Geometry in Nature

LINK TO PRESENTATION PART ONE RECORDING HERE

LINK TO PRESENTATION PART TWO RECORDING HERE

Ancient cultures demonstrated fascination and experimentation with geometry and its applications in understanding the relationship between the macrocosm and microcosm. Pythagoras wrote in the 6th century BCE that geometry supplied the means to understanding god, man, and nature. Geometric shapes have been used as religious symbols for thousands of years.

Megalithic sites like Stonehenge or Maes Howe show the use of geometry in measuring the movements of the sun and moon and the Egyptian pyramids demonstrate their use of the golden ratio. We'll look at the role of geometry in the art of natural history and at some of the ways that it is found in nature.

Celtic patterns

Celtic patterns are built on drawing with a compass and reflect the artists' understanding of the possibilities that emerge from changing the radius or shifting the center of the circle. Celtic artists chose to use stylized forms rather than natural ones to represent nature. The zoomorphic are based on the knot patterns using a technique is thought to be the final stage of a long apprenticeship of free-hand drawing with a compass.

Islamic art and geometry

Complex patterns can be developed with a compass and straight edge. These patterns can be contracted and expanded, suggesting the connection between the finite and infinite. A series of 8 pointed stars that expand and contract as they interlock emerges from a pattern built on a combination of horizontal and diagonal squares (pictured below). This pattern is sometimes referred to as The Breath of the Compassionate, based on the teachings of the Great Master Ibn al-'Arabi. He described the basis of creation as the Divine Breath from which came the four elements of Earth, Water, Air, and Fire, which are represented by the four sides or four corners of the square.

Arabesques

Botanical scrollwork has been used in art and architecture around the world and was especially common among the ancient Egyptians and Greeks.

Arabesques, called islimi in Persian, are found in different regions and eras of Islamic design. These patterns are representative of the expansion and contraction of creation and were used frequently in illuminated books, often filling in the space around the text.

Zillij

Zillij is the name given in Islamic design to the patterns that imitate fractals, or the repetition of smaller patterns to form the larger so that the smaller shapes resemble the whole.

Zillij patterns

In biomorphic ornamentation, plants are shown either in plan (from the top or bottom) or in elevation (from the side), not in perspective. At times, the top view of the flower is combined with the side view of the stalk and leaves.

Baroque ornamentation 1600-1750

A style of excessive ornamentation known as the Baroque emerged in the early 17th century as a product of the desire for opulence and extravagance. It appeared in architecture, ceramics, art and frames, furntiture, home decoration, jewelry, gardens and more. Motifs included natural objects like shells, cornucopias, flowers and foliage, as well as mythical beings. This style of ornamentation combines the man-made and the natural, just as the cabinets of curiosity did.

Japanese design often takes inspiration from the interactions of the elements over time, using references from nature like ripples in wet or wind-blown sand, cloud formations, water eddies, flames, and much more.

symmetry

The Golden Mean

The Golden Mean, also known as the Divine Proportion or the Golden Section or Golden Ratio, describes a proportion found in nature and imitated by humans in art and architecture. It is a proportion that can be reached through geometry, but can’t be expressed as a simple fraction. It is the ratio of a line segment cut into two pieces, so that the ratio of the shorter to the longer is the same as the longer to the whole.

Euclid (325-265 BCE) described this number in the 3rd century BCE as the “extreme and mean ratio” in his book The Elements of Geometry. To paraphrase Euclid: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.”

In mathematical notation it is identified as “phi,” the first letter of the name Phidias, the Greek sculptor who used the proportions in the Parthenon.

The Golden Ration is found in architecture (especially temples and tombs) and artifacts of ancient Egypt, Greece, MesoAmerica, and in neolithic stone structures in Britain and across Europe. References to the divine proportions are found in texts from ancient cultures, including the Talmud, I Ching, and Hindi texts.

