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Genetic code

The genetic code is the set of rules used by living cells to translate information encoded within genetic material (DNA or RNA sequences of nucleotide triplets, or codons) into proteins. Translation is accomplished by the ribosome, which links proteinogenic amino acids in an order specified by messenger RNA (mRNA), using transfer RNA (tRNA) molecules to carry amino acids and to read the mRNA three nucleotides at a time. The genetic code is highly similar among all organisms and can be expressed in a simple table with 64 entries.

The codons specify which amino acid will be added next during protein biosynthesis. With some exceptions, a three-nucleotide codon in a nucleic acid sequence specifies a single amino acid. The vast majority of genes are encoded with a single scheme (see the RNA codon table). That scheme is often referred to as the canonical or standard genetic code, or simply the genetic code, though variant codes (such as in mitochondria) exist.

History
Efforts to understand how proteins are encoded began after DNA's structure was discovered in 1953. The key discoverers, English biophysicist Francis Crick and American biologist James Watson, working together at the Cavendish Laboratory of the University of Cambridge, hypothesied that information flows from DNA and that there is a link between DNA and proteins. Soviet-American physicist George Gamow was the first to give a workable scheme for protein synthesis from DNA. He postulated that sets of three bases (triplets) must be employed to encode the 20 standard amino acids used by living cells to build proteins, which would allow a maximum of amino acids. He named this DNA–protein interaction (the original genetic code) as the "diamond code".

In 1954, Gamow created an informal scientific organisation the RNA Tie Club, as suggested by Watson, for scientists of different persuasions who were interested in how proteins were synthesised from genes. However, the club could have only 20 permanent members to represent each of the 20 amino acids; and four additional honorary members to represent the four nucleotides of DNA.

The first scientific contribution of the club, later recorded as "one of the most important unpublished articles in the history of science" and "the most famous unpublished paper in the annals of molecular biology", was made by Crick. Crick presented a type-written paper titled "On Degenerate Templates and the Adaptor Hypothesis: A Note for the RNA Tie Club" to the members of the club in January 1955, which "totally changed the way we thought about protein synthesis", as Watson recalled. The hypothesis states that the triplet code was not passed on to amino acids as Gamow thought, but carried by a different molecule, an adaptor, that interacts with amino acids. The adaptor was later identified as tRNA.

Codons
The Crick, Brenner, Barnett and Watts-Tobin experiment first demonstrated that codons consist of three DNA bases.

Marshall Nirenberg and J. Heinrich Matthaei were the first to reveal the nature of a codon in 1961. They used a cell-free system to translate a poly-uracil RNA sequence (i.e., UUUUU...) and discovered that the polypeptide that they had synthesized consisted of only the amino acid phenylalanine. They thereby deduced that the codon UUU specified the amino acid phenylalanine.

This was followed by experiments in Severo Ochoa's laboratory that demonstrated that the poly-adenine RNA sequence (AAAAA...) coded for the polypeptide poly-lysine and that the poly-cytosine RNA sequence (CCCCC...) coded for the polypeptide poly-proline. Therefore, the codon AAA specified the amino acid lysine, and the codon CCC specified the amino acid proline. Using various copolymers most of the remaining codons were then determined.

Subsequent work by Har Gobind Khorana identified the rest of the genetic code. Shortly thereafter, Robert W. Holley determined the structure of transfer RNA (tRNA), the adapter molecule that facilitates the process of translating RNA into protein. This work was based upon Ochoa's earlier studies, yielding the latter the Nobel Prize in Physiology or Medicine in 1959 for work on the enzymology of RNA synthesis.

Extending this work, Nirenberg and Philip Leder revealed the code's triplet nature and deciphered its codons. In these experiments, various combinations of mRNA were passed through a filter that contained ribosomes, the components of cells that translate RNA into protein. Unique triplets promoted the binding of specific tRNAs to the ribosome. Leder and Nirenberg were able to determine the sequences of 54 out of 64 codons in their experiments. Khorana, Holley and Nirenberg received the Nobel Prize (1968) for their work.

The three stop codons were named by discoverers Richard Epstein and Charles Steinberg. "Amber" was named after their friend Harris Bernstein, whose last name means "amber" in German. The other two stop codons were named "ochre" and "opal" in order to keep the "color names" theme.

Expanded genetic codes (synthetic biology)
In a broad academic audience, the concept of the evolution of the genetic code from the original and ambiguous genetic code to a well-defined ("frozen") code with the repertoire of 20 (+2) canonical amino acids is widely accepted. However, there are different opinions, concepts, approaches and ideas, which is the best way to change it experimentally. Even models are proposed that predict "entry points" for synthetic amino acid invasion of the genetic code.

Since 2001, 40 non-natural amino acids have been added into proteins by creating a unique codon (recoding) and a corresponding transfer-RNA:aminoacyl – tRNA-synthetase pair to encode it with diverse physicochemical and biological properties in order to be used as a tool to exploring protein structure and function or to create novel or enhanced proteins.

H. Murakami and M. Sisido extended some codons to have four and five bases. Steven A. Benner constructed a functional 65th (in vivo) codon.

In 2015 N. Budisa, D. Söll and co-workers reported the full substitution of all 20,899 tryptophan residues (UGG codons) with unnatural thienopyrrole-alanine in the genetic code of the bacterium Escherichia coli.

In 2016 the first stable semisynthetic organism was created. It was a (single cell) bacterium with two synthetic bases (called X and Y). The bases survived cell division.

In 2017, researchers in South Korea reported that they had engineered a mouse with an extended genetic code that can produce proteins with unnatural amino acids.

In May 2019, researchers reported the creation of a new "Syn61" strain of the bacterium Escherichia coli. This strain has a fully synthetic genome that is refactored (all overlaps expanded), recoded (removing the use of three out of 64 codons completely), and further modified to remove the now unnecessary tRNAs and release factors. It is fully viable and grows 1.6× slower than its wild-type counterpart "MDS42".

Reading frame
A reading frame is defined by the initial triplet of nucleotides from which translation starts. It sets the frame for a run of successive, non-overlapping codons, which is known as an "open reading frame" (ORF). For example, the string 5'-AAATGAACG-3' (see figure), if read from the first position, contains the codons AAA, TGA, and ACG ; if read from the second position, it contains the codons AAT and GAA ; and if read from the third position, it contains the codons ATG and AAC. Every sequence can, thus, be read in its 5' → 3' direction in three reading frames, each producing a possibly distinct amino acid sequence: in the given example, Lys (K)-Trp (W)-Thr (T), Asn (N)-Glu (E), or Met (M)-Asn (N), respectively (when translating with the vertebrate mitochondrial code). When DNA is double-stranded, six possible reading frames are defined, three in the forward orientation on one strand and three reverse on the opposite strand. Protein-coding frames are defined by a start codon, usually the first AUG (ATG) codon in the RNA (DNA) sequence.

In eukaryotes, ORFs in exons are often interrupted by introns.

Start and stop codons
Translation starts with a chain-initiation codon or start codon. The start codon alone is not sufficient to begin the process. Nearby sequences such as the Shine-Dalgarno sequence in E. coli and initiation factors are also required to start translation. The most common start codon is AUG, which is read as methionine or as formylmethionine (in bacteria, mitochondria, and plastids). Alternative start codons depending on the organism include "GUG" or "UUG"; these codons normally represent valine and leucine, respectively, but as start codons they are translated as methionine or formylmethionine.

The three stop codons have names: UAG is amber, UGA is opal (sometimes also called umber), and UAA is ochre. Stop codons are also called "termination" or "nonsense" codons. They signal release of the nascent polypeptide from the ribosome because no cognate tRNA has anticodons complementary to these stop signals, allowing a release factor to bind to the ribosome instead.

