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Thursday, August 11, 2022

Eye

From Wikipedia, the free encyclopedia

Eye
Schematic diagram of the human eye en.svg
 
Krilleyekils.jpg
 
Details
SystemNervous
Identifiers
LatinOculus
TA98A15.2.00.001
A01.1.00.007
TA2113, 6734

Eyes are organs of the visual system. They provide living organisms with vision, the ability to receive and process visual detail, as well as enabling several photo response functions that are independent of vision. Eyes detect light and convert it into electro-chemical impulses in neurons. In higher organisms, the eye is a complex optical system which collects light from the surrounding environment, regulates its intensity through a diaphragm, focuses it through an adjustable assembly of lenses to form an image, converts this image into a set of electrical signals, and transmits these signals to the brain through complex neural pathways that connect the eye via the optic nerve to the visual cortex and other areas of the brain. Eyes with resolving power have come in ten fundamentally different forms, and 96% of animal species possess a complex optical system. Image-resolving eyes are present in molluscs, chordates and arthropods.

The most simple eyes, pit eyes, are eye-spots which may be set into a pit to reduce the angles of light that enters and affects the eye-spot, to allow the organism to deduce the angle of incoming light. From more complex eyes, retinal photosensitive ganglion cells send signals along the retinohypothalamic tract to the suprachiasmatic nuclei to effect circadian adjustment and to the pretectal area to control the pupillary light reflex.

Overview

Complex eyes distinguish shapes and colours. The visual fields of many organisms, especially predators, involve large areas of binocular vision for depth perception. In other organisms, particularly prey animals, eyes are located to maximise the field of view, such as in rabbits and horses, which have monocular vision.

The first proto-eyes evolved among animals 600 million years ago about the time of the Cambrian explosion. The last common ancestor of animals possessed the biochemical toolkit necessary for vision, and more advanced eyes have evolved in 96% of animal species in six of the ~35 main phyla. In most vertebrates and some molluscs, the eye allows light to enter and project onto a light-sensitive layer of cells known as the retina. The cone cells (for colour) and the rod cells (for low-light contrasts) in the retina detect and convert light into neural signals which are transmitted to the brain via the optic nerve to produce vision. Such eyes are typically spheroid, filled with the transparent gel-like vitreous humour, possess a focusing lens, and often an iris. Muscles around the iris change the size of the pupil, regulating the amount of light that enters the eye and reducing aberrations when there is enough light. The eyes of most cephalopods, fish, amphibians and snakes have fixed lens shapes, and focusing is achieved by telescoping the lens in a similar manner to that of a camera.

The compound eyes of the arthropods are composed of many simple facets which, depending on anatomical detail, may give either a single pixelated image or multiple images per eye. Each sensor has its own lens and photosensitive cell(s). Some eyes have up to 28,000 such sensors arranged hexagonally, which can give a full 360° field of vision. Compound eyes are very sensitive to motion. Some arthropods, including many Strepsiptera, have compound eyes of only a few facets, each with a retina capable of creating an image. With each eye producing a different image, a fused, high-resolution image is produced in the brain.

The eyes of mantis shrimps (here Odontodactylus scyllarus) are considered the best in the whole animal kingdom.

Possessing detailed hyperspectral colour vision, the Mantis shrimp has the world's most complex colour vision system. Trilobites, now extinct, had unique compound eyes. Clear calcite crystals formed the lenses of their eyes. They differ in this from most other arthropods, which have soft eyes. The number of lenses in such an eye varied widely; some trilobites had only one while others had thousands of lenses per eye.

In contrast to compound eyes, simple eyes have a single lens. Jumping spiders have one pair of large simple eyes with a narrow field of view, augmented by an array of smaller eyes for peripheral vision. Some insect larvae, like caterpillars, have a type of simple eye (stemmata) which usually provides only a rough image, but (as in sawfly larvae) can possess resolving powers of 4 degrees of arc, be polarization-sensitive, and capable of increasing its absolute sensitivity at night by a factor of 1,000 or more. Ocelli, some of the simplest eyes, are found in animals such as some of the snails. They have photosensitive cells but no lens or other means of projecting an image onto those cells. They can distinguish between light and dark but no more, enabling them to avoid direct sunlight. In organisms dwelling near deep-sea vents, compound eyes are adapted to see the infra-red light produced by the hot vents; enabling the creatures to avoid being boiled alive.