Illustrating Fibonacci’s introduction of Arabic numerals to Europe,

from Gregor Reisch’s 1503 work

Margarita Philosophica

Leonardo Bonacci (c.1170-c.1250)

Leonardo Bonacci was born in Pisa, around 1170. He is better known by his nick-name, Fibonacci, which means "son of the bull." He published several books, established the Tuscan school of math, and recognized the recurring mathematical sequence found in natural forms. A a mathematician named Edouard Lucas named this phenomenon the “Fibonacci Sequence" in the 19th century, though the sequence had been described in Indian texts several centuries before Fibonacci's birth.

Ancient Greeks had an advanced understanding of geometry but were limited by the lack of a zero in their system. After the fall of ancient Greece, math came to a standstill in Europe, but continued to flourish in the Arabic-speaking world.

Fibonacci's father worked as a city official in charge of taxation of trade between Pisa and North Africa and so their family was stationed in Algeria. He used the current system of Roman numerals in accounting, which made adding, subtracting, and multiplication very complicated. It was here that Fibonacci learned that Arab mathematicians had their own numeral system, based on the system described by the 7th century Hindi scientist Brahmagupta. Their system, now called the Hindu-Arabic system, included a zero, which made the advancement of mathematics possible.

Portrait of Luca Pacioli, By Jacopo de Barbieri, 1495

In this portrait, Pacioli is staring at one of the Platonic solids and writing on a slate showing Euclid’s name

Fibonacci recognized the possibilities of the Arabic numeral system and traveled throughout the Mediterranean region (Egypt, Greece, Syria, Sicily, and Southern France) to study math. He published the Book of Calculation in 1202 in order to spread the new system throughout Europe.

His book had a greater impact on commerce than science when it was first published. Several chapters were devoted to explaining the calculations of profit, interest, and currency conversion, as well as rules and formulas in math and algebra, largely influenced by mathematicians from Persia, Egypt, and Baghdad.

After his fame spread across Europe Frederick II, the Holy Roman emperor, issued a set of algebraic challenges. Fibonacci published the solutions in his book of 1225, called Flos (Flowers).

Fibonacci is most famous for identifying the mathematical phenomenon found in nature by observing the breeding of rabbits. He posed this question in his book: If a pair of rabbits produces a new pair of rabbits each month that in turn can produce a new pair, what is the rate of reproduction? The answer is found by adding the two preceding numbers to find the next, like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

This ratio is found frequently in nature, especially in seed and petal growth patterns. Flower petals almost always correspond to numbers in the series. Fir cones grow in series of 3 and 5 or 5 and 8 intertwining spirals; pineapple scales wind in 8 rows of scales in one direction and 13 rows in the other.

Each time one number in the sequence is divided by the previous number, you’ll get closer to the golden ratio (approx. 1.618).

This phenomenon is often interpreted as proof of the divine and has been used for centuries to illustrate the connections between all living things.

Conrad Gessner

Naturalists found the pattern occurring commonly in nature and is most easily recognized in plants. Da Vinci also observed that leaves were often spiraled in an arrangement that followed the Fibonacci sequence. Johannes Kepler noted in the late 16th century that most wildflowers are pentagonal and the leaves occur in the sequence. In 1754 the term phyllotaxis was coined by Charles Bonnet, after the Greek words phyllo for “leaf” and taxis for “arrangement.”

80% of the so-called “higher” plants (those with vascular tissue) are arranged in spirals following this series. This pattern aids photosynthesis. Each leaf receives maximum sun and rain, spirals the moisture to the roots efficiently, and offers maximum exposure for pollination.

From Plato through the Middle Ages, a liberal arts education was built on seven subjects. The trivium, or foundation, included grammar, logic, and rhetoric. The quadrivium included music (number in time), geometry (number in space), mathematics (number in the abstract), and astronomy (number is space and time). The golden section was a common theme of the quadrivium.

The golden ratio is found in music and sound waves. Modern scales and pure harmonic intervals fall on numbers in the Fibonacci series. For example, the octave includes 8 notes taken from 13 chromatic notes.