Effect of mutations

During the process of DNA replication, errors occasionally occur in the polymerization of the second strand. These errors, mutations, can affect an organism's phenotype, especially if they occur within the protein coding sequence of a gene. Error rates are typically 1 error in every 10–100 million bases—due to the "proofreading" ability of DNA polymerases.

Missense mutations and nonsense mutations are examples of point mutations that can cause genetic diseases such as sickle-cell disease and thalassemia respectively. Clinically important missense mutations generally change the properties of the coded amino acid residue among basic, acidic, polar or non-polar states, whereas nonsense mutations result in a stop codon.

Mutations that disrupt the reading frame sequence by indels (insertions or deletions) of a non-multiple of 3 nucleotide bases are known as frameshift mutations. These mutations usually result in a completely different translation from the original, and likely cause a stop codon to be read, which truncates the protein. These mutations may impair the protein's function and are thus rare in in vivo protein-coding sequences. One reason inheritance of frameshift mutations is rare is that, if the protein being translated is essential for growth under the selective pressures the organism faces, absence of a functional protein may cause death before the organism becomes viable. Frameshift mutations may result in severe genetic diseases such as Tay–Sachs disease.

Although most mutations that change protein sequences are harmful or neutral, some mutations have benefits. These mutations may enable the mutant organism to withstand particular environmental stresses better than wild type organisms, or reproduce more quickly. In these cases a mutation will tend to become more common in a population through natural selection. Viruses that use RNA as their genetic material have rapid mutation rates, which can be an advantage, since these viruses thereby evolve rapidly, and thus evade the immune system defensive responses. In large populations of asexually reproducing organisms, for example, E. coli, multiple beneficial mutations may co-occur. This phenomenon is called clonal interference and causes competition among the mutations.

Degeneracy
Degeneracy is the redundancy of the genetic code. This term was given by Bernfield and Nirenberg. The genetic code has redundancy but no ambiguity (see the codon tables below for the full correlation). For example, although codons GAA and GAG both specify glutamic acid (redundancy), neither specifies another amino acid (no ambiguity). The codons encoding one amino acid may differ in any of their three positions. For example, the amino acid leucine is specified by YUR or CUN (UUA, UUG, CUU, CUC, CUA, or CUG) codons (difference in the first or third position indicated using IUPAC notation), while the amino acid serine is specified by UCN or AGY (UCA, UCG, UCC, UCU, AGU, or AGC) codons (difference in the first, second, or third position). A practical consequence of redundancy is that errors in the third position of the triplet codon cause only a silent mutation or an error that would not affect the protein because the hydrophilicity or hydrophobicity is maintained by equivalent substitution of amino acids; for example, a codon of NUN (where N = any nucleotide) tends to code for hydrophobic amino acids. NCN yields amino acid residues that are small in size and moderate in hydropathicity; NAN encodes average size hydrophilic residues. The genetic code is so well-structured for hydropathicity that a mathematical analysis (Singular Value Decomposition) of 12 variables (4 nucleotides x 3 positions) yields a remarkable correlation (C = 0.95) for predicting the hydropathicity of the encoded amino acid directly from the triplet nucleotide sequence, without translation. Note in the table, below, eight amino acids are not affected at all by mutations at the third position of the codon, whereas in the figure above, a mutation at the second position is likely to cause a radical change in the physicochemical properties of the encoded amino acid. Nevertheless, changes in the first position of the codons are more important than changes in the second position on a global scale. The reason may be that charge reversal (from a positive to a negative charge or vice versa) can only occur upon mutations in the first position of certain codons, but not upon changes in the second position of any codon. Such charge reversal may have dramatic consequences for the structure or function of a protein. This aspect may have been largely underestimated by previous studies.

Codon usage bias
The frequency of codons, also known as codon usage bias, can vary from species to species with functional implications for the control of translation. The codon varies by organism; for example, most common proline codon in E. coli is CCG, whereas in humans this is the least used proline codon.

Non-standard amino acids
In some proteins, non-standard amino acids are substituted for standard stop codons, depending on associated signal sequences in the messenger RNA. For example, UGA can code for selenocysteine and UAG can code for pyrrolysine. Selenocysteine came to be seen as the 21st amino acid, and pyrrolysine as the 22nd. Both selenocysteine and pyrrolysine may be present in the same organism. Although the genetic code is normally fixed in an organism, the achaeal prokaryote Acetohalobium arabaticum can expand its genetic code from 20 to 21 amino acids (by including pyrrolysine) under different conditions of growth.

Variations
There was originally a simple and widely accepted argument that the genetic code should be universal: namely, that any variation in the genetic code would be lethal to the organism (although Crick had stated that viruses were an exception). This is known as the "frozen accident" argument for the universality of the genetic code. However, in his seminal paper on the origins of the genetic code in 1968, Francis Crick still stated that the universality of the genetic code in all organisms was an unproven assumption, and was probably not true in some instances. He predicted that "The code is universal (the same in all organisms) or nearly so". The first variation was discovered in 1979, by researchers studying human mitochondrial genes. Many slight variants were discovered thereafter, including various alternative mitochondrial codes. These minor variants for example involve translation of the codon UGA as tryptophan in Mycoplasma species, and translation of CUG as a serine rather than leucine in yeasts of the "CTG clade" (such as Candida albicans). Because viruses must use the same genetic code as their hosts, modifications to the standard genetic code could interfere with viral protein synthesis or functioning. However, viruses such as totiviruses have adapted to the host's genetic code modification. In bacteria and archaea, GUG and UUG are common start codons. In rare cases, certain proteins may use alternative start codons. Surprisingly, variations in the interpretation of the genetic code exist also in human nuclear-encoded genes: In 2016, researchers studying the translation of malate dehydrogenase found that in about 4% of the mRNAs encoding this enzyme the stop codon is naturally used to encode the amino acids tryptophan and arginine. This type of recoding is induced by a high-readthrough stop codon context and it is referred to as functional translational readthrough.

Despite these differences, all known naturally occurring codes are very similar. The coding mechanism is the same for all organisms: three-base codons, tRNA, ribosomes, single direction reading and translating single codons into single amino acids. The most extreme variations occur in certain ciliates where the meaning of stop codons depends on their position within mRNA. When close to the 3' end they act as terminators while in internal positions they either code for amino acids as in Condylostoma magnum or trigger ribosomal frameshifting as in Euplotes.

The origins and variation of the genetic code, including the mechanisms behind the evolvability of the genetic code, have been widely studied, and some studies have been done experimentally evolving the genetic code of some organisms.

Inference
Variant genetic codes used by an organism can be inferred by identifying highly conserved genes encoded in that genome, and comparing its codon usage to the amino acids in homologous proteins of other organisms. For example, the program FACIL infers a genetic code by searching which amino acids in homologous protein domains are most often aligned to every codon. The resulting amino acid (or stop codon) probabilities for each codon are displayed in a genetic code logo.

As of January 2022, the most complete survey of genetic codes is done by Shulgina and Eddy, who screened 250,000 prokaryotic genomes using their Codetta tool. This tool uses a similar approach to FACIL with a larger Pfam database. Despite the NCBI already providing 27 translation tables, the authors were able to find new 5 genetic code variations (corroborated by tRNA mutations) and correct several misattributions. Codetta was later used to analyze genetic code change in ciliates.

Origin
The genetic code is a key part of the history of life, according to one version of which self-replicating RNA molecules preceded life as we know it. This is the RNA world hypothesis. Under this hypothesis, any model for the emergence of the genetic code is intimately related to a model of the transfer from ribozymes (RNA enzymes) to proteins as the principal enzymes in cells. In line with the RNA world hypothesis, transfer RNA molecules appear to have evolved before modern aminoacyl-tRNA synthetases, so the latter cannot be part of the explanation of its patterns.