Types

There are ten different eye layouts—indeed every technological method of capturing an optical image commonly used by human beings, with the exceptions of zoom and Fresnel lenses, occur in nature. Eye types can be categorised into "simple eyes", with one concave photoreceptive surface, and "compound eyes", which comprise a number of individual lenses laid out on a convex surface. Note that "simple" does not imply a reduced level of complexity or acuity. Indeed, any eye type can be adapted for almost any behaviour or environment. The only limitations specific to eye types are that of resolution—the physics of compound eyes prevents them from achieving a resolution better than 1°. Also, superposition eyes can achieve greater sensitivity than apposition eyes, so are better suited to dark-dwelling creatures. Eyes also fall into two groups on the basis of their photoreceptor's cellular construction, with the photoreceptor cells either being cilliated (as in the vertebrates) or rhabdomeric. These two groups are not monophyletic; the cnidaria also possess cilliated cells, and some gastropods, as well as some annelids possess both.

Some organisms have photosensitive cells that do nothing but detect whether the surroundings are light or dark, which is sufficient for the entrainment of circadian rhythms. These are not considered eyes because they lack enough structure to be considered an organ, and do not produce an image.

Non-compound eyes

Simple eyes are rather ubiquitous, and lens-bearing eyes have evolved at least seven times in vertebrates, cephalopods, annelids, crustaceans and cubozoa.

Pit eyes

Pit eyes, also known as stemma, are eye-spots which may be set into a pit to reduce the angles of light that enters and affects the eye-spot, to allow the organism to deduce the angle of incoming light. Found in about 85% of phyla, these basic forms were probably the precursors to more advanced types of "simple eyes". They are small, comprising up to about 100 cells covering about 100 µm. The directionality can be improved by reducing the size of the aperture, by incorporating a reflective layer behind the receptor cells, or by filling the pit with a refractile material.

Pit vipers have developed pits that function as eyes by sensing thermal infra-red radiation, in addition to their optical wavelength eyes like those of other vertebrates (see infrared sensing in snakes). However, pit organs are fitted with receptors rather different from photoreceptors, namely a specific transient receptor potential channel (TRP channels) called TRPV1. The main difference is that photoreceptors are G-protein coupled receptors but TRP are ion channels.

Spherical lens eye

The resolution of pit eyes can be greatly improved by incorporating a material with a higher refractive index to form a lens, which may greatly reduce the blur radius encountered—hence increasing the resolution obtainable. The most basic form, seen in some gastropods and annelids, consists of a lens of one refractive index. A far sharper image can be obtained using materials with a high refractive index, decreasing to the edges; this decreases the focal length and thus allows a sharp image to form on the retina. This also allows a larger aperture for a given sharpness of image, allowing more light to enter the lens; and a flatter lens, reducing spherical aberration. Such a non-homogeneous lens is necessary for the focal length to drop from about 4 times the lens radius, to 2.5 radii.

Heterogeneous eyes have evolved at least nine times: four or more times in gastropods, once in the copepods, once in the annelids, once in the cephalopods, and once in the chitons, which have aragonite lenses. No extant aquatic organisms possess homogeneous lenses; presumably the evolutionary pressure for a heterogeneous lens is great enough for this stage to be quickly "outgrown".

This eye creates an image that is sharp enough that motion of the eye can cause significant blurring. To minimise the effect of eye motion while the animal moves, most such eyes have stabilising eye muscles.

The ocelli of insects bear a simple lens, but their focal point usually lies behind the retina; consequently, those can't form a sharp image. Ocelli (pit-type eyes of arthropods) blur the image across the whole retina, and are consequently excellent at responding to rapid changes in light intensity across the whole visual field; this fast response is further accelerated by the large nerve bundles which rush the information to the brain. Focusing the image would also cause the sun's image to be focused on a few receptors, with the possibility of damage under the intense light; shielding the receptors would block out some light and thus reduce their sensitivity. This fast response has led to suggestions that the ocelli of insects are used mainly in flight, because they can be used to detect sudden changes in which way is up (because light, especially UV light which is absorbed by vegetation, usually comes from above).

Multiple lenses

Some marine organisms bear more than one lens; for instance the copepod Pontella has three. The outer has a parabolic surface, countering the effects of spherical aberration while allowing a sharp image to be formed. Another copepod, Copilia, has two lenses in each eye, arranged like those in a telescope. Such arrangements are rare and poorly understood, but represent an alternative construction.

Multiple lenses are seen in some hunters such as eagles and jumping spiders, which have a refractive cornea: these have a negative lens, enlarging the observed image by up to 50% over the receptor cells, thus increasing their optical resolution.