In 1680, Robert Hooke experimented with running a bow along the edge of a glass plate sprinkeld with flour to induce vibrations and noted the "nodal" patterns that emerged. A German physicist and musician named Ernst Chaladni would continue these experiments and perfect the technique in 1787.

Ernst Chaladni

"Chladni patterns"

see how they are formed here

Ernst Florens Friedrich Chladni ( 1756 –1827)

Chaladni is best know for his research on acoustics, his experiments with vibrating places, and for calculating the speed of sound of various gases. He perfected a technique to demonstrate the modes of vibration on a rigid surface that Robert Hooke had initiated. Chladni applied Hooke's process to metal plates covered in sand and recorded the patterns that emerged. Both men wanted to demonstrate their theory that sound travels in waves by devising a way to visually demonstrate this. Patterns are formed as the sand, flour, salt, or other particles move away from the antinodes, where the amplitude of the standing wave is at its maximum and toward the nodal lines, where the amplitude is at a minimum. The shapes are called Chaladni figures.

Paolo Uccello (1397-1475)

Uccello was a painter and mathematician who pioneered the use of perspective in painting before Leon Baptista Alberti wrote his treatise on perspective drawing. Uccello's perspective study of the vase (left) marks the shift in Renaissance art towards a more scientific approach to composing a picture.

LEON BAPTISTA ALBERTI (1404-1472)

Perspective is the representation of the spatial relation of objects as they appear to the eye, with parallel lines converging in order to create the illusion of depth and distance. The knowledge of the Greeks and Romans regarding perspective was lost. Architect Filippo Brunelleschi rediscovered the laws of perspective in the 15th century. He demonstrated a mathematical approach to rendering forms that decreased in size according to their distance from the viewer’s eye. His theories have roots in Euclid’s writings in his text called Optica, circa 300 B.C. It was the first texts on geometrical optics, in which he defined the terms visual ray and visual cone.

Leon Baptista Alberti

Leon Battista Alberti developed a theory of linear perspective, which he published in 1435 in a treatise called Della Pictura (On Painting).

His analysis of composition, color, and perspective influenced artists of the Renaissance. He view mathematics as the foundation for discussing art and science. Medieval painters like Giotto di Bondone of the 13th century and Filippo Brunilleschi of the 14th century experimented in more naturalistic approaches to conveying perspective. Alberti accurately described the mathematical approach to drawing in perspective, which impacted the way that Reniassance painters conveyed space. His understanding of perspective was informed by a treatise on optics written in the 11th century by the Arab polymath Alhazen Ibn al-Haytham.

He studied ancient architecture and theories of harmony in design and was influenced by the work of Vitruvius- the first century BCE Roman architect. Vitruvius was influenced by the work of Pythagoras, who held that geometry was the key to understanding the cosmos and wrote that the proportions of a temple were like that of a human, and that all parts of the human body are “measures of the whole.”

Alberti was more interested in theory and the social effects of architecture than in the practicalities of building. His treatise, On the Art of Building, was the foremost guide to architecture of the Renaissance and addressed issues like town planning and his philosophy of beauty. Informed by the work of the ancients, he introduced a Pythagorean system of ratios into architecture and outlined these ratios in the dimensions of the human body.

Birth of St. John the Baptist, Giovanni di Paolo, 1454, egg tempera on panel

rt: Flagellation of Christ, Piero della Francesca, c. 1470

Piero della Francesca further explained these techniques in his book, Della Picture, published in 1470. Once these theories were published, the ideas took hold immediately and changed painting throughout Europe, beginning in Italy with Paolo Uccello, Donatello, and Masaccio, for example.

Perspectograph by da Vinci

Da Vinci developed a tool called a Perspectograph- a plane of glass fit into a vertical frame with a viewing slot. The glass would be set in front of the scene to be traced. This tracing could be used as a rough sketch for a final painting and greatly simplified the process of foreshortening and perspective.