A hypothetical randomly evolved genetic code further motivates a biochemical or evolutionary model for its origin. If amino acids were randomly assigned to triplet codons, there would be 1.5 × 1084 possible genetic codes. This number is found by calculating the number of ways that 21 items (20 amino acids plus one stop) can be placed in 64 bins, wherein each item is used at least once. However, the distribution of codon assignments in the genetic code is nonrandom. In particular, the genetic code clusters certain amino acid assignments.

Amino acids that share the same biosynthetic pathway tend to have the same first base in their codons. This could be an evolutionary relic of an early, simpler genetic code with fewer amino acids that later evolved to code a larger set of amino acids. It could also reflect steric and chemical properties that had another effect on the codon during its evolution. Amino acids with similar physical properties also tend to have similar codons, reducing the problems caused by point mutations and mistranslations.

Given the non-random genetic triplet coding scheme, a tenable hypothesis for the origin of genetic code could address multiple aspects of the codon table, such as absence of codons for D-amino acids, secondary codon patterns for some amino acids, confinement of synonymous positions to third position, the small set of only 20 amino acids (instead of a number approaching 64), and the relation of stop codon patterns to amino acid coding patterns.

Three main hypotheses address the origin of the genetic code. Many models belong to one of them or to a hybrid:

 * Random freeze: the genetic code was randomly created. For example, early tRNA-like ribozymes may have had different affinities for amino acids, with codons emerging from another part of the ribozyme that exhibited random variability. Once enough peptides were coded for, any major random change in the genetic code would have been lethal; hence it became "frozen".
 * Stereochemical affinity: the genetic code is a result of a high affinity between each amino acid and its codon or anti-codon; the latter option implies that pre-tRNA molecules matched their corresponding amino acids by this affinity. Later during evolution, this matching was gradually replaced with matching by aminoacyl-tRNA synthetases.
 * Optimality: the genetic code continued to evolve after its initial creation, so that the current code maximizes some fitness function, usually some kind of error minimization.

Hypotheses have addressed a variety of scenarios:
 * Chemical principles govern specific RNA interaction with amino acids. Experiments with aptamers showed that some amino acids have a selective chemical affinity for their codons. Experiments showed that of 8 amino acids tested, 6 show some RNA triplet-amino acid association.
 * Biosynthetic expansion. The genetic code grew from a simpler earlier code through a process of "biosynthetic expansion". Primordial life "discovered" new amino acids (for example, as by-products of metabolism) and later incorporated some of these into the machinery of genetic coding. Although much circumstantial evidence has been found to suggest that fewer amino acid types were used in the past, precise and detailed hypotheses about which amino acids entered the code in what order are controversial. However, several studies have suggested that Gly, Ala, Asp, Val, Ser, Pro, Glu, Leu, Thr may belong to a group of early-addition amino acids, whereas Cys, Met, Tyr, Trp, His, Phe may belong to a group of later-addition amino acids.
 * Natural selection has led to codon assignments of the genetic code that minimize the effects of mutations. A recent hypothesis suggests that the triplet code was derived from codes that used longer than triplet codons (such as quadruplet codons). Longer than triplet decoding would increase codon redundancy and would be more error resistant. This feature could allow accurate decoding absent complex translational machinery such as the ribosome, such as before cells began making ribosomes.
 * Information channels: Information-theoretic approaches model the process of translating the genetic code into corresponding amino acids as an error-prone information channel. The inherent noise (that is, the error) in the channel poses the organism with a fundamental question: how can a genetic code be constructed to withstand noise while accurately and efficiently translating information? These "rate-distortion" models suggest that the genetic code originated as a result of the interplay of the three conflicting evolutionary forces: the needs for diverse amino acids, for error-tolerance and for minimal resource cost. The code emerges at a transition when the mapping of codons to amino acids becomes nonrandom. The code's emergence is governed by the topology defined by the probable errors and is related to the map coloring problem.
 * Game theory: Models based on signaling games combine elements of game theory, natural selection and information channels. Such models have been used to suggest that the first polypeptides were likely short and had non-enzymatic function. Game theoretic models suggested that the organization of RNA strings into cells may have been necessary to prevent "deceptive" use of the genetic code, i.e. preventing the ancient equivalent of viruses from overwhelming the RNA world.
 * Stop codons: Codons for translational stops are also an interesting aspect to the problem of the origin of the genetic code. As an example for addressing stop codon evolution, it has been suggested that the stop codons are such that they are most likely to terminate translation early in the case of a frame shift error. In contrast, some stereochemical molecular models explain the origin of stop codons as "unassignable".

title: Genetic code


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Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities $$a$$ and $$b$$ with $$a > b > 0$$, $$a$$ is in a golden ratio to $$b$$ if

where the Greek letter phi ($\varphi$ or $$\phi$$) denotes the golden ratio. The constant $$\varphi$$ satisfies the quadratic equation $$\varphi^2 = \varphi + 1$$ and is an irrational number with a value of

The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of $$\varphi$$—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.

Calculation
Two quantities $$a$$ and $$b$$ are in the golden ratio $$\varphi$$ if

One method for finding a closed form for $$\varphi$$ starts with the left fraction. Simplifying the fraction and substituting the reciprocal $$b/a = 1/\varphi$$,

Therefore,

Multiplying by $$\varphi$$ gives

which can be rearranged to

The quadratic formula yields two solutions:

Because $$\varphi$$ is a ratio between positive quantities, $$\varphi$$ is necessarily the positive root. The negative root is in fact the negative inverse $$-\frac{1}{\varphi}$$, which shares many properties with the golden ratio.

History
According to Mario Livio,

"Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics."

Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans. Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:

"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."

The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.

Luca Pacioli named his book Divina proportione (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about $$0.6180340$$" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:"Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel."

Eighteenth-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875.

By 1910, inventor Mark Barr began using the Greek letter phi ($\varphi$) as a symbol for the golden ratio. It has also been represented by tau ($\tau$), the first letter of the ancient Greek τομή ('cut' or 'section').

The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern. This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.

Irrationality
The golden ratio is an irrational number. Below are two short proofs of irrationality:

Contradiction from an expression in lowest terms

This is a proof by infinite descent. Recall that:

If we call the whole $$n$$ and the longer part $$m,$$ then the second statement above becomes

To say that the golden ratio $$\varphi$$ is rational means that $$\varphi$$ is a fraction $$n/m$$ where $$n$$ and $$m$$ are integers. We may take $$n/m$$ to be in lowest terms and $$n$$ and $$m$$ to be positive. But if $$n/m$$ is in lowest terms, then the equally valued $$m/(n-m)$$ is in still lower terms. That is a contradiction that follows from the assumption that $$\varphi$$ is rational.

By irrationality of $a + b$
Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If $$\varphi = \tfrac12(1 + \sqrt5)$$ is rational, then $$2\varphi - 1 = \sqrt5$$ is also rational, which is a contradiction if it is already known that the square roots of all non-square natural numbers are irrational.

Minimal polynomial
The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial

This quadratic polynomial has two roots, $$\varphi$$ and $$-\varphi^{-1}.$$

The golden ratio is also closely related to the polynomial

which has roots $$-\varphi$$ and $$\varphi^{-1}.$$ As the root of a quadratic polynomial, the golden ratio is a constructible number.

Golden ratio conjugate and powers
The conjugate root to the minimal polynomial $$x^2-x-1$$ is

The absolute value of this quantity ($0.618\ldots$) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, $$b/a$$).