Refractive cornea

In the eyes of most mammals, birds, reptiles, and most other terrestrial vertebrates (along with spiders and some insect larvae) the vitreous fluid has a higher refractive index than the air. In general, the lens is not spherical. Spherical lenses produce spherical aberration. In refractive corneas, the lens tissue is corrected with inhomogeneous lens material (see Luneburg lens), or with an aspheric shape. Flattening the lens has a disadvantage; the quality of vision is diminished away from the main line of focus. Thus, animals that have evolved with a wide field-of-view often have eyes that make use of an inhomogeneous lens.

As mentioned above, a refractive cornea is only useful out of water. In water, there is little difference in refractive index between the vitreous fluid and the surrounding water. Hence creatures that have returned to the water—penguins and seals, for example—lose their highly curved cornea and return to lens-based vision. An alternative solution, borne by some divers, is to have a very strongly focusing cornea.

Reflector eyes

An alternative to a lens is to line the inside of the eye with "mirrors", and reflect the image to focus at a central point. The nature of these eyes means that if one were to peer into the pupil of an eye, one would see the same image that the organism would see, reflected back out.

Many small organisms such as rotifers, copepods and flatworms use such organs, but these are too small to produce usable images. Some larger organisms, such as scallops, also use reflector eyes. The scallop Pecten has up to 100 millimetre-scale reflector eyes fringing the edge of its shell. It detects moving objects as they pass successive lenses.

There is at least one vertebrate, the spookfish, whose eyes include reflective optics for focusing of light. Each of the two eyes of a spookfish collects light from both above and below; the light coming from above is focused by a lens, while that coming from below, by a curved mirror composed of many layers of small reflective plates made of guanine crystals.

Compound eyes

An image of a house fly compound eye surface by using scanning electron microscope
 
Anatomy of the compound eye of an insect
 
Arthropods such as this bluebottle fly have compound eyes

A compound eye may consist of thousands of individual photoreceptor units or ommatidia (ommatidium, singular). The image perceived is a combination of inputs from the numerous ommatidia (individual "eye units"), which are located on a convex surface, thus pointing in slightly different directions. Compared with simple eyes, compound eyes possess a very large view angle, and can detect fast movement and, in some cases, the polarisation of light. Because the individual lenses are so small, the effects of diffraction impose a limit on the possible resolution that can be obtained (assuming that they do not function as phased arrays). This can only be countered by increasing lens size and number. To see with a resolution comparable to our simple eyes, humans would require very large compound eyes, around 11 metres (36 ft) in radius.

Compound eyes fall into two groups: apposition eyes, which form multiple inverted images, and superposition eyes, which form a single erect image. Compound eyes are common in arthropods, annelids and some bivalved molluscs. Compound eyes in arthropods grow at their margins by the addition of new ommatidia.[22]

Apposition eyes

Apposition eyes are the most common form of eyes and are presumably the ancestral form of compound eyes. They are found in all arthropod groups, although they may have evolved more than once within this phylum. Some annelids and bivalves also have apposition eyes. They are also possessed by Limulus, the horseshoe crab, and there are suggestions that other chelicerates developed their simple eyes by reduction from a compound starting point. (Some caterpillars appear to have evolved compound eyes from simple eyes in the opposite fashion.)

Apposition eyes work by gathering a number of images, one from each eye, and combining them in the brain, with each eye typically contributing a single point of information. The typical apposition eye has a lens focusing light from one direction on the rhabdom, while light from other directions is absorbed by the dark wall of the ommatidium.

Superposition eyes

The second type is named the superposition eye. The superposition eye is divided into three types:

  • refracting,
  • reflecting and
  • parabolic superposition

The refracting superposition eye has a gap between the lens and the rhabdom, and no side wall. Each lens takes light at an angle to its axis and reflects it to the same angle on the other side. The result is an image at half the radius of the eye, which is where the tips of the rhabdoms are. This type of compound eye, for which a minimal size exists below which effective superposition cannot occur, is normally found in nocturnal insects, because it can create images up to 1000 times brighter than equivalent apposition eyes, though at the cost of reduced resolution. In the parabolic superposition compound eye type, seen in arthropods such as mayflies, the parabolic surfaces of the inside of each facet focus light from a reflector to a sensor array. Long-bodied decapod crustaceans such as shrimp, prawns, crayfish and lobsters are alone in having reflecting superposition eyes, which also have a transparent gap but use corner mirrors instead of lenses.

Parabolic superposition

This eye type functions by refracting light, then using a parabolic mirror to focus the image; it combines features of superposition and apposition eyes.

Other

Another kind of compound eye, found in males of Order Strepsiptera, employs a series of simple eyes—eyes having one opening that provides light for an entire image-forming retina. Several of these eyelets together form the strepsipteran compound eye, which is similar to the 'schizochroal' compound eyes of some trilobites. Because each eyelet is a simple eye, it produces an inverted image; those images are combined in the brain to form one unified image. Because the aperture of an eyelet is larger than the facets of a compound eye, this arrangement allows vision under low light levels.