Vitruvian Man by Leonardo da Vinci, c.1490

Leonardo da Vinci (1452-1519)

Vitruvius (Marcus Vitruvius Pollio) (1st century BCE) was a Roman architect and engineer. His treatise, On Architecture, was influenced by theories of his predecessors, especially by the Greek architect, Hermogenes. Vitruvius's treatise was considered the authoritative text on architecture and design by artists and engineers of the Renaissance. It included descriptions of proportions and scale for city planning, temple construction, public spaces like theatres and bath houses, the proper use of the Greek orders and other guidelines for decoration, and much more. The architects of the Renaissance were influenced by these guidelines.

Vitruvius proposed that the human figure would fit perfectly inside a circle and a square. He explained that the height of the human figure was equal to the length with outstretched arms, in the proportions of a square, and that when placed inside a circle, the navel was at the center and the extremities touched the perimeter of a circle. The notion of a canon of ideal proportions dates back to ancient Egypt and was also addressed by Galen. The circle had, for centuries, been associated with the cosmic and divine, and the square represented the earth and the secular. The placement of the figure within the two reflected the Renaissance humanist belief that the human body was a microcosm of the universe.

Several artists tried to illustrate Vitruvius' concept. Leonardo da Vinci tested this theory for many months. He measured the proportions of many models, sketching them within the circle and square to see if Vitruvius' theory held up. Da Vinci created this drawing during his apprenticeship in Andrea del Verocchio's workshop, where he learned about architectural and technological design. He applied his knowledge of anatomy and proportions to art, hoping to elevate the status of the fine arts, which was regarded during this period as a handicraft. Da Vinci's drawings of the Vitruvian Man served to illustrate the book on divine proportions written by Luca Pacioli.

for more on Da Vinci's writings on proportions, click here

Da Vinci's plates for Luca Pacioli's

De Divina Proportione, 1509

Da Vinci studied mathematics with the Franciscan Friar, Luca Pacioli, and illustrated the friar's book, De divina proportione (pub 1509). Pacioli discussed the "golden section" and its applications in art, the human proportions, and architecture in his book, but focused mostly on polyhedra.

Plato proposed that the elements (earth, fire, water, air) were composed of these convex regular polyhedrons, which are 3-dimensional forms. The size and shape of the faces and the angles and edges of each form are identical. Like (the same shape) solids can be fitted together in three dimensions.

Euclid and the Ancient Greeks called these five solids the atoms of the Universe. They believed that all matter is made of these solids and each has a mystical side represented by its connection to earth, air, fire, or water.

Today we believe that all matter is made of a combination of atoms with a nucleus surrounded by electrons. The Greeks believed these solids also had a spherical property, where one Solid fits into a sphere that then fits into another solid and so on.

Polygons are 2 dimensional, and have an infinite number of possible edges, but polyhedrons are 3 dimensional, and the possibilities are limited to 5 shapes- the tetrahedron (4 sides); cube (6); octahedron (8); dodecahedron (12); and icosahedron (20).

Da Vinci was more interested in the shape, size, and perspective of these shapes than in their theoretical foundations. He used the mathemetical principles of perspective to create the illusion of space. After the publication of Pacioli’s book, the artists of the Renaissance applied the golden section to their compositions, including painting, sculpture, lettering, and more.

Left: Mona Lisa by da Vinci

right: Sandro Botticelli, Birth of Venus, 1485

Albrecht Durer (1471-1528)

Albrecht Durer (1471-1528) wrote a four volume series on geometry intended for artists and craftsmen that provided instruction and applications in construction and design. It was published in 1535 with the title Institutionum Geometricarum . The books covered linear geometry; 2 dimensional geometry and the construction of polygons; the principles of geometry applied to architecture, design, and typography; and polyhedra (including Platonic and Archimedean solids)

Durer included a perspective drawing tool similar to the one designed by da Vinci in his book, the Artist's Manual, published in 1525

Melancholia by Albrecht Durer, engraving, 1514

His engraving, “Melancholia,” was made in 1514. It has been interpreted in many ways, though his intentions are not clear. He had lost several friends and his mother shortly before it was made, so it may be purely emotional. Some think it is shows a muse wondering when and if inspiration will return. She is surrounded by carpenters tools and in the background looms a large polyhedron that has been the source of debate. Though the meaning of the polyhedra and tools is unclear, it demonstrates Durer's interest in geometry.