This illustrates the unique property of the golden ratio among positive numbers, that

or its inverse:

The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with $$\varphi$$:

The sequence of powers of $$\varphi$$ contains these values $$0.618033\ldots,$$ $$1.0,$$ $$1.618033\ldots,$$ $$2.618033\ldots;$$ more generally, any power of $$\varphi$$ is equal to the sum of the two immediately preceding powers:

As a result, one can easily decompose any power of $$\varphi$$ into a multiple of $$\varphi$$ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of $$\varphi$$:

If $$ \lfloor n/2 - 1 \rfloor = m,$$ then:

Continued fraction and square root

The formula $$\varphi = 1 + 1/\varphi$$ can be expanded recursively to obtain a continued fraction for the golden ratio:

It is in fact the simplest form of a continued fraction, alongside its reciprocal form:

The convergents of these continued fractions ($1/1,$ $$2/1,$$ $$2/1,$$ $$3/2,$$ $$5/3,$$ $$8/5,$$ $$13/8,$$ ... or $$1/1,$$ $$1/2,$$ $$2/3,$$ $$3/5,$$ $$5/8,$$ $8/13,$ ...) are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational $$\xi$$, there are infinitely many distinct fractions $$p/q$$ such that, "\xi-\frac{p}{q}\right" This means that the constant $$\sqrt{5}$$ cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.

A continued square root form for $$\varphi$$ can be obtained from $$\varphi^2 = 1 + \varphi$$, yielding:

Relationship to Fibonacci and Lucas numbers
Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence $$0,1$$:

The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in which each term is the sum of the previous two, however instead starts with $$2,1$$:

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates $$\varphi$$.

For example, $$\frac{F_{16}}{F_{15}} = \frac{987}{610} = 1.6180327\ldots, $$ and $$\frac{L_{16}}{L_{15}} = \frac{2207}{1364} = 1.6180351\ldots.$$

These approximations are alternately lower and higher than $$\varphi,$$ and converge to $$\varphi$$ as the Fibonacci and Lucas numbers increase.

Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:

Combining both formulas above, one obtains a formula for $$\varphi^n$$ that involves both Fibonacci and Lucas numbers:

Between Fibonacci and Lucas numbers one can deduce $$L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n,$$ which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:

Indeed, much stronger statements are true:
 * $$ \vert L_n - \sqrt{5} F_n \vert = \frac{2}{\varphi^n} \to 0, $$
 * $$ (L_{3n}/2)^2 = 5 (F_{3n}/2)^2 + (-1)^n. $$

These values describe $$\varphi$$ as a fundamental unit of the algebraic number field $$\mathbb{Q}(\sqrt5)$$.

Successive powers of the golden ratio obey the Fibonacci recurrence, i.e. $$\varphi^{n+1} = \varphi^n + \varphi^{n-1}.$$

The reduction to a linear expression can be accomplished in one step by using:

This identity allows any polynomial in $$\varphi$$ to be reduced to a linear expression, as in:

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation:

In particular, the powers of $$\varphi$$ themselves round to Lucas numbers (in order, except for the first two powers, $$\varphi^0$$ and $$\varphi$$, are in reverse order):

and so forth. The Lucas numbers also directly generate powers of the golden ratio; for $$n \ge 2$$:

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is $$L_n = F_{n-1}+F_{n+1}$$; and, importantly, that $$ L_n = \frac{F_{2n}}{F_n}$$.

Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.

Geometry
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

Construction
Dividing by interior division
 * 1) Having a line segment $$AB,$$ construct a perpendicular $$BC$$ at point $$B,$$ with $$BC$$ half the length of $$AB.$$ Draw the hypotenuse $$AC.$$
 * 2) Draw an arc with center $$C$$ and radius $$BC.$$ This arc intersects the hypotenuse $$AC$$ at point $$D.$$
 * 3) Draw an arc with center $$A$$ and radius $$AD.$$ This arc intersects the original line segment $$AB$$ at point $$S.$$ Point $$S$$ divides the original line segment $$AB$$ into line segments $$AS$$ and $$SB$$ with lengths in the golden ratio.

Dividing by exterior division Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.
 * 1) Draw a line segment $$AS$$ and construct off the point $$S$$ a segment $$SC$$ perpendicular to $$AS$$ and with the same length as $$AS.$$
 * 2) Do bisect the line segment $$AS$$ with $$M.$$
 * 3) A circular arc around $$M$$ with radius $$MC$$ intersects in point $$B$$ the straight line through points $$A$$ and $$S$$ (also known as the extension of $$AS$$). The ratio of $$AS$$ to the constructed segment $$SB$$ is the golden ratio.

Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.

Golden angle
When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure $g\colon$

This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.

Pentagon and pentagram

In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are $$a,$$ and short edges are $$b,$$ then Ptolemy's theorem gives $$a^2 = b^2 + ab.$$ Dividing both sides by $$ab$$ yields (see above),

The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by $$\varphi$$. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is $$\varphi,$$ as the four-color illustration shows.

Pentagonal and pentagrammic geometry permits us to calculate the following values for $$\varphi$$:

Golden triangle and golden gnomon
The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle 36° and base angles 72°. Its two equal sides are in the golden ratio to its base. The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle 108° and base angle 36°. Its base is in the golden ratio to its two equal sides. The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles, as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.

If the apex angle of the golden gnomon is trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.

Penrose tilings
The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together. Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
 * Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.
 * The kite and dart Penrose tiling uses kites with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.
 * The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context Robinson triangles, can be used as the prototiles for a form of the Penrose tiling.
 * The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals $$1:\varphi$$, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.

Odom's construction

George Odom found a construction for $$\varphi$$ involving an equilateral triangle: if the line segment joining the midpoints of two sides is extended to intersect the circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.

Kepler triangle
The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression:

These side lengths are the three Pythagorean means of the two numbers $$\varphi \pm 1$$. The three squares on its sides have areas in the golden geometric progression $$1\mathbin{:}\varphi\mathbin{:}\varphi^2$$.

Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two reflected copies of the Kepler triangle, sharing the longer of their two legs. The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.

For a Kepler triangle with smallest side length $$s$$, the area and acute internal angles are:

Golden rectangle

The golden ratio proportions the adjacent side lengths of a golden rectangle in $$1:\varphi$$ ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in $$\varphi$$ ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).

Golden rhombus
A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly $$1:\varphi$$. For a rhombus of such proportions, its acute angle and obtuse angles are:

The lengths of its short and long diagonals $$d$$ and $$D$$, in terms of side length $$a$$ are:

Its area, in terms of $$a$$,and $$d$$:

Its inradius, in terms of side $$a$$:

Golden rhombi form the faces of the rhombic triacontahedron, the two golden rhombohedra, the Bilinski dodecahedron, and the rhombic hexecontahedron.

Golden spiral
Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio, or their approximations generated from Fibonacci numbers, often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the polar equation with $$(r,\theta)$$:

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each 108° that it turns, instead of the 90° turning angle of the golden spiral. Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.

In the dodecahedron and icosahedron

The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. A dodecahedron has $$12$$ regular pentagonal faces, whereas an icosahedron has $$20$$ equilateral triangles; both have $$30$$ edges.

For a dodecahedron of side $$a$$, the radius of a circumscribed and inscribed sphere, and midradius are ($$r_u, $$ $$r_i,$$ and $$r_m,$$ respectively):

While for an icosahedron of side $$a$$, the radius of a circumscribed and inscribed sphere, and midradius are:

The volume and surface area of the dodecahedron can be expressed in terms of $$\varphi$$:

As well as for the icosahedron:

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving $$\varphi$$. The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of:

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings. In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain $$12$$ vertices of the icosahedron, or equivalently, intersect the centers of $$12$$ of the dodecahedron's faces.

A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is $$\tfrac{2}{2+\varphi}$$ times that of the dodecahedron's. In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in $$\varphi : \varphi^{2}$$ ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's $$12$$ vertices touch the $$12$$ edges of an octahedron at points that divide its edges in golden ratio.

Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the compound of five cubes, compound of five octahedra, compound of five tetrahedra, the compound of ten tetrahedra, rhombic triacontahedron, icosidodecahedron, truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, rhombic enneacontahedron, and Kepler-Poinsot polyhedra, and rhombic hexecontahedron. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the golden ratio.

Other properties
The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation $$x^2-x-1=0$$ or on $$x^2-5=0$$ (to compute $$\sqrt{5}$$ first). The time needed to compute $$n$$ digits of the golden ratio using Newton's method is essentially $$O(M(n))$$, where $$M(n)$$ is the time complexity of multiplying two $$n$$-digit numbers. This is considerably faster than known algorithms for $\pi$ and $e$. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers $$F_{25001}$$ and $$F_{25000},$$ each over $$5000$$ digits, yields over $$10{,}000$$ significant digits of the golden ratio. The decimal expansion of the golden ratio $$\varphi$$ has been calculated to an accuracy of ten trillion ($1 \times 10^{13} = 10{,}000{,}000{,}000{,}000$) digits.

In the complex plane, the fifth roots of unity $$z = e^{2\pi k i/5}$$ (for an integer $k$ ) satisfying $$z^5 = 1$$ are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, $$z + \bar z,$$ is a quadratic integer, an element of $\mathbb{Z}[\varphi].$ Specifically,

This also holds for the remaining tenth roots of unity satisfying $$z^{10} = 1,$$

For the gamma function $$\Gamma$$, the only solutions to the equation $$\Gamma(z-1) = \Gamma(z+1)$$ are $$z = \varphi$$ and $$z = -\varphi^{-1}$$.

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or $$\varphi$$-nary), quadratic integers in the ring $$\mathbb{Z}[\varphi]$$ – that is, numbers of the form $$a + b\varphi$$ for $$a, b \in \mathbb{Z}$$ – have terminating representations, but rational fractions have non-terminating representations.

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is $$4\log(\varphi).$$

The golden ratio appears in the theory of modular functions as well. For $$\left|q\right|<1$$, let

Then

and

where $$\operatorname{Im}\tau>0$$ and $$(e^z)^{1/5}$$ in the continued fraction should be evaluated as $$e^{z/5}$$. The function $$\tau\mapsto R(e^{2\pi i\tau})$$ is invariant under $$\Gamma (5)$$, a congruence subgroup of the modular group. Also for positive real numbers $$a, b \in \mathbb{R}^+$$ and $$ab = \pi^2,$$ then

$$\varphi$$ is a Pisot–Vijayaraghavan number.

Architecture
The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.

In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

Art
Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although Leonardo's Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.

Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is $$1.34,$$ with averages for individual artists ranging from $$1.04$$ (Goya) to $$1.46$$ (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and $$\sqrt5$$ proportions, and others with proportions like $$\sqrt2,$$ $$3,$$ $$4,$$ and $$6.$$

Books and design
According to Jan Tschichold,

"There was a time when deviations from the truly beautiful page proportions $2\mathbin{:}3,$ $1\mathbin{:}\sqrt3,$ and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter."

According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.

Flags

The aspect ratio (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer.

Music
Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals $a + b⁄a = a⁄b = φ.$ $1.618....$ $√5$ and $−φ^{−1}$ and the main climax sits at the phi position".

The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.

Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, a musical scale based on using the golden ratio as its fundamental musical interval. When measured in cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.

Nature

Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".

The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.

Physics
The quasi-one-dimensional Ising ferromagnet (cobalt niobate) has 8 predicted excitation states (with E8 symmetry), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its paramagnetic phase; revealing, just below its critical field, a spin dynamics with sharp modes at low energies approaching the golden mean.

Optimization
There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem or Tammes problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. $$360^\circ/\varphi \approx 222.5^\circ.$$ This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.

The golden ratio is a critical element to golden-section search as well.

Disputed observations
Examples of disputed observations of the golden ratio include the following:
 * Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive phalangeal and metacarpal bones (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.
 * The shells of mollusks such as the nautilus are often claimed to be in the golden ratio. The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio, or sometimes claimed that each new chamber is golden-proportioned relative to the previous one. However, measurements of nautilus shells do not support this claim.
 * Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is $$1.45.$$
 * Studies by psychologists, starting with Gustav Fechner c. 1876, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.
 * In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio. The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.

Egyptian pyramids
The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.

The Parthenon

The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries. Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

Modern art
The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism. Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat. (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat's writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.) The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception". However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not, and Marcel Duchamp said as much in an interview. On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition. Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier Bandeaux d'Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have discredited these claims.

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title: Golden ratio


text:
Genome

In the fields of molecular biology and genetics, a genome is all the genetic information of an organism. It consists of nucleotide sequences of DNA (or RNA in RNA viruses). The nuclear genome includes protein-coding genes and non-coding genes, other functional regions of the genome such as regulatory sequences (see non-coding DNA), and often a substantial fraction of junk DNA with no evident function. Almost all eukaryotes have mitochondria and a small mitochondrial genome. Algae and plants also contain chloroplasts with a chloroplast genome.

The study of the genome is called genomics. The genomes of many organisms have been sequenced and various regions have been annotated. The Human Genome Project was started in October 1990, and then reported the sequence of the human genome in April 2003, although the initial "finished" sequence was missing 8% of the genome consisting mostly of repetitive sequences.

With advancements in technology that could handle sequencing of the many repetitive sequences found in human DNA that were not fully uncovered by the original Human Genome Project study, scientists reported the first end-to-end human genome sequence in March 2022.

Origin of the term
The term genome was created in 1920 by Hans Winkler, professor of botany at the University of Hamburg, Germany. The website Oxford Dictionaries and the Online Etymology Dictionary suggest the name is a blend of the words gene and chromosome. However, see omics for a more thorough discussion. A few related -ome words already existed, such as biome and rhizome, forming a vocabulary into which genome fits systematically.

Definition
It's very difficult to come up with a precise definition of "genome." It usually refers to the DNA (or sometimes RNA) molecules that carry the genetic information in an organism but sometimes it is difficult to decide which molecules to include in the definition; for example, bacteria usually have one or two large DNA molecules (chromosomes) that contain all of the essential genetic material but they also contain smaller extrachromosomal plasmid molecules that carry important genetic information. The definition of 'genome' that's commonly used in the scientific literature is usually restricted to the large chromosomal DNA molecules in bacteria.

Nuclear genome
Eukaryotic genomes are even more difficult to define because almost all eukaryotic species contain nuclear chromosomes plus extra DNA molecules in the mitochondria. In addition, algae and plants have chloroplast DNA. Most textbooks make a distinction between the nuclear genome and the organelle (mitochondria and chloroplast) genomes so when they speak of, say, the human genome, they are only referring to the genetic material in the nucleus. This is the most common use of 'genome' in the scientific literature.

Ploidy
Most eukaryotes are diploid, meaning that there are two of each chromosome in the nucleus but the 'genome' refers to only one copy of each chromosome. Some eukaryotes have distinctive sex chromosomes, such as the X and Y chromosomes of mammals, so the technical definition of the genome must include both copies of the sex chromosomes. For example, the standard reference genome of humans consists of one copy of each of the 22 autosomes plus one X chromosome and one Y chromosome.

Sequencing and mapping
A genome sequence is the complete list of the nucleotides (A, C, G, and T for DNA genomes) that make up all the chromosomes of an individual or a species. Within a species, the vast majority of nucleotides are identical between individuals, but sequencing multiple individuals is necessary to understand the genetic diversity. In 1976, Walter Fiers at the University of Ghent (Belgium) was the first to establish the complete nucleotide sequence of a viral RNA-genome (Bacteriophage MS2). The next year, Fred Sanger completed the first DNA-genome sequence: Phage Φ-X174, of 5386 base pairs. The first bacterial genome to be sequenced was that of Haemophilus influenzae, completed by a team at The Institute for Genomic Research in 1995. A few months later, the first eukaryotic genome was completed, with sequences of the 16 chromosomes of budding yeast Saccharomyces cerevisiae published as the result of a European-led effort begun in the mid-1980s. The first genome sequence for an archaeon, Methanococcus jannaschii, was completed in 1996, again by The Institute for Genomic Research.