Good fliers such as flies or honey bees, or prey-catching insects such as praying mantis or dragonflies, have specialised zones of ommatidia organised into a fovea area which gives acute vision. In the acute zone, the eyes are flattened and the facets larger. The flattening allows more ommatidia to receive light from a spot and therefore higher resolution. The black spot that can be seen on the compound eyes of such insects, which always seems to look directly at the observer, is called a pseudopupil. This occurs because the ommatidia which one observes "head-on" (along their optical axes) absorb the incident light, while those to one side reflect it.

There are some exceptions from the types mentioned above. Some insects have a so-called single lens compound eye, a transitional type which is something between a superposition type of the multi-lens compound eye and the single lens eye found in animals with simple eyes. Then there is the mysid shrimp, Dioptromysis paucispinosa. The shrimp has an eye of the refracting superposition type, in the rear behind this in each eye there is a single large facet that is three times in diameter the others in the eye and behind this is an enlarged crystalline cone. This projects an upright image on a specialised retina. The resulting eye is a mixture of a simple eye within a compound eye.

Another version is a compound eye often referred to as "pseudofaceted", as seen in Scutigera. This type of eye consists of a cluster of numerous ommatidia on each side of the head, organised in a way that resembles a true compound eye.

The body of Ophiocoma wendtii, a type of brittle star, is covered with ommatidia, turning its whole skin into a compound eye. The same is true of many chitons. The tube feet of sea urchins contain photoreceptor proteins, which together act as a compound eye; they lack screening pigments, but can detect the directionality of light by the shadow cast by its opaque body.

Nutrients

The ciliary body is triangular in horizontal section and is coated by a double layer, the ciliary epithelium. The inner layer is transparent and covers the vitreous body, and is continuous from the neural tissue of the retina. The outer layer is highly pigmented, continuous with the retinal pigment epithelium, and constitutes the cells of the dilator muscle.

The vitreous is the transparent, colourless, gelatinous mass that fills the space between the lens of the eye and the retina lining the back of the eye. It is produced by certain retinal cells. It is of rather similar composition to the cornea, but contains very few cells (mostly phagocytes which remove unwanted cellular debris in the visual field, as well as the hyalocytes of Balazs of the surface of the vitreous, which reprocess the hyaluronic acid), no blood vessels, and 98–99% of its volume is water (as opposed to 75% in the cornea) with salts, sugars, vitrosin (a type of collagen), a network of collagen type II fibres with the mucopolysaccharide hyaluronic acid, and also a wide array of proteins in micro amounts. Amazingly, with so little solid matter, it tautly holds the eye.

Evolution

Evolution of the mollusc eye

Photoreception is phylogenetically very old, with various theories of phylogenesis. The common origin (monophyly) of all animal eyes is now widely accepted as fact. This is based upon the shared genetic features of all eyes; that is, all modern eyes, varied as they are, have their origins in a proto-eye believed to have evolved some 650-600 million years ago, and the PAX6 gene is considered a key factor in this. The majority of the advancements in early eyes are believed to have taken only a few million years to develop, since the first predator to gain true imaging would have touched off an "arms race" among all species that did not flee the photopic environment. Prey animals and competing predators alike would be at a distinct disadvantage without such capabilities and would be less likely to survive and reproduce. Hence multiple eye types and subtypes developed in parallel (except those of groups, such as the vertebrates, that were only forced into the photopic environment at a late stage).

Eyes in various animals show adaptation to their requirements. For example, the eye of a bird of prey has much greater visual acuity than a human eye, and in some cases can detect ultraviolet radiation. The different forms of eye in, for example, vertebrates and molluscs are examples of parallel evolution, despite their distant common ancestry. Phenotypic convergence of the geometry of cephalopod and most vertebrate eyes creates the impression that the vertebrate eye evolved from an imaging cephalopod eye, but this is not the case, as the reversed roles of their respective ciliary and rhabdomeric opsin classes and different lens crystallins show.

The very earliest "eyes", called eye-spots, were simple patches of photoreceptor protein in unicellular animals. In multicellular beings, multicellular eyespots evolved, physically similar to the receptor patches for taste and smell. These eyespots could only sense ambient brightness: they could distinguish light and dark, but not the direction of the light source.

