Commelina communis

Commelina communis, commonly known as the Asiatic dayflower, is an herbaceous annual plant in the dayflower family. It gets its name because the blooms last for only one day. It is native throughout much of East Asia and northern parts of Southeast Asia. In China, the plant is known as yazhicao (), roughly translating to "duckfoot herb", while in Japan it is known as , meaning "dew herb". It has also been introduced to parts of central and southeastern Europe and much of eastern North America, where it has spread to become a noxious weed. It is common in disturbed sites and in moist soil. The flowers emerge from summer through fall and are distinctive with two relatively large blue petals and one very reduced white petal.
The Asiatic dayflower plant serves as the type species for its large genus. Linnaeus picked the name "Commelina" in honour of the two Dutch botanists of the Commelijn family, using the two large showy petals of "Commelina communis" to symbolise them. Linnaeus described the species in the first edition of his landmark work, "Species Plantarum", in 1753. Long before the plant was studied in Europe, however, it had been used for generations in traditional Chinese medicine. The flowers have also been used in Japan to produce a dye and a pigment that was used in many world-renowned Ukiyo-e woodcuts from the 18th and early 19th centuries. In the modern era the plant has found limited use as a model organism in the field of plant physiology due to its complex pigment chemistry and the ease of viewing its stomata.
The Asiatic dayflower is considered a weed both in areas where it was introduced and in certain parts of its native range. The flowers' interactions with pollinators have been well studied and have helped to support important hypotheses about pollination in the field of plant ecology. Recent research has also revealed that the Asiatic dayflower can bioaccumulate a number of metals, making it a candidate for revegetating and essentially cleaning spoiled copper mines. Several animals and fungi use the plant as a food source, with a few species feeding upon it exclusively.
Description.
The Asiatic dayflower is an annual herb with stems that are typically decumbent, meaning that they are prostrate at the base but become erect towards the tips, but some individuals may be simply erect. The diffusely branched stems tend to root at the basal nodes. The pubescence on the stems is variable, but common patterns include a line of hair continuous with the leaf sheath, or they may be glabrous basally, meaning hairless, and puberulent towards the extremities, that is covered with fine hairs. The leaves are sessile: they lack a leaf stalk, also known as a petiole; or they may be subpetiolate, meaning they have very small petioles. The leaf sheaths are cylindrical, sometimes striped with red, and typically glabrous, but usually have margins that are puberulent or pilose, meaning lined with fine, soft hairs. The leaf blades range from narrowly lanceolate, or lance-shaped, to ovate-elliptic, between egg-shaped and ellipse-shaped. They measure by wide. The blades range from glabrous to puberulent and have scabrescent, or slightly rough, margins. Their tips are acute, meaning they come to a point quickly, to acuminate, meaning the point develops gradually. The leaf bases are oblique, or uneven.
The flowers are arranged on inflorescences called cincinni (singular: cincinnus), which are also called scorpioid cymes. This is a form of a monochasium where the lateral branches arise alternately. The cincinni are subtended by a spathe, a modified leaf. The solitary spathes usually measure long, but some may be up to in length, while they are tall, but sometimes up to . The uncurved spathes typically have a cordate, or heart-shaped, whitish base, which contrasts with its dark green veins. Their margins lack hairs, are somewhat scabrous, or rough, and are unfused, meaning they are distinct to the base. Their apices are acute to acuminate while the surfaces are glabrous, puberulent, or hirsute-ciliate, meaning with longer, shaggier hairs. The spathes are borne on peduncles, or stalks, that measure and sometimes up to long.
There are often two cincinni present, though the upper, or distal, cincinnus may be vestigial. The lower, or proximal, cincinnus bears 1 to 4 bisexual flowers and is nearly included in the spathe, while the upper cincinnus has 1 to 2 male flowers and is about long. The individual flowers are subtended by bracteoles that fall off early in development. The pedicels supporting single flowers, and later the fruits, are erect initially but curve when in fruit. They measure about . The 3 concave, membranous sepals are inconspicuous, but persist after the fruit develops; the lateral pair are fused basally, measure only long by wide, and are elliptic and glabrous. The lower sepal is lanceolate and about long by about wide. The 2 upper petals are blue to indigo in colour, while the much smaller lower petal is white. The upper two petals measure long by wide, while the lower petal is long by about wide. The 2 upper petals are composed of a claw about long and a broadly ovate limb with an acute apex and a cuneate-cordate base.
There are three anticous fertile stamens, meaning they are on the lower part of the flower, and three posticous infertile stamens, meaning they are on the upper part of the flower. These infertile stamens are termed staminodes. The fertile stamens are dimorphic: the lateral pair have maroon to indigo anthers that measure about long and are elliptic with a base that is sagittate or arrowhead-shaped. Their filaments are about long. The central fertile stamen has a yellow, elliptic anther with a maroon connective and a base that is hastate or spearhead-shaped, but with the lobes at right angles. The anther measures about long while its filament is about long. The three staminodes are all alike with yellow, cruciform, or cross-shaped, antherodes that are about long on filaments about long. Sometimes the antherodes will have a central maroon spot. Each antherode has two abortive lateral pollen sacks. The ovary is ellipsoid, about long and has a style that is about long.
The fruit is a dehiscent, ellipsoid capsule with two locules each containing two seeds. The capsule is glabrous, brown, measures long, and dehisces into two valves. The seeds are brown or brownish yellow in colour and deltoid, or roughly triangular in outline. They are dorsiventral, meaning they have distinct upper and lower surfaces, with the ventral, or lower, surface being planar and the dorsal, or upper, surface being convex. Seeds range in length from , but seeds as short as can occur, while they are across. The surfaces are rugose pitted-reticulate and are densely covered with smaller farinose granules with sparse larger farinose granules.
Taxonomy.

Isotropic line

In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form.
Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point that depend on the imaginary unit :
Laguerre then interpreted these lines as geodesics:
In the complex projective plane, points are represented by homogeneous coordinates formula_3 and lines by homogeneous coordinates formula_4. An isotropic line in the complex projective plane satisfies the equation:
In terms of the affine subspace , an isotropic line through the origin is
In projective geometry, the isotropic lines are the ones passing through the circular points at infinity.
In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs:
Relativity.
Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of photons: "The worldline of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line."
For isotropic lines through the origin, a particular point is a null vector, and the collection of all such isotropic lines forms the light cone at the origin.
Élie Cartan expanded the concept of isotropic lines to multivectors in his book on spinors in three dimensions.