The PENTAGRAM

Also called the pentangle or pentacle. This ancient symbol has had an array of meanings and only became a symbol of the occult in the last two centuries. It has appeared on pot shards from ancient Babylonia, around 3500 BCE. In Mesopotamian cuneiform writing it was used to represent the four cardinal points, with the fifth point representing “above." In ancient Egyptian hieroglyphs, it's often associated with the goddess Sopdet; it was used as a symbol of wellbeing by the ancient Greeks. In ancient Christian symbolism it can represent the five wounds of Christ or the five senses- and at times is used to represent the Holy Spirit. During the Renaissance it represented the five Platonic solids. It’s seen in Hindu tantric writings and art; it's used by ancient Hebrews as a symbol of Truth or as reference to the 5 books of the Pentateuch; and it plays a role in Arthurian legend- appearing on Sir Gawain’s shield as a symbol of the five virtues.

During the Renaissance, it became a symbol of magic and by the mid-19th century was adopted by various occultists and given various meanings. With a single point at the top, it has been used to represent the spirit presiding over the four elements of matter, and when reversed, is sometimes thought to symbolize evil, as the proper order of things had been turned upside down.

Wentzel Jamnitzer (c.1507-1585)

Engravings by Jost Amman for Wentzel Jamnitzer’s Perspectiva Corporum Regularium

Wenzel Jamnitzer was a famous German goldsmith and artist who worked for several Holy Roman Emperors, including Charles V, Ferdinand I, Maximilian II, and Rudolf II. He wrote a book called Perspectiva Corporum Regularium (Perspective of Regular Solids), pub. 1560, which explored the writings of Plato's Timaeus and Euclid's Elements. Each chapter outlined the connection between each of the Platonic solids with the elements of medieval cosmology. The tetrahedron represents fire; the cube is the symbol of earth; the octahedron is air; the icosahedron is water, and the dodecahedron represents the heavens, with each of the twelve planes representing a sign of the zodiac.

Each chapter includes four illustrations of the polyhedra in their stellated and truncated form to demonstrate the theory that all life forms are a combination of these basic elements.

Wentzel Jamnitzer

Wentzel Jamnitzer Making a Perspective Drawing,

engraved by Jost Amman, c.1565

Jamnitzer built models of the polyhedra from paper and wood and used a mechanical device to trace them in perspective. He then gave the drawings to engraver Jost Amman to complete.

Jamnitzer was part of the German Mannerist movement and was involved in a metaphysical exploration of the “architecture of the universe.” His book influenced - or at least gained the interest of- Tycho Brahe and Johannes Keppler.

Engravings by Jost Ammans,

from drawings by Wentzel Jamnitzer

This illustration from Kepler's Mysterium Cosmographicum, represents the heavens as a series of celestial spheres with the spaces in between filled by Platonic solids.

Johannes Kepler (1571-1630)

Kepler is most famous for his discovery of the laws of planetary motion. He made important contributions to the study of optics and geometry and his work also provided the foundation for Newton’s theory of gravitation. He tried to demonstrate what he believed were the geometric underpinnings of the universe.

He discovered that the triangular progressions of Jupiter and Saturn would (or so he thought), over time, sketch out a circular shape that resembled a planetary orbit and wondered if this explained the order and spacing of celestial bodies. He developed this 2 dimensional concept (above, left) into a three dimensional geometric form (below, left) representing the heavens as a series of celestial spheres with the spaces in between filled by Platonic solids.

His book, Mysterium Cosmographicum, pub. in 1597, defended the theories of Copernicus. Copernicus explained that it was the sun and not the earth at the center of the universe, which was controversial because it conflicted with the Church's interpretation of planetary movement taken from the Bible, which described the sun moving across the sky and not the earth rotating around it.