Through gradual change, the eye-spots of species living in well-lit environments depressed into a shallow "cup" shape. The ability to slightly discriminate directional brightness was achieved by using the angle at which the light hit certain cells to identify the source. The pit deepened over time, the opening diminished in size, and the number of photoreceptor cells increased, forming an effective pinhole camera that was capable of dimly distinguishing shapes. However, the ancestors of modern hagfish, thought to be the protovertebrate, were evidently pushed to very deep, dark waters, where they were less vulnerable to sighted predators, and where it is advantageous to have a convex eye-spot, which gathers more light than a flat or concave one. This would have led to a somewhat different evolutionary trajectory for the vertebrate eye than for other animal eyes.

The thin overgrowth of transparent cells over the eye's aperture, originally formed to prevent damage to the eyespot, allowed the segregated contents of the eye chamber to specialise into a transparent humour that optimised colour filtering, blocked harmful radiation, improved the eye's refractive index, and allowed functionality outside of water. The transparent protective cells eventually split into two layers, with circulatory fluid in between that allowed wider viewing angles and greater imaging resolution, and the thickness of the transparent layer gradually increased, in most species with the transparent crystallin protein.

The gap between tissue layers naturally formed a biconvex shape, an optimally ideal structure for a normal refractive index. Independently, a transparent layer and a nontransparent layer split forward from the lens: the cornea and iris. Separation of the forward layer again formed a humour, the aqueous humour. This increased refractive power and again eased circulatory problems. Formation of a nontransparent ring allowed more blood vessels, more circulation, and larger eye sizes.

Relationship to life requirements

Eyes are generally adapted to the environment and life requirements of the organism which bears them. For instance, the distribution of photoreceptors tends to match the area in which the highest acuity is required, with horizon-scanning organisms, such as those that live on the African plains, having a horizontal line of high-density ganglia, while tree-dwelling creatures which require good all-round vision tend to have a symmetrical distribution of ganglia, with acuity decreasing outwards from the centre.

Of course, for most eye types, it is impossible to diverge from a spherical form, so only the density of optical receptors can be altered. In organisms with compound eyes, it is the number of ommatidia rather than ganglia that reflects the region of highest data acquisition. Optical superposition eyes are constrained to a spherical shape, but other forms of compound eyes may deform to a shape where more ommatidia are aligned to, say, the horizon, without altering the size or density of individual ommatidia. Eyes of horizon-scanning organisms have stalks so they can be easily aligned to the horizon when this is inclined, for example, if the animal is on a slope.

An extension of this concept is that the eyes of predators typically have a zone of very acute vision at their centre, to assist in the identification of prey. In deep water organisms, it may not be the centre of the eye that is enlarged. The hyperiid amphipods are deep water animals that feed on organisms above them. Their eyes are almost divided into two, with the upper region thought to be involved in detecting the silhouettes of potential prey—or predators—against the faint light of the sky above. Accordingly, deeper water hyperiids, where the light against which the silhouettes must be compared is dimmer, have larger "upper-eyes", and may lose the lower portion of their eyes altogether. In the giant Antarctic isopod Glyptonotus a small ventral compound eye is physically completely separated from the much larger dorsal compound eye. Depth perception can be enhanced by having eyes which are enlarged in one direction; distorting the eye slightly allows the distance to the object to be estimated with a high degree of accuracy.

Acuity is higher among male organisms that mate in mid-air, as they need to be able to spot and assess potential mates against a very large backdrop. On the other hand, the eyes of organisms which operate in low light levels, such as around dawn and dusk or in deep water, tend to be larger to increase the amount of light that can be captured.

It is not only the shape of the eye that may be affected by lifestyle. Eyes can be the most visible parts of organisms, and this can act as a pressure on organisms to have more transparent eyes at the cost of function.

Eyes may be mounted on stalks to provide better all-round vision, by lifting them above an organism's carapace; this also allows them to track predators or prey without moving the head.

Physiology

Visual acuity

The eye of a red-tailed hawk

Visual acuity, or resolving power, is "the ability to distinguish fine detail" and is the property of cone cells. It is often measured in cycles per degree (CPD), which measures an angular resolution, or how much an eye can differentiate one object from another in terms of visual angles. Resolution in CPD can be measured by bar charts of different numbers of white/black stripe cycles. For example, if each pattern is 1.75 cm wide and is placed at 1 m distance from the eye, it will subtend an angle of 1 degree, so the number of white/black bar pairs on the pattern will be a measure of the cycles per degree of that pattern. The highest such number that the eye can resolve as stripes, or distinguish from a grey block, is then the measurement of visual acuity of the eye.

For a human eye with excellent acuity, the maximum theoretical resolution is 50 CPD (1.2 arcminute per line pair, or a 0.35 mm line pair, at 1 m). A rat can resolve only about 1 to 2 CPD. A horse has higher acuity through most of the visual field of its eyes than a human has, but does not match the high acuity of the human eye's central fovea region.

