the top of the stalk, the opened crown chakra. The symbolism is quite good as it shows the fountain-like effect of the energy when [the chakras are fully activated].”10

Many people, of course, associate this experience with psychedelic mushrooms and associated visionary experiences.

However, it is of the utmost importance to realize that no drugs or exterior substances are needed to experience our own spiritual natures. One can choose to use psychedelic substances as a tool to enhance spiritual seeking but it can be a very limiting and fragmented existence if a human believes their only route inwards is through psychedelic substances. This will close them off to a wide variety of spiritual experiences that occur in waking, sleeping, dreaming, social, creative and contemplative states.

First you must know yourself. The only way to know yourself is to explore the inner workings of your own consciousness while sober and awake through contemplation, meditation, emotions, thoughts, beliefs and dreams.

Fairy Rings

Fairy Rings are naturally occurring rings or arcs of mushrooms.

There are around 60 mushroom species which can grow in the ring pattern. Some may grow to over 33 feet in diameter. These are associated with various myths and folklore.

“For thousands of years, peoples of various cultures have regarded fairy rings with a strong sense of curiosity and fear, believing them to be mystical, supernatural places. Folklore tells us that cultures across Europe have traditionally believed that fairy rings are the dwelling place of fairies, elves, witches, and other magical beings– and that in some cases, they may be dangerous to enter. Even though fairy rings were thought to be little realms where fairies dance and play, lore suggested that if a human interrupted the fun, the cost could be deadly serious.

Science tells us that fairy rings– or patterns of certain types of mushrooms that grow in circular formations– are naturally-occurring phenomena that usually appear year after year on lawns, in fields, and in forests. Fairy rings occur when a mushroom spore falls in a favorable spot, grows a mycelium (the vegetative part of a fungus), and spreads out an underground network of fine, tubular threads called hyphae. Mushroom caps then appear at the edges of this network. The formations continue to expand outward, using up all the nutrients within them as they grow larger. A ring found in Belfort, France– the largest ever seen– measures approximately 2,000 feet in diameter, and is an astounding 700 years old.”11

Once again, with fairy rings, we have growth along a geometric matrix. In this case, the circle – the fundamental shape out of which all other polygons emerge.