The development of new technologies has made genome sequencing dramatically cheaper and easier, and the number of complete genome sequences is growing rapidly. The US National Institutes of Health maintains one of several comprehensive databases of genomic information. Among the thousands of completed genome sequencing projects include those for rice, a mouse, the plant Arabidopsis thaliana, the puffer fish, and the bacteria E. coli. In December 2013, scientists first sequenced the entire genome of a Neanderthal, an extinct species of humans. The genome was extracted from the toe bone of a 130,000-year-old Neanderthal found in a Siberian cave.

New sequencing technologies, such as massive parallel sequencing have also opened up the prospect of personal genome sequencing as a diagnostic tool, as pioneered by Manteia Predictive Medicine. A major step toward that goal was the completion in 2007 of the full genome of James D. Watson, one of the co-discoverers of the structure of DNA.

Whereas a genome sequence lists the order of every DNA base in a genome, a genome map identifies the landmarks. A genome map is less detailed than a genome sequence and aids in navigating around the genome. The Human Genome Project was organized to map and to sequence the human genome. A fundamental step in the project was the release of a detailed genomic map by Jean Weissenbach and his team at the Genoscope in Paris.

Reference genome sequences and maps continue to be updated, removing errors and clarifying regions of high allelic complexity. The decreasing cost of genomic mapping has permitted genealogical sites to offer it as a service, to the extent that one may submit one's genome to crowdsourced scientific endeavours such as DNA.LAND at the New York Genome Center, an example both of the economies of scale and of citizen science.

Viral genomes
Viral genomes can be composed of either RNA or DNA. The genomes of RNA viruses can be either single-stranded RNA or double-stranded RNA, and may contain one or more separate RNA molecules (segments: monopartit or multipartit genome). DNA viruses can have either single-stranded or double-stranded genomes. Most DNA virus genomes are composed of a single, linear molecule of DNA, but some are made up of a circular DNA molecule.

Prokaryotic genomes
Prokaryotes and eukaryotes have DNA genomes. Archaea and most bacteria have a single circular chromosome, however, some bacterial species have linear or multiple chromosomes. If the DNA is replicated faster than the bacterial cells divide, multiple copies of the chromosome can be present in a single cell, and if the cells divide faster than the DNA can be replicated, multiple replication of the chromosome is initiated before the division occurs, allowing daughter cells to inherit complete genomes and already partially replicated chromosomes. Most prokaryotes have very little repetitive DNA in their genomes. However, some symbiotic bacteria (e.g. Serratia symbiotica) have reduced genomes and a high fraction of pseudogenes: only ~40% of their DNA encodes proteins.

Some bacteria have auxiliary genetic material, also part of their genome, which is carried in plasmids. For this, the word genome should not be used as a synonym of chromosome.

Eukaryotic genomes
Eukaryotic genomes are composed of one or more linear DNA chromosomes. The number of chromosomes varies widely from Jack jumper ants and an asexual nemotode, which each have only one pair, to a fern species that has 720 pairs. It is surprising the amount of DNA that eukaryotic genomes contain compared to other genomes. The amount is even more than what is necessary for DNA protein-coding and noncoding genes due to the fact that eukaryotic genomes show as much as 64,000-fold variation in their sizes. However, this special characteristic is caused by the presence of repetitive DNA, and transposable elements (TEs).

A typical human cell has two copies of each of 22 autosomes, one inherited from each parent, plus two sex chromosomes, making it diploid. Gametes, such as ova, sperm, spores, and pollen, are haploid, meaning they carry only one copy of each chromosome. In addition to the chromosomes in the nucleus, organelles such as the chloroplasts and mitochondria have their own DNA. Mitochondria are sometimes said to have their own genome often referred to as the "mitochondrial genome". The DNA found within the chloroplast may be referred to as the "plastome". Like the bacteria they originated from, mitochondria and chloroplasts have a circular chromosome.

Unlike prokaryotes where exon-intron organization of protein coding genes exists but is rather exceptional, eukaryotes generally have these features in their genes and their genomes contain variable amounts of repetitive DNA. In mammals and plants, the majority of the genome is composed of repetitive DNA.

DNA sequencing
High-throughput technology makes sequencing to assemble new genomes accessible to everyone. Sequence polymorphisms are typically discovered by comparing resequenced isolates to a reference, whereas analyses of coverage depth and mapping topology can provide details regarding structural variations such as chromosomal translocations and segmental duplications.

Coding sequences
DNA sequences that carry the instructions to make proteins are referred to as coding sequences. The proportion of the genome occupied by coding sequences varies widely. A larger genome does not necessarily contain more genes, and the proportion of non-repetitive DNA decreases along with increasing genome size in complex eukaryotes.

Noncoding sequences
Noncoding sequences include introns, sequences for non-coding RNAs, regulatory regions, and repetitive DNA. Noncoding sequences make up 98% of the human genome. There are two categories of repetitive DNA in the genome: tandem repeats and interspersed repeats.

Tandem repeats
Short, non-coding sequences that are repeated head-to-tail are called tandem repeats. Microsatellites consisting of 2–5 basepair repeats, while minisatellite repeats are 30–35 bp. Tandem repeats make up about 4% of the human genome and 9% of the fruit fly genome. Tandem repeats can be functional. For example, telomeres are composed of the tandem repeat TTAGGG in mammals, and they play an important role in protecting the ends of the chromosome.

In other cases, expansions in the number of tandem repeats in exons or introns can cause disease. For example, the human gene huntingtin (Htt) typically contains 6–29 tandem repeats of the nucleotides CAG (encoding a polyglutamine tract). An expansion to over 36 repeats results in Huntington's disease, a neurodegenerative disease. Twenty human disorders are known to result from similar tandem repeat expansions in various genes. The mechanism by which proteins with expanded polygulatamine tracts cause death of neurons is not fully understood. One possibility is that the proteins fail to fold properly and avoid degradation, instead accumulating in aggregates that also sequester important transcription factors, thereby altering gene expression.

Tandem repeats are usually caused by slippage during replication, unequal crossing-over and gene conversion.

Transposable elements
Transposable elements (TEs) are sequences of DNA with a defined structure that are able to change their location in the genome. TEs are categorized as either as a mechanism that replicates by copy-and-paste or as a mechanism that can be excised from the genome and inserted at a new location. In the human genome, there are three important classes of TEs that make up more than 45% of the human DNA; these classes are The long interspersed nuclear elements (LINEs), The interspersed nuclear elements (SINEs), and endogenous retroviruses. These elements have a big potential to modify the genetic control in a host organism.

The movement of TEs is a driving force of genome evolution in eukaryotes because their insertion can disrupt gene functions, homologous recombination between TEs can produce duplications, and TE can shuffle exons and regulatory sequences to new locations.

Retrotransposons
Retrotransposons are found mostly in eukaryotes but not found in prokaryotes. Retrotransposons form a large portion of the genomes of many eukaryotes. A retrotransposon is a transposable element that transposes through an RNA intermediate. Retrotransposons are composed of DNA, but are transcribed into RNA for transposition, then the RNA transcript is copied back to DNA formation with the help of a specific enzyme called reverse transcriptase. A retrotransposon that carries reverse transcriptase in its sequence can trigger its own transposition but retrotransposons that lack a reverse transcriptase must use reverse transcriptase synthesized by another retrotransposon. Retrotransposons can be transcribed into RNA, which are then duplicated at another site into the genome. Retrotransposons can be divided into long terminal repeats (LTRs) and non-long terminal repeats (Non-LTRs).