Spherical aberration limits the resolution of a 7 mm pupil to about 3 arcminutes per line pair. At a pupil diameter of 3 mm, the spherical aberration is greatly reduced, resulting in an improved resolution of approximately 1.7 arcminutes per line pair. A resolution of 2 arcminutes per line pair, equivalent to a 1 arcminute gap in an optotype, corresponds to 20/20 (normal vision) in humans.

However, in the compound eye, the resolution is related to the size of individual ommatidia and the distance between neighbouring ommatidia. Physically these cannot be reduced in size to achieve the acuity seen with single lensed eyes as in mammals. Compound eyes have a much lower acuity than vertebrate eyes.

Colour perception

"Colour vision is the faculty of the organism to distinguish lights of different spectral qualities." All organisms are restricted to a small range of electromagnetic spectrum; this varies from creature to creature, but is mainly between wavelengths of 400 and 700 nm. This is a rather small section of the electromagnetic spectrum, probably reflecting the submarine evolution of the organ: water blocks out all but two small windows of the EM spectrum, and there has been no evolutionary pressure among land animals to broaden this range.

The most sensitive pigment, rhodopsin, has a peak response at 500 nm. Small changes to the genes coding for this protein can tweak the peak response by a few nm; pigments in the lens can also filter incoming light, changing the peak response. Many organisms are unable to discriminate between colours, seeing instead in shades of grey; colour vision necessitates a range of pigment cells which are primarily sensitive to smaller ranges of the spectrum. In primates, geckos, and other organisms, these take the form of cone cells, from which the more sensitive rod cells evolved. Even if organisms are physically capable of discriminating different colours, this does not necessarily mean that they can perceive the different colours; only with behavioural tests can this be deduced.

Most organisms with colour vision can detect ultraviolet light. This high energy light can be damaging to receptor cells. With a few exceptions (snakes, placental mammals), most organisms avoid these effects by having absorbent oil droplets around their cone cells. The alternative, developed by organisms that had lost these oil droplets in the course of evolution, is to make the lens impervious to UV light—this precludes the possibility of any UV light being detected, as it does not even reach the retina.

Rods and cones

The retina contains two major types of light-sensitive photoreceptor cells used for vision: the rods and the cones.

Rods cannot distinguish colours, but are responsible for low-light (scotopic) monochrome (black-and-white) vision; they work well in dim light as they contain a pigment, rhodopsin (visual purple), which is sensitive at low light intensity, but saturates at higher (photopic) intensities. Rods are distributed throughout the retina but there are none at the fovea and none at the blind spot. Rod density is greater in the peripheral retina than in the central retina.

Cones are responsible for colour vision. They require brighter light to function than rods require. In humans, there are three types of cones, maximally sensitive to long-wavelength, medium-wavelength, and short-wavelength light (often referred to as red, green, and blue, respectively, though the sensitivity peaks are not actually at these colours). The colour seen is the combined effect of stimuli to, and responses from, these three types of cone cells. Cones are mostly concentrated in and near the fovea. Only a few are present at the sides of the retina. Objects are seen most sharply in focus when their images fall on the fovea, as when one looks at an object directly. Cone cells and rods are connected through intermediate cells in the retina to nerve fibres of the optic nerve. When rods and cones are stimulated by light, they connect through adjoining cells within the retina to send an electrical signal to the optic nerve fibres. The optic nerves send off impulses through these fibres to the brain.

Pigmentation

The pigment molecules used in the eye are various, but can be used to define the evolutionary distance between different groups, and can also be an aid in determining which are closely related—although problems of convergence do exist.

Opsins are the pigments involved in photoreception. Other pigments, such as melanin, are used to shield the photoreceptor cells from light leaking in from the sides. The opsin protein group evolved long before the last common ancestor of animals, and has continued to diversify since.

There are two types of opsin involved in vision; c-opsins, which are associated with ciliary-type photoreceptor cells, and r-opsins, associated with rhabdomeric photoreceptor cells. The eyes of vertebrates usually contain ciliary cells with c-opsins, and (bilaterian) invertebrates have rhabdomeric cells in the eye with r-opsins. However, some ganglion cells of vertebrates express r-opsins, suggesting that their ancestors used this pigment in vision, and that remnants survive in the eyes. Likewise, c-opsins have been found to be expressed in the brain of some invertebrates. They may have been expressed in ciliary cells of larval eyes, which were subsequently resorbed into the brain on metamorphosis to the adult form. C-opsins are also found in some derived bilaterian-invertebrate eyes, such as the pallial eyes of the bivalve molluscs; however, the lateral eyes (which were presumably the ancestral type for this group, if eyes evolved once there) always use r-opsins. Cnidaria, which are an outgroup to the taxa mentioned above, express c-opsins—but r-opsins are yet to be found in this group. Incidentally, the melanin produced in the cnidaria is produced in the same fashion as that in vertebrates, suggesting the common descent of this pigment.