Kepler presented the idea that the Platonic solids could be inscribed and circumscribed by spherical orbits, which would correspond with the spaces between the 6 known planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn) of the solar system and demonstrated the spacing of the planets based on this idea.

Kepler took an apprenticeship with astronomer Tycho Brahe in Denmark and learned through Brahe's observations that the planets moved on an elliptical path, not a circular one. Kepler used mathematics to formulate three laws of planetary movement- the first being that planets follow elliptical paths around the sun. The second is that the planets move faster when closer to the sun, and the third explains the relationship between a planet's distance from the sun and the amount of time it takes to complete an orbit. His laws provided the foundation for Isaac Newton's theory of gravity.

Below: the Platonic solids as depicted in Kepler's book.

Each of the Platonic solids corresponded with one of the elements.

the Platonic Solids

and their corresponding elements

Johannes Kepler

(1571-1630)

Nehemiah Grew (1641-1712)

Nehemiah Grew began studying the anatomy of plants in 1664. He was a colleague of the important Italian biologist and physician Marcello Malpighi, who was a leader of microscopical anatomy, and the two shared ideas and information. Grew's essay on the Anatomy of Vegetables earned him an invitation to join the Royal Society of London. He was guided by the belief that there were similarities between plants and animals and he used the microscope to find them. He recognized the organs and structures of plants and published a series of pamphlets which would eventually be compiled into his four volume book, The Anatomy of Plants. It included his illustrations and observations of vegetables, roots, trunks, leaves, flowers, fruits, and seeds.

Grew's important contributions to botany include descriptions of the morphology of stems and roots. He also correctly theorized that stamens are male organs and provided the first known microscopic description of pollen. Through the microscope he was able to observe the folding of unexpanded leaves in a flower bud and noted that sap circulated through plant tissue, delivering material to aid in its growth. In addition to botany, he was interested in exploring the geometric principles of nature and wrote an essay on snowflakes for the Royal Society in 1673.

Grew was made secretary of the Royal Society. When their ever-growing collection of naturalia was given a wing at Gresham College in London, it became his responsibility to oversee the production of a catalog of the collection. He contributed some of the illustrations, but the majority were contributed by several unidentified artists and naturalists.

Thomas Wright (1711-1786)

“The Object of that incomprehensible Being, which alone and in himself comprehends and constitutes supreme Perfection”.

engraving

“You may if you please, call a partial View of Immensity, or without much Impropriety perhaps, a finite View of Infinity”.

mezzotint

“A perspective View of the visible Creation, including the regions around our Sun, Syrius and Rigel. The rest is a promiscuous Disposition of all the Variety of other Systems within our finite Vision, as they are supposed to be posited behind one another, in the infinite Space, and round every visible Star.”

mezzotint

Thomas Wright is the author of An Original Theory or New Hypothesis of the Universe (London: 1750). He studied math and astronomy, but earned his income as a tutor and gardener. His book is written in an unusual style, in the form of a series of letters to a friend, interspersed with fragments of poetry. He delivered lectures on his theories about the Milky Way to the Royal Society of London, but was never made a member because of their strict policy not to included language related to divinity, metaphysics, or morals in their publications.

Dominique François Jean Arago ( 1786 – 1853)

Movement of Vénus relative to the Earth, after Cassini, from 1708 a 1716

Francois Arago,

Astronomie Populaire, 1854

Eclipse drawing from the 1858 book "Popular Astronomy" by Francois Arago

Dominique François Jean Arago ( 1786 – 1853) was a French mathemetician, astronomer, physicist, and freemason. Francois was recommended to the position of secretary to the Paris Observatory and then commissioned to complete the meridian arc measurements when had been initiated by another astronomer (named JBJ Delambre). The purpose of measuring the arc was to determine the exact length of a meter.