Long terminal repeats (LTRs) are derived from ancient retroviral infections, so they encode proteins related to retroviral proteins including gag (structural proteins of the virus), pol (reverse transcriptase and integrase), pro (protease), and in some cases env (envelope) genes. These genes are flanked by long repeats at both 5' and 3' ends. It has been reported that LTRs consist of the largest fraction in most plant genome and might account for the huge variation in genome size.

Non-long terminal repeats (Non-LTRs) are classified as long interspersed nuclear elements (LINEs), short interspersed nuclear elements (SINEs), and Penelope-like elements (PLEs). In Dictyostelium discoideum, there is another DIRS-like elements belong to Non-LTRs. Non-LTRs are widely spread in eukaryotic genomes.

Long interspersed elements (LINEs) encode genes for reverse transcriptase and endonuclease, making them autonomous transposable elements. The human genome has around 500,000 LINEs, taking around 17% of the genome.

Short interspersed elements (SINEs) are usually less than 500 base pairs and are non-autonomous, so they rely on the proteins encoded by LINEs for transposition. The Alu element is the most common SINE found in primates. It is about 350 base pairs and occupies about 11% of the human genome with around 1,500,000 copies.

DNA transposons
DNA transposons encode a transposase enzyme between inverted terminal repeats. When expressed, the transposase recognizes the terminal inverted repeats that flank the transposon and catalyzes its excision and reinsertion in a new site. This cut-and-paste mechanism typically reinserts transposons near their original location (within 100 kb). DNA transposons are found in bacteria and make up 3% of the human genome and 12% of the genome of the roundworm C. elegans.

Genome size
Genome size is the total number of the DNA base pairs in one copy of a haploid genome. Genome size varies widely across species. Invertebrates have small genomes, this is also correlated to a small number of transposable elements. Fish and Amphibians have intermediate-size genomes, and birds have relatively small genomes but it has been suggested that birds lost a substantial portion of their genomes during the phase of transition to flight. Before this loss, DNA methylation allows the adequate expansion of the genome.

In humans, the nuclear genome comprises approximately 3.1 billion nucleotides of DNA, divided into 24 linear molecules, the shortest 45 000 000 nucleotides in length and the longest 248 000 000 nucleotides, each contained in a different chromosome. There is no clear and consistent correlation between morphological complexity and genome size in either prokaryotes or lower eukaryotes. Genome size is largely a function of the expansion and contraction of repetitive DNA elements.

Since genomes are very complex, one research strategy is to reduce the number of genes in a genome to the bare minimum and still have the organism in question survive. There is experimental work being done on minimal genomes for single cell organisms as well as minimal genomes for multi-cellular organisms (see developmental biology). The work is both in vivo and in silico.

Genome size differences due to transposable elements

There are many enormous differences in size in genomes, specially mentioned before in the multicellular eukaryotic genomes. Much of this is due to the differing abundances of transposable elements, which evolve by creating new copies of themselves in the chromosomes. Eukaryote genomes often contain many thousands of copies of these elements, most of which have acquired mutations that make them defective. Here is a table of some significant or representative genomes. See for lists of sequenced genomes.

Genomic alterations
All the cells of an organism originate from a single cell, so they are expected to have identical genomes; however, in some cases, differences arise. Both the process of copying DNA during cell division and exposure to environmental mutagens can result in mutations in somatic cells. In some cases, such mutations lead to cancer because they cause cells to divide more quickly and invade surrounding tissues. In certain lymphocytes in the human immune system, V(D)J recombination generates different genomic sequences such that each cell produces a unique antibody or T cell receptors.

During meiosis, diploid cells divide twice to produce haploid germ cells. During this process, recombination results in a reshuffling of the genetic material from homologous chromosomes so each gamete has a unique genome.

Genome-wide reprogramming
Genome-wide reprogramming in mouse primordial germ cells involves epigenetic imprint erasure leading to totipotency. Reprogramming is facilitated by active DNA demethylation, a process that entails the DNA base excision repair pathway. This pathway is employed in the erasure of CpG methylation (5mC) in primordial germ cells. The erasure of 5mC occurs via its conversion to 5-hydroxymethylcytosine (5hmC) driven by high levels of the ten-eleven dioxygenase enzymes TET1 and TET2.

Genome evolution
Genomes are more than the sum of an organism's genes and have traits that may be measured and studied without reference to the details of any particular genes and their products. Researchers compare traits such as karyotype (chromosome number), genome size, gene order, codon usage bias, and GC-content to determine what mechanisms could have produced the great variety of genomes that exist today (for recent overviews, see Brown 2002; Saccone and Pesole 2003; Benfey and Protopapas 2004; Gibson and Muse 2004; Reese 2004; Gregory 2005).

Duplications play a major role in shaping the genome. Duplication may range from extension of short tandem repeats, to duplication of a cluster of genes, and all the way to duplication of entire chromosomes or even entire genomes. Such duplications are probably fundamental to the creation of genetic novelty.

Horizontal gene transfer is invoked to explain how there is often an extreme similarity between small portions of the genomes of two organisms that are otherwise very distantly related. Horizontal gene transfer seems to be common among many microbes. Also, eukaryotic cells seem to have experienced a transfer of some genetic material from their chloroplast and mitochondrial genomes to their nuclear chromosomes. Recent empirical data suggest an important role of viruses and sub-viral RNA-networks to represent a main driving role to generate genetic novelty and natural genome editing.

In fiction
Works of science fiction illustrate concerns about the availability of genome sequences.

Michael Crichton's 1990 novel Jurassic Park and the subsequent film tell the story of a billionaire who creates a theme park of cloned dinosaurs on a remote island, with disastrous outcomes. A geneticist extracts dinosaur DNA from the blood of ancient mosquitoes and fills in the gaps with DNA from modern species to create several species of dinosaurs. A chaos theorist is asked to give his expert opinion on the safety of engineering an ecosystem with the dinosaurs, and he repeatedly warns that the outcomes of the project will be unpredictable and ultimately uncontrollable. These warnings about the perils of using genomic information are a major theme of the book.

The 1997 film Gattaca is set in a futurist society where genomes of children are engineered to contain the most ideal combination of their parents' traits, and metrics such as risk of heart disease and predicted life expectancy are documented for each person based on their genome. People conceived outside of the eugenics program, known as "In-Valids" suffer discrimination and are relegated to menial occupations. The protagonist of the film is an In-Valid who works to defy the supposed genetic odds and achieve his dream of working as a space navigator. The film warns against a future where genomic information fuels prejudice and extreme class differences between those who can and cannot afford genetically engineered children.

title: Genome


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Gaia philosophy

Gaia philosophy (named after Gaia, Greek goddess of the Earth) is a broadly inclusive term for relating concepts about, humanity as an effect of the life of this planet.

The Gaia hypothesis holds that all organisms on a life-giving planet regulate the biosphere in such a way as to promote its habitability. Gaia concepts draw a connection between the survivability of a species (hence its evolutionary course) and its usefulness to the survival of other species. While there were a number of precursors to Gaia hypothesis, the first scientific form of this idea was proposed as the Gaia hypothesis by James Lovelock, a UK chemist, in 1970. The Gaia hypothesis deals with the concept of biological homeostasis, and claims the resident life forms of a host planet coupled with their environment have acted and act like a single, self-regulating system. This system includes the near-surface rocks, the soil, and the atmosphere. Gaia hypothesis criticism]]). These theories are, however, significant in green politics.

Predecessors to the Gaia theory
There are some mystical, scientific and religious predecessors to the Gaia philosophy, which had a Gaia-like conceptual basis. Many religious mythologies had a view of Earth as being a whole that is greater than the sum of its parts (e.g. some Native American religions and various forms of shamanism).