Additional images

Straightedge and compass construction

Creating a regular hexagon with a straightedge and compass

Straightedge-and-compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.

The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, the only permissible constructions are those granted by the first three postulates of Euclid's Elements.

It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone, or by straightedge alone if given a single circle and its center.

The ancient Greek mathematicians first conceived straightedge-and-compass constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.

In spite of existing proofs of impossibility, some persist in trying to solve these problems.[1] Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone.

In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots.

Straightedge and compass tools

Straightedge and compass
 
A compass

The "straightedge" and "compass" of straightedge-and-compass constructions are idealizations of rulers and compasses in the real world:

  • The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. The line drawn is infinitesimally thin point-width. It can only be used to draw a line segment between two points, with infinite precision to those points, or to extend an existing segment.
  • The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn starting from two given points: the centre and a point on the circle, and aligned to those points with infinite precision. The arc that is drawn is infinitesimally thin point-width. The compass may or may not collapse when it is not drawing a circle.

Actual compasses do not collapse and modern geometric constructions often use this feature. A 'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history. In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass.

Each construction must be exact. "Eyeballing" it (essentially looking at the construction and guessing at its accuracy, or using some form of measurement, such as the units of measure on a ruler) and getting close does not count as a solution.

Each construction must terminate. That is, it must have a finite number of steps, and not be the limit of ever closer approximations.

Stated this way, straightedge-and-compass constructions appear to be a parlour game, rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be proved to be exactly correct.

History

The ancient Greek mathematicians first attempted straightedge-and-compass constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and a regular polygon with 3, 4, or 5 sides (or one with twice the number of sides of a given polygon). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides. Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side.

Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle; but these methods also cannot be followed with just straightedge and compass.

No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for a regular polygon of n sides to be constructible.

In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary.

Then in 1882 Lindemann showed that is a transcendental number, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.

The basic constructions

The basic constructions

All straightedge-and-compass constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are:

  • Creating the line through two existing points
  • Creating the circle through one point with centre another point
  • Creating the point which is the intersection of two existing, non-parallel lines
  • Creating the one or two points in the intersection of a line and a circle (if they intersect)
  • Creating the one or two points in the intersection of two circles (if they intersect).

For example, starting with just two distinct points, we can create a line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle.

Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry is equivalent to an axiomatic algebra, replacing its elements by symbols. Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry.

Much used straightedge-and-compass constructions

The most-used straightedge-and-compass constructions include:

Constructible points

Straightedge-and-compass constructions corresponding to algebraic operations
x = a·b   (intercept theorem)
x = a/b   (intercept theorem)
x=a   (Pythagorean theorem)

One can associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors. Finally we can write these vectors as complex numbers.

Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x +yk, where x, y, and k are in F.

Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an algebraic number, though not every algebraic number is constructible; for example, 32 is algebraic but not constructible.

Constructible angles

There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an abelian group under addition modulo 2π (which corresponds to multiplication of the points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular heptadecagon (the seventeen-sided regular polygon) is constructible because

as discovered by Gauss.

The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in the complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes. In addition there is a dense set of constructible angles of infinite order.

Relation to complex arithmetic

Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.

Given any such interpretation of a set of points as complex numbers, the points constructible using valid straightedge-and-compass constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π). The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of addition, subtraction, multiplication, division, complex conjugate, and square root, which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple straightedge-and-compass construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations.

Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using straightedge and compass from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots.

For example, the real part, imaginary part and modulus of a point or ratio z (taking one of the two viewpoints above) are constructible as these may be expressed as

Doubling the cube and trisection of an angle (except for special angles such as any φ such that φ/(2π) is a rational number with denominator not divisible by 3) require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. None of these are in the fields described, hence no straightedge-and-compass construction for these exists.

Impossible constructions

The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable. With modern methods, however, these straightedge-and-compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and the Greeks knew how to solve them without the constraint of working only with straightedge and compass.)

Squaring the circle

The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass.

Squaring the circle has been proved impossible, as it involves generating a transcendental number, that is, π. Only certain algebraic numbers can be constructed with ruler and compass alone, namely those constructed from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.

Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity.