He and his partner entered Spain in order to make measurements along the mountains for their project, but were mistaken for spies and put in jail. He managed to preserve his measurements and as soon as he was freed, he deposited them in the Bureau of Longitudes in Paris. He was elected to the French Academy of Sciences and became the chair of analytical geometry. He was appointed as one of the astronomers of the Paris Observatory, where he delivered a series of lectures in astronomy and published them in his book Astronomie Populaire, in 1854.

His researched the velocity of sound, the pressure of steam at different temperatures, and magnetism. He discovered a phenomenon that would be named Arago's rotations, and that involves the interactions between a magnetized need and a moving metal disk.

M.C. Escher (1898-1972)

Escher is an extraordinary artist who explored the possibilities of geometry. He was very interested in the Platonic solids and used them as basis for interlocking patterns. He trained as an architect, which provided the skills that were so useful in inventing and depicting space, but never practiced architecture. Instead, he committed himself to the graphic arts. He travelled through the Abruzzo region of Italy by donkey and found inspiration in the narrow, winding streets of Italian cities. He used the shifting vantage points of the zig-zagging roads to create impossible landscapes and interiors.

Alfadena, 1929 middle: Goriano Sicoli, 1929

Belvedere, 1958

*the woman in the black dress at the foot of the stairs is taken from the Heironymous Bosch triptych, Garden of Earthly Delights.

MC Escher explored the geometry and logic of space and used the Platonic solids as reference for many of his drawings. He began to explore the possibilities of tessellations after seeing the Moorish tiles that adorned the mosques and buildings of Toledo and Granada, Spain. Escher experimented with patterns based the square, triangle, and hexagon, using them in glide reflections, translations, and rotations.

Contrast (Order and Chaos), 1950

Plane Symmetry, Escher,

He was also interested in observing and recording nature- especially in capturing the transparent and reflective qualities of water, as seen below.

clockwise from left: Prickly Flower, Puddle, Three Worlds, Dewdrop Leaf

Chris Jordan (1963-present)

Jordan is a former corporate lawyer turned contemporary artist/activist that uses mathematics to digitally compose images that convey the scale of consumption of particular resources to his audience. His series, Running the Numbers, addresses some contemporary environmental challenges. While he is not using geometry, I find his use of mathematics to convey scale too compelling to omit from a discussion on interpreting nature through visual media.

Jordan's imitation of George Seurat's Grande Jatte depicts 400,000 plastic bottle caps- the average number consumed in the US every minute.

Toxic Forest represents the estimated half-million cigarettes smoked every minute in America (in 2013). Cigarette butts are the #1 pollutant of American beaches, parks, and cities.

The Great Wave (after Hokusai) depicts 2.4 million pieces of plastic- an estimation of the number entering oceans around the world every hour.

symmetry

Reflection symmetry, sometimes called mirror symmetry: “An object has reflection symmetry if there exists a line that the object can be flipped (reflected) over such that it looks exactly the same before and after the flip. The line that the object flips across is called the axis of symmetry (plural: axes of symmetry). An object may have 0, 1 or more than one axes of symmetry. An axis of symmetry is also sometimes called a line of symmetry.

“Rotational symmetry: An object has rotational symmetry if it looks unchanged after being turned (rotated) by some specific amount less than a full 360-degree spin. The center-point of the spin is called the point of symmetry of the rotation”

The number of times an object can be rotated in a 360 degree turn and look the same is its order number.

Radial symmetry

The number of times an object can be rotated in a 360 degree turn and look the same is its order number.

Translation symmetry: Translational symmetry is when a pattern looks the same before and after it has been moved ("translated") a particular distance in a specified direction.

In translation symmetry, the shape is repeated at equal intervals, as if stamped along a straight line across the surface.

Phyllotaxis

This term , based on the Greek words phullon (leaf) and taxis (arrangement) was coined by Charles Bonnet in 1754 to describe the arrangement of leaves on a plant.

Our class project will involve drawing with a compass and ruler and developing a series of sketches that demonstrate three different types of symmetry found in nature.

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