Isaac Newton wrote of the earth, "Thus this Earth resembles a great animal or rather inanimate vegetable, draws in æthereall breath for its dayly refreshment & vitall ferment & transpires again with gross exhalations, And according to the condition of all other things living ought to have its times of beginning youth old age & perishing."

Pierre Teilhard de Chardin, a paleontologist and geologist, believed that evolution fractally unfolded from cell to organism to planet to solar system and ultimately the whole universe, as we humans see it from our limited perspective. Teilhard later influenced Thomas Berry and many Catholic humanist thinkers of the 20th century.

Lewis Thomas believed that Earth should be viewed as a single cell; he derived this view from Johannes Kepler's view of Earth as a single round organism.

Buckminster Fuller is generally credited with making the idea respectable in Western scientific circles in the 20th century. Building to some degree on his observations and artifacts, e.g. the Dymaxion map of the Earth he created, others began to ask if there was a way to make the Gaia theory scientifically sound.

In 1931, L.G.M. Baas Becking delivered an inaugural lecture about Gaia in the sense of life and earth.

Oberon Zell-Ravenheart in 1970 in an article in Green Egg Magazine, independently articulated the Gaia Thesis.

Many believe that these ideas cannot be considered scientific hypotheses; by definition a scientific hypothesis must make testable predictions. As the above claims are not currently testable, they are outside the bounds of current science. This does not mean that these ideas are not theoretically testable. As one can postulate tests that could be applied, given enough time and space, then these ideas should be seen as scientific hypotheses.

These are conjectures and perhaps can only be considered as social and maybe political philosophy; they may have implications for theology, or thealogy as Zell-Ravenheart and Isaac Bonewits put it.

Range of views
According to James Kirchner there is a spectrum of Gaia hypotheses, ranging from the undeniable to radical. At one end is the undeniable statement that the organisms on the Earth have radically altered its composition. A stronger position is that the Earth's biosphere effectively acts as if it is a self-organizing system which works in such a way as to keep its systems in some kind of equilibrium that is conducive to life. Today many scientists consider that such a view (and any stronger views) are unlikely to be correct. An even stronger claim is that all lifeforms are part of a single planetary being, called Gaia. In this view, the atmosphere, the seas, the terrestrial crust would be the result of interventions carried out by Gaia, through the coevolving diversity of living organisms.

The most extreme form of Gaia theory is that the entire Earth is a single unified organism with a highly intelligent mind that arose as an emergent property of the whole biosphere. In this view, the Earth's biosphere is consciously manipulating the climate in order to make conditions more conducive to life. Scientists contend that there is no evidence at all to support this last point of view, and it has come about because many people do not understand the concept of homeostasis. Many non-scientists instinctively and incorrectly see homeostasis as a process that requires conscious control.

The more speculative versions of Gaia, including versions in which it is believed that the Earth is actually conscious, sentient, and highly intelligent, are usually considered outside the bounds of what is usually considered science.

Gaia in biology and science
Buckminster Fuller has been credited as the first to incorporate scientific ideas into a Gaia theory, which he did with his Dymaxion map of the Earth.

The first scientifically rigorous theory was the Gaia hypothesis by James Lovelock, a UK chemist.

A variant of this hypothesis was developed by Lynn Margulis, a microbiologist, in 1979. Her version is sometimes called the "Gaia Theory" (note uppercase-T). Her model is more limited in scope than the one that Lovelock proposed.

Whether this sort of system is present on Earth is still open to debate. Some relatively simple homeostatic mechanisms are generally accepted. For example, when atmospheric carbon dioxide levels rise, plants are able to grow better and thus remove more carbon dioxide from the atmosphere. Other biological effects and feedbacks exist, but the extent to which these mechanisms have stabilized and modified the Earth's overall climate is largely not known.

The Gaia hypothesis is sometimes viewed from significantly different philosophical perspectives. Some environmentalists view it as an almost conscious process, in which the Earth's ecosystem is literally viewed as a single unified organism. Some evolutionary biologists, on the other hand, view it as an undirected emergent property of the ecosystem: as each individual species pursues its own self-interest, their combined actions tend to have counterbalancing effects on environmental change. Proponents of this view sometimes point to examples of life's actions in the past that have resulted in dramatic change rather than stable equilibrium, such as the conversion of the Earth's atmosphere from a reducing environment to an oxygen-rich one.

Depending on how strongly the case is stated, the hypothesis conflicts with mainstream neo-Darwinism. Most biologists would accept Daisyworld-style homeostasis as possible, but would certainly not accept the idea that this equates to the whole biosphere acting as one organism.

A very small number of scientists, and a much larger number of environmental activists, claim that Earth's biosphere is consciously manipulating the climate in order to make conditions more conducive to life. Scientists contend that there is no evidence to support this belief.

Gaia in the social sciences and politics
A social science view of Gaia theory is the role of humans as a keystone species who may be able to accomplish global homeostasis. Whilst a few social scientists who draw inspiration from 'organic' views of society have embraced Gaia philosophy as a way to explain the human-nature interconnections, most professional social scientists are more involved in reflecting upon the way Gaia philosophy is used and engaged with within sub-sections of society. Alan Marshall, in the Department of Social Sciences at Mahidol University, for example, reflects upon the way Gaia philosophy has been used and advocated in various societal settings by environmentalists, spiritualists, managers, economists, and scientists and engineers. As Marshall explains, most social scientists had already given up on systems ideas of society in the 1960s before Gaia philosophy was born under James Lovelock's ideas since such ideas were interpreted as supporting conservatism and traditionalism. Gaia theory also influenced the dynamics of green politics.

Gaia in religion
Rosemary Radford Ruether, the American feminist scholar and theologian, wrote a book called Gaia and God: An Ecofeminist Theology of Earth Healing.

A book edited by Allan Hunt Badiner called Dharma Gaia explores the ground where Buddhism and ecology meet through writings by the Dalai Lama, Gary Snyder, Thich Nhat Hanh, Allen Ginsberg, David Abram, Joanna Macy, Robert Aitken, and 25 other Buddhists and ecologists.

Gaianism, an earth-centered philosophical, holistic, and spiritual belief that shares expressions with earth religions and paganism while not identifying exclusively with any specific religion, sprang from the gaia hypothesis.

Criticism
One of the most problematic issues with referring to Gaia as an organism is its apparent failure to meet the biological criterion of being able to reproduce. Richard Dawkins has asserted that the planet is not the offspring of any parents and is unable to reproduce.

Books on Gaia

 * Alan Marshall (2002), The Unity of Nature, Imperial College Press.
 * Mary Midgley (2007), Earthy realism: the meaning of Gaia
 * Mary Midgley (2001), Gaia: the next big idea
 * Lawrence E. Joseph (1991), Gaia: the growth of an idea
 * Stephen Henry Schneider (2004), Scientists debate gaia: the next century
 * Allan Hunt Badiner (1990), Dharma Gaia: A Harvest of Essays in Buddhism and Ecology
 * George Ronald Williams (1996), The molecular biology of Gaia
 * Tyler Volk (2003), Gaia's Body: Toward a Physiology of Earth
 * Norman Myers (1993), Gaia An Atlas of Planet Management
 * Anne Primavesi (2008), Gaia and Climate Change: A Theology of Gift Events
 * Anne Primavesi (2000), Sacred Gaia: holistic theology and earth system science
 * Anne Primavesi (2003), Gaia's gift: earth, ourselves, and God after Copernicus
 * Peter Bunyard (1996), Gaia in Action: Science of the Living Earth
 * Francesca Ciancimino Howell (2002), Making Magic with Gaia: Practices to Heal Ourselves and Our Planet
 * Pepper Lewis (2005), Gaia Speaks
 * Toby Tyrrell (2013), On Gaia

title: Gaia philosophy