A method which comes very close to approximating the "quadrature of the circle" can be achieved using a Kepler triangle.

Doubling the cube

Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over the rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass.

Angle trisection

Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2π/5 radians (72° = 360°/5) can be trisected, but the angle of π/3 radians (60°) cannot be trisected. The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a neusis construction).

Distance to an ellipse

The line segment from any point in the plane to the nearest point on a circle can be constructed, but the segment from any point in the plane to the nearest point on an ellipse of positive eccentricity cannot in general be constructed.

Alhazen's problem

In 1997, the Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient Alhazen's problem (billiard problem or reflection from a spherical mirror).

Constructing regular polygons

Construction of a regular pentagon

Some regular polygons (e.g. a pentagon) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass?

Carl Friedrich Gauss in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular n-sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was provided by Pierre Wantzel in 1837.

The first few constructible regular polygons have the following numbers of sides:

3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272... (sequence A003401 in the OEIS)

There are known to be an infinitude of constructible regular polygons with an even number of sides (because if a regular n-gon is constructible, then so is a regular 2n-gon and hence a regular 4n-gon, 8n-gon, etc.). However, there are only 31 known constructible regular n-gons with an odd number of sides.

Constructing a triangle from three given characteristic points or lengths

Sixteen key points of a triangle are its vertices, the midpoints of its sides, the feet of its altitudes, the feet of its internal angle bisectors, and its circumcenter, centroid, orthocenter, and incenter. These can be taken three at a time to yield 139 distinct nontrivial problems of constructing a triangle from three points. Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39 the required triangle exists but is not constructible.

Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined.

Restricted Constructions

Various attempts have been made to restrict the allowable tools for constructions under various rules, in order to determine what is still constructable and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and straightedge can.

Constructing with only ruler or only compass

It is possible (according to the Mohr–Mascheroni theorem) to construct anything with just a compass if it can be constructed with a ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). The truth of this theorem depends on the truth of Archimedes' axiom, which is not first-order in nature. Examples of compass-only constructions include Napoleon's problem.

It is impossible to take a square root with just a ruler, so some things that cannot be constructed with a ruler can be constructed with a compass; but (by the Poncelet–Steiner theorem) given a single circle and its center, they can be constructed.

Extended constructions

The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories. This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described above) has a planar construction. A complex number that includes also the extraction of cube roots has a solid construction.

In the language of fields, a complex number that is planar has degree a power of two, and lies in a field extension that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3.

Solid constructions

A point has a solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful.

The ancient Greeks knew that doubling the cube and trisecting an arbitrary angle both had solid constructions. Archimedes gave a solid construction of the regular 7-gon. The quadrature of the circle does not have a solid construction.

A regular n-gon has a solid construction if and only if n=2j3km where m is a product of distinct Pierpont primes (primes of the form 2r3s+1). The set of such n is the sequence

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97... (sequence A051913 in the OEIS)

The set of n for which a regular n-gon has no solid construction is the sequence

11, 22, 23, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100... (sequence A048136 in the OEIS)

Like the question with Fermat primes, it is an open question as to whether there are an infinite number of Pierpont primes.

Angle trisection

What if, together with the straightedge and compass, we had a tool that could (only) trisect an arbitrary angle? Such constructions are solid constructions, but there exist numbers with solid constructions that cannot be constructed using such a tool. For example, we cannot double the cube with such a tool. On the other hand, every regular n-gon that has a solid construction can be constructed using such a tool.

Origami

The mathematical theory of origami is more powerful than straightedge-and-compass construction. Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore, origami can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems.

Markable rulers

Archimedes, Nicomedes and Apollonius gave constructions involving the use of a markable ruler. This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects the two given lines, such that the distance between the points of intersection equals the given segment. This the Greeks called neusis ("inclination", "tendency" or "verging"), because the new line tends to the point. In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them.

The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions. In fact, using this tool one can solve some quintics that are not solvable using radicals. It is known that one cannot solve an irreducible polynomial of prime degree greater or equal to 7 using the neusis construction, so it is not possible to construct a regular 23-gon or 29-gon using this tool. Benjamin and Snyder proved that it is possible to construct the regular 11-gon, but did not give a construction. It is still open as to whether a regular 25-gon or 31-gon is constructible using this tool.

Trisect a straight segment

Trisection of a straight edge procedure.

Given a straight line segment called AB, could this be divided in three new equal segments and in many parts required by the use of intercept theorem

Computation of binary digits

In 1998 Simon Plouffe gave a ruler-and-compass algorithm that can be used to compute binary digits of certain numbers. The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits.

Extraterrestrial liquid water

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